Approximation of unbounded functions with linear positive operators

APPROXIMATION OF UNBOUNDED

FUNCTIONS WITH LINEAR

POSITIVE OPERATORS

R. K. S. RATHORE

Approximation of unbounded

functions with linear positive

operators

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XI N) CD BIBLIOTHEEK TU Delft P 1835 6267

Approximation of unbounded functions

with linear positive operators

PROEFSCHRIFT ter verkrijging van

de graad van doctor in de

technische wetenschappen

aan de Technische Hogeschool Delft,

op gezag van de rector magnificus

ir. H. B. Boerema, hoogleraar

in de afdeling der elektrotechniek,

voor een commissie aangewezen

door het college van dekanen

te verdedigen op

woensdag 27 november 1974

te 14.00 uur door

RAM KISHORE SINGH RATHORE

Ph. D. in Mathematics

(Indian Institute of Technology, Delhi)

geboren te Lakhimpur-Kheri (U.P.) India

r

Dit proefschrift is goedgekeurd door de promotor PROF. DR. P. C. SIKKEMA

CONTENTS

INTRODUCTION ^ 1

CHAPTER I METHOD OF TEST FUNCTIONS 8 1.1 Approximation of functions having at the most

a polynomial growth when the variable tends to

+ 00 8

1.1.1 Definitions and notations 8 1.1.2 Convergence of the operators L 10 1.1.3 Asymptotic formulae for twice differentiable

functions I8 1.1.i* A class of linear positive operator sequences 32

1.1.5 Generalizations for functions of several

vari-ables Ul 1.1.6 On a generalized sequence of linear positive

operators 55 1.1.T A method of constructing operators for

func-tions of several variables 61

1.2 The trigonometric case 63 1.2.1 Asymptotic formulae for twice differentiable

functions 6h

1.2.2 Generalizations for functions of several

va-riables 79

1.3 Approximation of functions of an exponential

growth 86 1.3.1 Single variable case 87

1.3.2 Generalizations for functions of several

CHAPTER 2 METHOD OF BOUNDING FUNCTIONS AND THE W-FUNCTIONS 97

2.1 Method of bounding functions 97

2.1.1 Basic convergence 97 2.1.2 Asymptotic estimates 99

2.1.3 Asymptotic formulae 1OU

2.2 Combining the techniques of bounding functions

and the W-functions 111

2.2.1 Asymptotic formulae 111 2.2.2 Asymptotic estimates 117

CHAPTER 3 APPROXIMATION OF UNBOUNDED FUNCTIONS BY

OPERA-TORS OF SUMMATION TYPE 121

3.1 A general outline 121 3.2 The Bernstein polynomials 122

CHAPTER k APPROXIMATION OF UNBOUNDED FUNCTION<=! BY

OPERA-TORS OF INTEGRAL TYPE 133

h.^ Approximation of \mbounded integrable functions 139

i+.l.l The general method 139

U.I.2 The generalized Jackson operators L lUo np-p

U.I.3 The Gamma operators G lU2 U.1.U Singular integrals W of Gauss-Weierstrass lU6

U.I.5 De La Vallee - Poussin integrals V lU8

REFERENCES 151

INTRODUCTION

1. Schurer [ 59-62] , Hsu [ 17-18] , Wang [ 18] , Wood [ 12-14] llttller [48] Eisenberg [l2-14]and several other

researchers have studied the possibility of

approximating a real (or complex) valued fiinction f(t), defined on the real line or on a subset of it and

unbounded as t — + oo (or to some other point or points), at its points of continuity by means of a suitable

sequence JL } (n=1,2,...) of linear operators ultimately positive for the points (or a single point) of continuity. In general, such a procedure assumes the convergence L (g;x) — g(x) as n — oo, where x is a fixed point ot it belongs to the set of points on which an approximation is desired, for some test fvmctions g(t) and, in

addition, the uniform boundedness (uniform with respect to n) or a kind of convergence as n -» oo ) of the

sequence JL ( Q ; X ) J (n=1,2,...), where Q(t) is a suitably chosen unbounded function. Under such assumptions we arrive at a class (for which Q(t) is called a bounding function) of unbounded functions f(t) which can be aporoximated at a point t=x of continuity by the senuence JL (f;x)j as n ->• oo . In this connection we remark that almost all the work of above authors does not use the full force of their assumptions and that it is possible to enlar^^en the classes of unbounded

fljinctions for \jixich their results hold under the same or even milder assumptions.

Recently l/alk [81] and Schmid [56] used the notion of a socalled \/-function h(t), defined for t £ [0,oo ) and satisfying h(t) s 0, t ^ 0 and h(t) / t - » o o a s t-'oo, and assuFiinj the uniform boundedness or a certain kind

of convergence of the sequence jL (h(|f|);x)j

(n=1,2,...) at a point x of continuity of the function f(t), proved some converi^ence properties of the sequence

jL (f;x)j (n=1,2,...) to f(x) as n -* oo . This approach does not require an explicit knowledge of the kind of unboundedness of f(t), Kov/ever, v/hereas the use of a bounding function gives rise to a v/iiole class of ujibounded function.s for v/Lich the approximation takes place, the use of a "W'-function h(t) requires computations for the sequence {L (h(|f|);x) }for each f. Tne concept of the bounding functions is certainly better than that of the V/-functions since in any case h(|f|) is a bounding function for the function f.

2_, \/hile developing a general theory of linear

combinations of linear positive oper-.tors, the author [54] introduced a concept of local unsaturation of linear positive operators on positive zero orders. Besides linear combinations, this concept plays a very useful role also in the theory of simulti.neous

approximation by linear positive oper .tors, linear combinations of trieir iterates and the approximation of unbjunded functions, Ilain results based on this concept are described as follov/s-r

Let>(z) denote a class of non-nei^ative functions F (t,x), (a ranging over an index set l) of two

variables t and x, \.'here each F (t,x) satisfies the

a

follov/ing properties: (a) F (x,x) = 0, (b) for an arbitrary 6 > 0» F (t,x) has a positive lo\/er bound

a

for all t satisfying | t-x | a 6 v/hile x retains a fixed value, (c) keeping x fixed, P (t,x) is continuous at t=x and (d) there exists a trcaisitive relation denoted by the "greater tl.an" sign (> ) sue. t]iat c(,p £ I,a > P

imply

F^(t,x)

(1) lim •• f,^\ = 0 (keeping x fixed) t -. X ^p^^'^-'

and that given a positive nuxiber if there exists a positive number M such that v/henever P. (t,x) g N (keeping x fixed), there holds P (t,x) g MP (t,x),

p ex

Let JL } (n=1,2,...) be a se'-^uence of linear positive operators such that for each a £ I and a fixed X, the function P (t.x) £ D, the domain of definition of the oijer-tor seouonce jL 1. Further assume that for each a £ I there holds

(2) lim L (P (t,x);x) = 0. \ / n^ a

n — CO

Let a,p £ I and a > p . There are the following three possibilities (if necessary, restricting to sub-sequences):

(i) L (F„;x) = o(L (F ;x)) (ii) L^/Fp;x) = 0(L^.^(P^;x)), and (iii) L ( F ; X ) = o(L (P„;x))

as n — CO , Here the symbol 0 denotes a strict capital order. The lollot/ing three theorems wore proved in [54] i/hich cover the possibilities (i)-(iii).

TilSORSi I If a > p , tliere does not hold the relation (3) L^(Fp;x) = o(L^(F^;x)) , (n -* oo )

provided the function P is uniformly bounded by P a

whenever P is unbounded (that is to say given an

arbitrary N > 0 there exists an k > 0 sucii that v/henever P (t,x) S K, there holds F (y,x) g HF (t,x), and without any condition of P is bounded. ..Iso (j) is false if

for some y > « > p, L^'^^Y''''' ^ ° '^^n^^p'""^ ^' ( n - - ) , without any further condition.

THEOREM II If L (P ;x) = e(-7--r) and L (P„;x) = 0( / v ) n^ ' ^cp(n)' n^ p 9(,n) ' where (p(n) is non-zero and tends to oo v;ith n, and a > p. then for no y > a can there hold

(4) L ( F ;x) = o(—7—r), (n -* 00 ) . TH'^ .REII III If for some a > p there holds (5) \^'^^^^-) = o(L^(Pp;x)), (n -^ c- ). Then for every |i < p ,

(6) LjPp;x) = o(LjF^;x)).

P^r convenience the functions F (a £ l) are called a

"positive zeros" and the subscripts a are called the corresponding "oraers". If P has a nigher zero-ordor than F (i.e. a > p ), from the ^oint of view of

r

approxima,tion it seems natural to expect that the

convergence of the sequence JL (p ;x)} to zero De faster than that of the sequence jL (F ;x) j . Ile-.ce \/e say tnat j L } is saturated at a(a £ l) if a higher oraer of zero does not im^^ly a better approximation, that is to say, for n O Y > o : ( Y £ I ) there holds

L (P ;x) = O ( L ( P ;x)), as n — 0 0 , On the other hand if n^ Y n a ^ "

for some Y > a there holds L (P ;x) = 0 (L (P ;x)), v/e ' n^ Y ^ n^ a

say that JL ! is unsaturated at a .

'' ' n'

In case P (t,x) and F„(t,x) have comparable zero a p

orders of same pairty, i.e. F (t,x)

(7) lim -Tf (^ „ \ = P

where p is a positive number and further if given an arbitrary positive II ttiere exists an II > 0 such that F^(t,x) g II and Pp(t,x) g il respectively imply

P ( t , x ) g M F „ ( t , x ) and P „ ( t ; x ) S HP ( t , x ) , we v ; r i t e a p p a a ~ p . The f o l l o v / i n g t h e o r e m of [ 5 4 ] d e a l s v/ith t h e c a s e a ~ p. TKE0REI4 IV I f a ~ P and {L j i s u n s a t u r a t e d a t p , t h a n t h e r e h o l d s F „ ( t , x ) L (F ; x ) ^Q) 1^"^ F ( t , x ) = ^^"^ L (P =x) • t - * x p ^ ' ^ n — oo n ^ p ' ^

A converse of theorem IV is also true. If for every a ~ p the relation (o) holds and if there exists a Y > a such that given an arbitrary N > 0 there exists an M > 0 such that F (t;x) g N implies P (t;x) g MP (t;x), then JL I is unsaturated at p. This converse proposition is not included in [54]» which, however, can be proved rather easily.

The definition of unsaturati^-n of {L } at p implies the existence of a Y > P such that L (F ;x) = 0

( L (p„;x)) as n -* oo, in this context it was remarked in [54] that the functions F and F_ in theorem IV may be replaced by more general functions f(t) and g(t) which have the same order of zero at t=x, are positive in a neighbourhood of this point (encluding the point x ) , are bounded by a function P outside this neighbourhood where F is such that fL | are not saturated before YI

Y ' n' ' ' has a zero of an order higher than that of f or g and has other properties of this notation,

An example of a class of the type ^ Z ) is given by the set of functions P = It-xI , a £ R , the positive real line excluding zero where a > P has the usual

significance.

The theory of local unsaturation of positive linear operators on positive zero orders (together with

theorems I-IV and the remark after theorem IV) provides a great deal of insight into the study of asymptotic approximation fcrnulae of Voi-onovskaja type (for tvj-ice differentiable (at x) functions) and those containing higher derivatives upto an even order. For instance a necessary and sufficient condition that a sequence

jL } of linear positive operators possesses an asymptotic formula for twice differentiable (at x) functions of a certain growth (if unbounded) is that the sequence JL } be unsaturated at a second order positive zero having a similar growth. For a higher asymptotic expansion conta.ining derivatives upto f^ (x) a similar necessary and sufficient condition is local unsaturation at a

th

2m ' oraer positive zero of an appropirate growth. Prom this point of vie^v various equivalent formulations of xureckii's theorem ([10], p. 76) are all obvious.

Above discussion of local unsaturation is usefull for having a macroscopic viev/ of various results obtained in tills thesis. It will also help in obtaining various equivalent foriaulations of tiie given results v/iiica may be easier to deal \/itii a particular situti-ti-n,

5. Contents of the thesis.

Chapters 1-4 deal v;lth the deternination of classes of unbounded functions, members of which may be a.pproxi-mable v/ith the help of a given sequence of linear

positive operators.

Chapter 1 is an extension of Schurer's work [62], Under the sane (or milder) hypothesis as in [62], we obtain results v/hich are ap_.licable to functions having an unboundedness of a higher order. These results can be generalized for otlier test f-onctions forming a

Tchebychev system (extended, in case of asymptotic formulae), To indicate some of the manipulations in-volved we also include, in this chapter, some results using the test functions 1, sin t, cos t,... and 1,

+ ott +2at (' -V n^ e— , e- ,... (a > 0;,

Chapter 2 generalizes some of the results of Kliller [48] Hiiller and V/alk [50], Eisenberg and Wood [14] and Sclimid [56] etc, based on the concept of bounding functions and the \/-functions. Besides giving an

independent treatment based on the concept of bounding functions we also make a unified treatment which

combines the two tecliniques of the bounding and the W-functions, This unified treatment generalizes several results of Sclimid [56].

Chapter 3 makes use of certain estimates connected with the theory of local unsaturation of linear positive operators (of summation type) on positive zero orders and envisages its application in determining some classes of unbounded functions which admit of an approximation by means of these operators. We limit ourselves to the case of the Bernstein polynomials only. It v/ould, however, be clear that the method introduced has a general applicability,

Chapter 4 deals v/ith the approximation of unbounded but integrable functions by means of sequences of linear positive operators of integral type. \/e limit our

attention to the generalized Jackson operators, the Vallee-Poussin integrals, the Gamma operators of Iluller and the singular integrals of of Gauss-Weierstra.ss, Nevertheless, the method introduced seems to be of a general applicability,

CHAPTER 1.

METHOD OP TEST FUNCTIONS.

The test functions used in this chapter are 1, t, t ,...; 1, s m t, cos t, ... and e (a > 0, m = 0, + 1, + 2,..,). Parallel formulations of various results proved can be obtained in terms of other test functions (e.g. those forming Tchebycheff and extended Tchebycheff systems). Some other useful test functions are {exp a-r^, a g 0|, Jexp a(s ) , ex £ E, the real

1 ^ s s line! f ll, t"'', t"^,...| andje"/"*^, a g OJ.The first three of these have been used by the author [54] in connection with several operator sequences generated by special functions. The last one provides a deeper

understanding of approximation properties of the Gamma operators of Miiller introduced in [45]•

1,1. Approximation of functions having at the most a polynomial growth when the variable tends to + QQ.

1,1,1.Definitions and notations

Let X, X be two given subsets of R and let L ( X ) , D ( X ) be two linear spaces of real (or complex) valued functions defined on X and X respectively (the scalar field being the set of real (or complex)

numbers). In the complex case it is assumed that if f £ D ( X ) then also its complex conjugate f £ D(x), Let U be an unbounded index set of positive real numbers and let {L , n £ Uj be a class of linear operators

mapping D ( X ) into D(X), The class {L , n £ Uj is said to be ultimately positive on a point set X (by definition X S.X) if to each f £ D(X) and satisfying f(t) g 0 for

all t £ X, there exists a positive number n , say, such that for all n £ U with n > n there holds L (f;x) > 0

o n^ ' ~ for all X £ X*. It would be convenient to call f(x) respectively L (f;x)as the function and the operator ( L ) value of f at the point x, ..'ith n g 0, \ic defrne H „: class of all rea,l (or' cor.plex)valued functions

n, X

f(t) defined on X, to each of \/Lich there exist constants A,B > 0 such that |f(t)| < A+3(t)^ for all t £ X.

H ,,(x): class of all f £ H ,. \;hich have an extension

m,X^ ' m,X

^ •

1 on R wnich is continuous at the point x.

H^''4(x) : class of all f £ H ^ which have an extension m,A m,X f on R which is twice differentiable at the point x. In the sequel f is used to denote both f and f. Similarly

the derivatives f'(x), f"(x),,., are denoted by f'(x), f"(x),,.., respectively, Ii this context we remark that the possibility of several extensions f will lead to no ambiguity in our results.

<a,b> : some open interval (c,d) containing the closed interval [a,b] .

H^ I <a,b> : class of all f £ H ^ which have an m,A ^ m,A extension f on R which is twice differentiable at each point of <a,b> with f" continuous at each point of

[a,b].

Let S be a subset of R. By K „(S) we denote m,A

the class of all f C H ^ such that for each x £ S, m,A

f £ H Y ( X ) ,

^y « Q , X ' « Q , x ( - ) ' « Q , X ^ ^ ) ' «^ Q,X<^'^> ^^^ H v ( S ) we d e n o t e t h e c l a s s e s of a l l f i i n c t i o n s f s u c h

(2) that for some positive m, f £ H ,,, H v(x), H^ v(x),

/„>, m,X' m,X ' m,X H^ ^<a,b> and H ^(.S) respectively,

m,A m,A (o\

Thus the classes H(x) and H^ '^(x) as defined by Schurer [62] are identical with our classes H rjCx)

(2) ' and E\ n(x) respectively,

2fR

1,1,2. Convergence of the operators L .

The following theorem is a generalization of a theorem of Schurer [62], valid for functions of the classes H_ Y ( ^ ) where m is a positive integer. Schurer's theorem is related to the class H„ ^^(x) and makes an improvement over a result of Bohman [5] and Korovkin [30] (dealing with the class H„ -,([a,b])) under the same premises. For m=1 we obtain the result of Schurer. Unless otherwise clear form the context, X will denote a fixed point.

THEOREM 1 Let m be a positive integer and m' an odd positive integer such that m' < 2m, Let JL , n £ U) be a class of linear operators defined on a common domain D(X) (X S R ) of functions into a domain

L ( X ) ( X S R ) of functions and ultimately positive on a set 5 f ^ X , Assuming that 1, t , t £ D ( X ) and writing

L (1;x) = 1 + a (x) n^ ' ^ n^ ' (1) L^(t'"';x) = x^' + p^(x)

T ('j_2m \ 2 m / \ n^ ;x} = X + Yj^(x) ,

where x £ X and n £ U, i f and o n l y i f t h e r e h o l d ( 2 ) l i m a ( x ) = l i m p ( x ) = lim Y ( X ) = 0, \ / n^ ' ^n^ ' 'n^ ' '

then for each f £ D(x) r> H„ y(^) ^^ have (3) lim L j f ; x ) = f(x).

n — oo

Further, let S ^ X be a compact set. Then for each f £ D(X) r\ H„ yC^) relation (3) holds uniformly in X £ S if and only if (2) hold uniformly in x £ S, Note. Since in order to construct L (f;x), n £ U the function values f(t) for t ^ X are not required, for an arbitrary f £ H„ -^{x) rs D ( X ) one may expect the

cm, A

convergence L (f;x) -» f(x) as n -» 00 only for the points X £ 5?, the closure of X (b^ virtue of the continuity of f ) . Tnus, at first sight it may seem strange that in the formulation of above theorem we do not include the hypothesis that x be a point or a

cluster point of X. In fact, such an explicit inclusion would be redundant as it is already implicit in the relations (1) and (2). To prove this assume that (1) and (2) hold without x £ X. We show a contradiction. Let p(t) be the polynomial as defined in (4) below. We have

shown in the sequel that for an arbitrary 6 > 0, p(t) has a positive lower bound, say m., on the set

jt : I t-x I g 6] , if X )^ X, there exists a 6 > 0 such that (x-6, X +6) r^ X = 0. Hence for all t £ X there holds the inequality

0 g 1 g m""" p(t).

By the ultimate positivity of JL ,n £ U] at the point X it follows that for all sufficiently large n £ U we have

From this, using (l) and (2), we reach at lim L^(l;x) = 0,

n -> 00

which IS a contradiction,

A similar remark would be seen to be applicable m all further results on the basic convergence of operator values,

Proof of theorem 1. In both the assertions, the

necessity part of the theorem is trivial. To prove that the conditions are suflicient consider the polynomial p(t) defined by

/ , \ / i_ ^ . .2m _ , m' 2m-m' / , , \ 2m

(4) P('t) = in "t - 2mt X + (2m-m')x . By Descartes' rule of signs it follo\7S that if x 4= 0 then p(t) has at the most two real zeros, multiple zeros counted after their multi^l"cities and tnat they have the same sign. Clearly t=x is a zero of p(t), ,/hen x=0, t=0 is the only zero of p(t) and we have p(t) > 0 if t 4= 0, If X + 0, v;e have p'(x) = 0 so that t=x is a double zero and tnerefore p(t) has no other zero and since p(t) IS of an even degree we nave p(t) > 0 if t 4= X, hence for every 5 > 0, p(t) has a positive lower bound on the set jt : | t-x| > b],

Since the op'--ra,tors L , (n £ U ) , are linear it IS sufficient to prove the result for real valued functions f £ D ( X ) rs H_ v(^)* '^hen the functions

^m, A

L (f) are also real valued. (This remark snail 'oe made n '

use of m all suosequent results). By the continuity of f(t) at t=x, given an arbitrary e > 0, there exists a 6 > 0 such that

Also there exists a jjositive real number A such that |f(t) - f(x) I < A p(t), ([b-x|g 6 , t £ X ) , 3y non-negativity of p(t), therefore

|f(t) - f(x)| < e + A p(t), (t £ X ) . The functions c + A p(t) + (f(t) - f(x)) belong to D ( X ) n, H„ y(^)« Hence by the ultimate positivity of

|L , n £ Uj there exists a number N > 0 such that L^( c + A p(t) + (f(t) - f(x)); X ) g 0, n > N^,

By the linearity of L we then have

|L^(f;x) - f(x)| g 1 L^(1.;x) - l||f(x)| + e L^(l;x) + A L^(p(t);x),

Using (l) and (2) we can find an Np > 0 such that

| L ^ ( 1 ; X ) - I| 1 f(x)| < z

L^(l;x) < 2 I n > N^ .

and A L^(p(t);x) < e

Let N = max JN , N j. Then

|L^(f;x) - f(x)| < 4£ , n > N,

By the arbitrariness of e > 0 it follows that lim L^(f;x) = f(x),

n — oo

proving the first assertion,

To prove the second assertion, assume on the contrary that under the given assumptions (3) does not hold uniformly in x £ S, Then given an arbitrary c > 0

there exists a sequence jn, , n, £ Uj of numbers and a sequence jx, , x, £ Sjof points satisfying n, - oo , k — oo and lim x, =x for some x £ S such that

k - oo

(5) |L (f;xj^) - f(x^)l > e , k=1,2,.,. .

On the domain D(x) of functions define at x a sequence {M, j (k=1,2,,..) of operators by the relation (6) M^(f;x) = L (r;xj^).

Then, writing

Mj^(l;x) E 1 + % ( x j ^ ) = 1 + cej^(x) (7) ^it'^'ix) = xj' + p ^ ( x ^ ) = x"^' + p'(x)

Mj^(t ^;x) = x^ + Yj, (xj^) = X ™ + Yj^(x) , (the primes are just a convenient notation and do not represent derivatives) since a , p , Y "* ^ uniformly on S as n -• oo and since x, — x as k — oo , it follows that

(8) lim aj[.(x) = lim P^(x) = lim y^M = 0.

k - » o o k - ' o o k - » o o

By the first part of the theorem and (6) it follows that

(9) lim Mj^(f;x) = f(x).

k -* oo

Also since f(t) is continuous at t=x, lira f(x, )

1 k

k — oo

= f(x). Hence there exists a natural number N, such that

Since (l0) contradicts (5), the second assertion of the theorem follows. This completes the proof of the theorem, Remark 1, In the statement of theorem 1, (2) can

equivalently be replaced by

(11) lim {L^(l;x) - If = lim L^( (t-x)^°'3x)- 0, n -* OO n -* oo

(assuming that (t-x) £ D(X)), To prove this fact we replace the polynomial p(t) by the polynomial (t-x) in the proof of theorem 1 and argue as before.

Remark 2. In the light of theorem 1 we point out to an improvement on the following result of Hsu [17] • THEOREM I Let JL j (n=1,2,,,,) be a sequence of linear operators such that for all large n and every f(t) belonging to the domain of definition of JL j and non-negative for -oo < t < oo we have L (f;x) g 0 for all X £ [-1,1]. Let {a j be a sequence of real numbers increasing to + oo with n(a 4= O) and let the following limit relation

(12) lim L ((a t ) ^ ; a"''x) = x^ \ / n ^ ^ n ' ' n ^

n -> oo

exist and hold uniformly for all values of x in every finite interval, where k = 0,1,2,m,m+1,m+2; and m is a non-negative even integer. Then for every function f(t) defined and continuous on (-00,00) and satisfying the condition

(15) f ( t ) = o d t D , ( t ->±00)

we have the limit relation

(14) lim L^(f(a^t); a~^x) = f(x), n -• 00

(_oo < X < oo). Moreover, this relation holds uniformly on every finite interval of x,

Our improvement on this theorem is as follows: if m=2, condition (12) is superfluous for k=3,4. If m g 4, condition (12) is superfluous for k=1,m+1,m+2, Also in the latter case condition (l2) for k=1 can be replaced by (12) for any odd positive integer k < m,

To prove this assertion, consider the sequence (L*j(n=1,2,,..) of operators defined by

(15) L* (f;x) = L (f(a t ) ; a~''x), n=1,2,...

\ ^ I n ^ ' ^ n ^ ^ n ^ ' n ^ ' ' '

where the L are as in the above theorem. Asstime that n

(12) holds uniformly for all values of x in every finite interval for k=0, m', m, where m' is an odd positive integer less than m. (in particular we may take m'=l), Replacing L by L* and 2m by m in theorem 1, the result of Hsu follows,

Remark 3. If the class {L , n £ Uj of operators is restricted to the non-negative real axis (i.e. to construct L (f;x), (n £ U ) , the values f(t) for t £ (- °°, 0) are not required or equivalently X :^R E [0, 00)), then theorem 1 can be generalized to the following,

THEOREM 2. Let m, m' be two pooitive numbers with m' < m. Let JL , n £ UJ be a class of linear operators defined on a common domain D(X) ( X ^ R ) of functions into a domain D ( X ) ( X C R ) of functions and ultimately positive on a set X ^ X , Assuming that 1,t ,t £ D(Xj and writing

L (l;x) = 1 + a (x) n^ n^ ' (16) L^(t"'';x) = x""' + p^(x)

L (t ;x) = X + Y (x), n^ ' 'n^ '

where x £ X and n £ U, if and only if there hold

(17) lim a^(x) = lim p^(x) = lim Y ^ C X ) = 0,

n - > o o n — oo n - * o o

then we have for each f £ D(x) n. H ^(x) ^ m,X ' (18) lim L^(f;x) = f(x),

n — oo

Further, let S ^ X be a compact set. Then for each f £ D ( X ) ^ H ^(S) relation (l8) holds uniformly

m, A

in x £ S if ond only if (17) hold uniformly in x £ S. Proof. Consider the function

I ,\ ,,m ,m' m-m' / • \ m q(t) = m't - mt X + (m-m')x ,

for t g 0 and for a fixed x g 0. Then t=x is a zero of q(t). If y g 0 (y 4^ x) would we another zero of q(t),

m' m'

then u = X ,y would be two distinet zeros of the function

/ N , m/m' m-m' / . \ m v(uj = m'u - mux + (m-m'Jx ,

Applying Rolle's theorem, there would then exist a m' m'

positive number ^ lying bet'..een x and y , such that • ^ - 1

,/„N „ m' m-m' ^ v ( U = ' n ^ - m x = 0 ,

m'

Clearly this implies that ^ = x , vAich is a

contradiction. Hence the only non-negative real zero of q(t) is t=x. It is clear that because m > m' we have q(t) > 0 if t 4= X and that q(t) has a positive lov/er bound on the set ft :It-xI g 6} for each 6 > 0,

Replacing the polynomial p(t) by the function q(t) in the proof of theorem 1 and proceeding analogously we can complete the proof of theorem 2,

1,1,3. Asymptotic formulae for twice differentiable functions.

Next we consider the existence of an asymptotic formula giving rate of convergence of L (f;x) to f(,x) for twice differentiable functions. In connection with Theorem 1 of [62], Schurer remarked that the

corresponding theorem of Bohman-Korovkin, although 2

utilizing the unbounded functions t and t , gives the convergence only for bounded functions. Interestingly enough, his next result. Theorem 2 [62] utilizes the function (t-x) for some positive integer m, which has the unboundedness of the order t , 11 | - oo and yet produces a result only for f(t) = 0(t ) , |tj -* <», With the help of the following theorem, with no extra assumptions we can obtain the result for f(t) = 0(t ),

I t I — o o ,

THEOREM 1, Let m > 2 be a positive number. Let JL , n £ uj be a class of linear operators defined on a common domain D ( X ) ( X ^ R ) of functions into a domain L ( X ) ( X ' ^ R ) of functions and ultimately positive on a set X c ^ x . Assume that x £ X and that the functions 1,t,t^, It-xl"^ £ D(X), If and only if there hold

and ^o(x) ^ L ( l ; x ) = 1 + —7—r + o(—7—v) n^ 9 ( n ) > C n ) ' '? ( x ) ( 1 ) L ( t ; x ) = X + "'/ ^ + o( / \ ) 2 2 2 ^"'^'^ 1 L ( t ; x ) = X + 7—r + o(—7—v)

(2) L^(|t-xr;x) = o ( ^ ) ,

a s n -> 00, v/here (p(n) 4= 0, q)(n) -> 00 a s n -* oo, t h e n f o r a l l f £ D ( X ) r^ H^^^(x) t h e r e e x i s t s t h e m,X asymptotic relation (3) L^(f;x) -f(x) = - ^ [f(.),^(x) + f'(x){T/x) - xw^(x)j + ^ ^ {?2(x) -2x?^(x)+x2l'^(x)j] + o ( ^ ^ ) , n -' 00 .

Further, if [a,b] is an interval ( [ a , b ] ^ X ) such that for each x £ [a,b], the function |t-x| £ D(X) then for all f £ L(X) rs H^^^<a,b>, the formula (3)

' m,X ' ' ^^'

holds uniformly in x £ [a,b] if and only if (1) and (2) hold uniformly in x £ [a,b] , provided that the function ?2(x) - 2x'?.(x) + X V (x) is bounded for x £ [a,b], Remark 1. If x is not a cluster point of X, for an

(2)

f £ D ( X ) r\ VL- Y ( X ) we can have many extensions of f which will assign arbitrary values to f'(x) and f"(x)

(the value f(x) being an exception if x £ X without being a limit point of X ) . However, this fact does not lead to a contradiction in relation (3)» since in such a case 'I'.(x) - x^F^(x) and 5' (x) - 2x1'(x) + x 'If (x) are automatically both zero. To prove this v/e observe that, in this case, there exist positive constantsA and B

such

and

that for all t £ X t-x < ii t-x 1

/-J. ^ 2 , -o , im

(t-xj < B t-x Using (2) we have then

L^(t-x;x) = o ( ; j | ^ ) , (n -> 00 ) n- ^fsf\

2 1

and L ^ ( ( t - x ) ; x ) = o(rnn") »(n -^ °° ) ^

from which the assertion follows.

A similar remark is applicable in all further results on asymptotic formulae and estimates and therefore any more reference to this point will be omitted,

Remark 2. If v;e assume that the function 2

l'p(x) - 2x?^(x) + X ? (x) is unbounded on [a,b] but that (1) and (2) hold uniformly in x £ [a,b] then an application of Holder's inequality shows that V (x)

0

must be unbounded on [a,b] . Consequently for all sufficiently large n the function L (l;x) is unbounded and therefore discontinuous on [a,b]. Thus in a practical approximation method the unboundedness of ^^(x) - 2x^1' (x) + X 1' (x) is unlikely to arise, Remark 3' In the second part of tneorem 1, dealing with the uniformity of relation (3) on an interval [a,b], we have made the assumption (in the definition

(2)

of the class H -5.<a,b>) that the seG>.nd derivative f" m, A

be continuou" ,",t each point of the interv::l ^a,b].

However, Suzuki [74] (Theorem B, p, 451) in his formulation (wnich is given v/ithout a proof) of Mamedov [36] and Scnurer's tneorem (Theorem 2, [o2]) does not

assume this continuity. His statement, which we reproduce exo.ctly, is as follows: (The sequence {L j of linear positive operators, occuring in the following theorem, is assumed to be mapping C[a,b] into C [a,b], where [a,b] is a finite interval).

THE OH EIM I Assume that the sequence of linear positive operators {L (f;x)j has the property that

L^(l;x) = 1 , X £ [a,b] ,

L (t;x) = X + — 7 — V + o(—7—v)f uniformly over n^ ' 9(n) ^cp(n)" ^

'^o(x)

[a,b], L^(t ;x) = X + /^\ + o ( - 7 ^ ) . uniformly over

[a,b], If there exists a positive integer m ( > 1) such that

Lj^((t-x) ";x) = °('^(7T)> uniformly over [a,bj ,

then for each f(x) £ J^ [a,b ], we have

2f'(x)Y (x)+f"(x){ Y (x)-2xT. (x) j

L (f;x)-f(x)= ^-^-7-.; ^ + o ( - 7 ^ )

n^ ' ^ ^ ' 29(n) ^ ( n ) ^

uniformly on [a ,b ] , a < a < b < b, where C^ •^[a,b] is the set of all real functions f(x) of which the

second derivatives f"(x) exist in [a,b] and are bounded. In the same paper Suzuki used theorem I to prove Proposition 1, p. 432; Proposition 4> P« 434;

Corollary 1, p. 437; Corollary 2, p. 437 and Theorem 1, p. 438 (in which actually Poposition 1 is used).

In Theorem 1 the operators L considered are such that n

they preserve linear functions and '{'p(x) is boundea, twice continuously differentiable and not equal to zero

on ( a , b ) .

In fact the statement of tneorem I is not true. We give an example of a sequence L j of linear positive operators mapping C[-2,2] into C[-2,2]preserving

linear functions and satisfying the assumptions of theorem I witn ^p(x) bounded, tv/ice continuously differentiable and not equal to zero on (-2,2) for

which the asymptotic formula given by theorem I does not hold unii .irnly on [-1,1] for a function

f £ C^2) [_2,2].

Let jL j (n=1,2,...) be the Sequence of linear

positive operators, miapping C[-2,2] into C[-2,2J, defined in the follov/ing way:

i[„,

_Jisdi^Js^^

, ,(,. vSujUi!;

(2n+ ^JTt L (f;: n^ (2n+ -l):: 1 < |x •) < 2 ^ [i(x+ (2n+ ^)n 1,2,... , We have r(x-L^(l;x) = 1, x £ [-2,2] , L v"fc;x) n^ ' ^ e [-2,2] , (2n+ •^)n T rx2 N 2 y (x) . 1 -, L^(t ;x) = X + — ^ + o(-2) , 4rt "n n uniformly over [-2,2j, where

f |X| S 1, 'P(x) = 1, i x| g 1 Also 1-(| x ! - l ) ^ , 1 < Ixl g 2 . . ((t-x)'^'^;x) = o ( ^ ) , uniformly over [-2,2] , n j^^

where m > 1 is an arbitrary positive integer. Assuming that theorem I

f £ c'-^^[-2,2] we would have

Assuming that theorem I is true, for each (2),

1 t^ \ ff -S ^(x)f"(x) ^ / I N

8-11 n n

uniformly on [a^,b.] where -2 < a < b. < 2, Taking a =-1, b.=1, for an arbitrary e > 0 we can then find an integer N > 0 such that for all n > N. and x£ [-1,1] there holds

|m2{f(x-t^) + f(x-^) - 2f(x)j - f"(x)| < e -1

where m = (2n+T-)7i and f is a fixed element of

(2) 1 C^ "^[-2,2], Choosing x=0 and —, in turn, we have

I 2

f{^) + f(-^) - 2f(o)j - f"(o)| < e , and f(^) + f(o) - 2f(-i)j - f"(^)| < £ ,

whenever n > N., It follows from these two inequalities that

|m2|f(|) + 3f(o) - 3f(^) - fi-i)]-{f"ii) - f(o)}|<2e whenever n > N., By L'Hospital's rule

lim m2(f(|) + 3f(o) - 3f(^) - f(-i)! m — 00

= lim f (2f(^) - 3f'(-) + f'(-^)} 2 ' ^m' ^m' ^ m'' m -* 00

= lim m!f'(|) - f'(o)j - lim -^{f' (^)-f' (o)j m -• 00 m -• 00

+ lim f (f(-l) - f'(o)j m -* 00

Hence there exists a positive integer Np such that

|m^{f(^) + 3f(o) - 3f(-) - f(--)j| < £ for all n > Np. Thus for all n > N ,Np we have

f "( \ ) - f"(o)| < 3e . (2n+2)Ti

Consider the following choice of the function f.

0, if x=0, f(x) = ^

x^sin -, if X £ [-2,2] and x 4= 0. It is easy to check that f(x) £ C^^[-2,2] and that on the interval [-2,2] its second derivative f"(x) exists and is given by

0 , if x=0,

f"(x) = j

(12x^-1)sin - - 6x cos -,if x £[-2,2],x40. Clearly we can choose a positive integer N, such that for all n > N, we have

3

|12( ——)^sin(2n-4)7i - — ^ — c o s ( 2 n - 4 ) T i | < e. (2n-t^)7i (2n4^)K Thus for all n > N , Np, N, we have

I 1 |sin (2n+2)7i| < 4e •

As e > 0 is arbitrary, this gives us a desired contradiction.

(4) f(t) = f(x) + (t-x)f'(x) + % ^ f"(x)

+ {(t-x)2 + It-xrjh^(t)

and putting h (x) = 0, the continuity of h (t) at t=x

DC 3C

fellows. Thus given an arbitrary £ >0, there exists a 6 > 0 such that

(5) |h (t)I < e , for all t £ X with jt-xj < b . Since there exist positive constants A and B such that

|f(t)| < A + Bltl"^ for ail t £ X, it follows from (4) that there exists a positive constant M such that (6) Ih (t)I < M, for all t £ X with |t-x| g b . Obviously we can choose M so large that both of the inequalities

(7) l\(t)| < M, and

ra\ lu ^ ^ I / Mlt-xT"^ (8) h (t) < e + —' hi— ^ ' ' x^ ^ ' ,m-2 o are valid for all t £ X.

By linearity and the positivity of L , using (7) and (8) we have the inequality

(9) |L^(f;x)-f(x)L^(l;x)-f'(x){L^(t;x)-xL^(l;x) - ^^{L^(tSx)-2xL^(t;x)+x^L^(l;; n' ;x;

= lLj{(t-x)2+|t-xrjh^(t);x)|

S e L^((t-x)^x)+M(l4^j;^)L^(|t-x|"';x), 6

valid for all sufficiently large n.

It follows from (I) that, for all sufficiently large n,

"f^M ^ ^ ^^ ^^ ^^^ ^^T^(x)-xy^(x) 9^ (10) |r(,){-°_^+1 -L^(l;x)j+f.(x)i ^ -L^(t;x)+xL^(l;x)j+ n ^„(^) Y2(^)-2x^-,(x)+x^?o(x) 2 5 ^(^ - L (t ;x) + 2 x L (t;x) - x L (l;x)j| g -f-y n^ ' ^ n^ ^ n^ ' ^'I 9(n) and also that

(11) L^((t-x)2;x) g ^ (^)

where K is a positive number independent of n and e, Again by (2) for all sufficiently large n

(12) M(1 . - ^ ) L ^ ( l t - x r ; x ) g ^ .

Combining above inequalities, it follows that for all sufficiently large n

(13) |9(n)iL^(f;x) - f(x) - ^(7y[i"(x)y^(x)

+ f'(x)fY^(x) - xV^(x)j + ^ f"(x){'?2(x)-2x?^(x) + x^Y^(x)j]j| g e (2+K).

Since e > 0 is arbitrary (3) follows.

For the uniform convergence part, by a mean value theorem

(14) f(t)-f(x)-f'(x)(t-x) = -^^=1^ f " ( 0

for some E, lying between t and x, if f" exists at all (2)

points between t and x. If f £ H /. <a,b>, applying m, A

(Lemma 1, p,12, Korovkin [30]) to the function f", we find that given an arbitrary e > 0, there exists a 6 > 0, independent of x, such that

( 1 5 ) | f " ( t ) - f " ( x : S £

f o r a l l t £ X s u c h t h a t | t - x | < 6 , x £ [ a , b ] . Hence from ( 4 ) we have t h e i n e q u a l i t y

( 1 6 ) 2 | h ^ ( t ) | j l + | t - x r - 2 j = | f " ( 0 - f " ( x ) | § £

for all t £ X and satisfying |t-x| < 6 ,x £ [a,b]. It also follows that there exists a constant M such that (7) and (8) hold for all x £ [a,b]. By the boundedness of the function Y (x)-2xy (x) + x 'F (x), K in (II) can also be chosen to be independent of x. In this way the right hand side of (13) becomes independent of x and the uniform convergence follows.

As the necessity parts in both the assertions of the theorem are trivial this completes the proof of the theorem.

COROLLARY 1,1 Let m^ and mp be two positive numbers where m > 2, Let (L , n £ UJ be a class of linear operators defined on a common domain D(X) (X^=.R) of functions into a domain D(x) (X & R ) of functions and ultimately positive on a set X £^X, Assuming that

~ 2 I i™1 I * imi I | m 2 . ,

X £ X and 1,t,t , |t-x| and |t-x| |t| £ D ( X ) , if and only if (1) together with the conditions

m^ . (17) L (|t-x| ^ x ) = o{-i-v) ^ ' n^' I ' ' 9(n) and

(18) L^(it-x| ^ t ) 2;x) = o ( ^ )

hold as n ^ 00, where 9(n) 4= 0, (p(n) -* 00 as n -* 00 , then the asymptotic relation (3) holds for each

f £ D(X) r. H^2) u).

m^+mp.X ^ ^^

|t-x|"^'^ h i ^ G D ( X ) f o r a l l x £ [ a , b ] and t h a t t h e

p

function T„(x)--2x'l'. (x) + x '? (x) is bounded on [a,b],

(?) °

for each f £ D(x) r, E^ i ^<a.,-b>, (3) holds uniformly m.+mp,A

in X £ [a,b] if and only (l), (17) and (18) hold uniformly in x £ [a,b],

Proof. The necessity parts in both the assertions of the corollary are trivially true. In order to prove the sufficiency parts, we observe that two positive constants A and B can be found such that for all real values of t the following inequality holds

m.+m m m. m ( 1 9 ) | t - x | g A | t - x | + B | t - x | | t | , m +m2 f o r e a c h x £ [ a , b ] . Hence, a s s u m i n g t h a t | t - x | £ D ( X ) , f o r a l l n s u f f i c i e n t l y l a r g e m +mp m ( 2 0 ) L ^ ( l t - x l ; x ) g A L ^ ( | t - x | ' ; x ) m mp + B L ( I t - x l Itl ;x)

by (17) and (I8), Thus, in case (17) and (I8) hold uniformly in x £ [a,b], the o-term in (20) holds

uniformly in x £ [a,b]. Now with m=m^+m^, theorem 1 is applicable and the sufficiency parts of the corollary follow,

m.+mp

In case | t-x | ^ I^(x), we go back to the proof of theorem 1 and replace the function |t-x| everywhere by the function A|t-x| + B|t-x| |t| and proceed analogously. This completes the proof of the corollary. COROLLARY 1.2 Let m g 4 I^e an even positive integer, Let JL , n £ uj be a class of linear operators defined

on a common domain D ( X ) ( X £ R ) of functions into a domain D ( X ) ( X ^ R ) of functions and ultimately positive

r^ '- r^ 2 3 4

on a set X ^ X . Assume that x £ X and 1,t,t ,t ,t , ^m-4^^m-3^^m-2^^m-1 ^^^ ^m ^ ^^^^^ ^^^ ^^^^^^ ^^^^^ ^^^

Tp(x) be some functions of x. Then a necessary and sufficient condition for (3) to hold for each

(2)

f £ L(X) r> H^ ^(x) is that it holds for the functions i,u,,..,u ancL u ,,.., X .

Further, assuming that [ a , b j ^ X and that the 2

function ?p(x)-2xT (x) + x ¥ (x) is bounded on [a,b], for each f £ D ( X ) r^ H^^2<a,b>, (3) holds uniformly in

m,A

X £ [a,b] if and only if it holds so for the functions 1 + +4 ^ +m-4 .m

Proof. The necessity parts in both the assertions of the corollary are trivial. To prove the sufficiency parts,

2

we note that for f(t) = 1,t,t , the relation (3) is identical with the respective relations in (1). Now, since f,f',f" occur linearly in (3), choosing m.,=4 and mp=m-4 we find that (17) and (I8) are satisfied. Now corollary 1.1 is applicable, completining the proof of the corollary 1,2,

COROLLARY 1,3 Let m g 4 te an arbitrary positive integer (i.e. not necessarily even). Let {L ,n £ Uj be a class of linear operators defined on a common domain

D ( X ) ( X ^ R " ^ ) of functions into a domain D ( X ) ( X ^ R ' * ' )

of functions and ultimately positive on a set X . ^ X , Assume that x £ X and 1,t,,.., t and t ,...,t £ D ( X ) ,

Let Y (X),'? (x) and Tp(x) be some functions of x. Then a necessary and sufficient condition for (3) to hold for each f £ D(X) ^ H^^|(x) is that if holds for the ^ X • -IX x 4 •, x m - 4 xin

Proof. Choose m.=4, mp=m-4. Then (19) with |t|

mp '

replaced by t '^ is valid for all t g 0. Rest of the proof follows along the lines of the proofs of the corollary 1,1 and the corollary 1,2,

Remark 1. In the special case when ?p(x)-2x'i'^ (x) 2

+ X f (x) = 0, relation (2) is already fulfilled with m=2. The second derivative f"(x) then does not occur in (3) and this formula reduces to

(21) L^(f;x) - f(x) = ^ [ f ( x ) ^ o ( x )

+ f'(x){¥,,(x) - xf^(x)j] + o ( - ^ ) , In fact the existence of f"(x) is not necessary in this case and it is sufficient to assume that, with an

extension of f, f'(t) exists for t belonging to some neighbourhood, say (c,d), of the point x and that the first divided differences of f'(t), at the point x, are uniformly bounded for all sufficiently small

step-lengths h, say |h| < 6 where 6 > 0. To prove this we proceed as follows. By the assumptions on f'(t), for a sufficiently small & > 0, there exists a constant M > 0 such that

if'(x+h) - f'(x)i

I h 1 ^ ^ ^ '

whenever |h| < 6 (in this and the following step we are working with an extension of f and so we need not

specialize x+h,t and ^ to belong to X ) , By a mean value theorem if S > 0 is sufficiently small and I t-x | < 6, we have

where ^ is a point lying between t and x and therefore

|(t-x)(f'(0-f(x))| g(t-x)2|^l^lM^Iil|.

Now we can specialize t to belong to X and have (22) |f(t)-f(x)-(t-x)f'(x)I g M(t-x)^

wheij.ever |t-x| < 6 and t £ X, Rest of the proof follows on the lines of the proof of theorem 1,

Formulae of type (21) occur in certain

perturbations of the Baskakov sequences recently studied by Sikkema [70].

Remark 2 In the case when (2) or similar conditions do not hold or are not known to hold, the following

[l

1

result may, nevertheless be applicable. By H^ ^(x)

m, A

we denote the subclass of H „(x) consisting of the m,X

functions f which, with an extension, possess a first derivative at each point in a neighbourhood say (c,d) of the point x such that at the point x all the first divided differences of f'(t) with sufficiently small step-lengths h, say |h| < 6 where 6 > 0, are uniformly bounded by M„(x), say, i.e.

|f'(t)-f'(x)| < M^(x) |t-x|

for all t satisfying 0 < |t-x| < b for a sufficiently small 6 > 0. If for each x £ [a,b], f £ H'-''-!(X) and if

'- ' •'' m,X^ ' the set { M „ ( X ) , X £ [a,b]j of numbers is bounded, say by M > 0, then we write f £ H^- -! [a,b]. It is clear that such an M exists if for same 6 > 0, f' £ Lip,, on the interval [a-6,b+6] ; also that it is necessary to have f £ Lip.^ 1 on the interval [a,b].

f

be a class of linear operators defined on a common

domain D ( X ) ( X S - R ) of functions into a domain D ( J J ) ( X ^ R )

of functions and ultimately positive on a set S!^X. Assume that x £ X and 1,t,t ,t ~ ,t ~ and t £ D ( X ) ,

Then for each f £ D(x) r^ H'-''J(X) there holds (23) L^(f;x) - f(x) = O ( ^ ) ,

as n -* oo, where (p(n) 4= 0, q)(n) -^ooas n — oo, if and only if it holds for the functions 1,t,t ,t ,t and t .

Further, assiuning that [a,b] ^ X , for each

f £ D(X) ^ HL''^[a,b], (23) holds uniformly in x £ [a,b] m,A 2 _._p if and only if it holds so for the functions 1,t,t ,t ,

,m-1 , ,m t and t .

/^ +

Also, in the case when X,X ^ R , the above asseri assertions are true even when m is an odd integer g 2. A proof of theorem 2 can be given in a way similar to that of the proof of theorem 1 where instead of

using (4) we start with the inequality (22) of remark 1. We omit the details.

1.1.4 A class of linear positive operator sequences In tnis section we apply the results of sections 1.1.2-1.1.3 to the case of the Baskakov-eequences of linear positive operators and thereby extend an earlier study of them made by Schurer [62]. First we give a brief resume of the results of Schurer.

Schurer [62] considered the sequence jL j(n=1,2,...) of operators defined by

^ cp' '(x) X

(1) L (f;x) = s (-i)^-^Hn ^(zh\)'

_n

THEOREM III Let I9 (x)j be such that we have the special case

x(n) = n, 'i'(n,x) = n,

'?(m(n),x) = m(n), m(n) = n+c,

(for all sufficiently large values of n ) , where c is an integer indepenuent of n and a, (x) are independent

XV, n

of k, say, a (x). If the a (x) possess the property that at a fixed point x £ [o,b]

^6) «n^-) = ^ - "(7(^)

v/here T(n) has the prop rties: I) T(n) 4= 0; 2)

lim T(n) = 00; 3) T(n) = o(n), n -* 00 , then we have

n - oc , ,

for f £ Up ^+(x)

(7) L„(f,x) - r(,) . -P('^j;;(-) . o ( ^ ) .

Before obtaining convergence thev,rems for classes (2)

larger than the classes Kp _^(x) and Kp ^4.(x) of

functions, we note that no particular purpose is served by taking n to be an integer in the definiti -n of the operators L . We assume, therefore, that n £ U where U is an unbounded set of positive real numbers. With this stipulation the numbers m(n) also need not be integers. Further, it v/ould be sufficient to assume that the properties i)-iii) hold for all sufi'iently large values of n.

The follo\/ing theorem extends the result of theorem 1 of Scaurer to the classes H p+(x) and Ji„ -n+^^-fl^^j wituviut any extra assumption.

ThliOREM 1 Let f £ H p+(x), x £ [o,b]. Then for the

where jx(n) } (n=1,2,...) is a sequence of positive numbers increasing to infinity with n and the sequences

W (x)j(n=1,2,...) of functions possesses the following properties on an interval [o,b] (o < b < °o) s

i) V^{°) = 1

ii) 9 (x) is infinitely differentiable and (-l)^9^^\x) g 0, (k=0,1,2,...);

iii) there exists a positive integer m(n), not depending on k, such that

- 9i")(x) = .(n,x)cpif;))(x) il.a,^,(x)}, k=1,2,..., where

iii) a, (x) converges to zero uniformly in k when n -* oo, and

iii)p 'i'(n,x) satisfies the following properties:

(2) li,

Ii^4-

=1, and

n -• oo A\ /

v\ -, • 'y(m(n) .X,

3) lim — ^ — ^ Y ^ — ' n -* OO

Here and in theorems I-III it is assumed that f belongs to a certain class of functions for which(l) for n=1,2,... is meaningful. This depends on the nature of the functions 9 (x).

Remark. It is known that a function 9(x) satisfying (_l)^(pV^-'(x) g 0 (k=0,1 ,2,...; X £ [o,b]) has an

analytic continuation for |x-b| g b. Hence for 0 g a g b the series

00 , v k ( k ) / N k „ (-1) 9'- a a ) x

k' k=0

has a radius of convergence > a. It follows that for m=U,1,2,..., the series

°° / ^ \k (k) / N k, m (-1 ) 9 (x) X k

k'

k=0 ^•

is convergent for each x £ [o,b] . As a Cv.nsequence, if f(t) is a function bounded on each bounded subset of R and satisfies f(t) = 0(t"), (t — oo), for some m > 0, then (l) for n=1,2,... is meaningful for sucn a function.

Schurer proved the follov/ing tneorems l-III: THEOREM I If f £ H^ j^+(x) and if the sequence 9 (x) satisfies the conditions (i)-(iii) then the sequence { L (f;X)}(n=1,2,...) defined m (1) converges to f(x) when n — 00, if^ moreover, m (iii)-, a, (x) converges

I iC, n

to zero uniformly in x on [o,b] and if the relations (2) and (3) hold uniformly m x on [o,b], then the sequence JL (f;x)j (n=1,2,...) converges uniformly on [o,b] to f(x), assuming f(x) £ Hp j^+(x) (O g x g b ) . THEOHEIi II Let {9 (x)j be such that we have the special case

x(n) = n, 1'(n,x) = n

'i'(m(n),x) = m(n), m(n) = n+c,

(for all sufficiently large values of n ) , i/htre c is an integer independent of n and a, (x) are independent

rC I n

of k, say a (x). If the a (x) possess the property that at a fixed point x £ [o,b]

(4) a (x) = ^ ^ + o(l) ' n^ ' n ^n^

t h e n we have l o r f £ lip n + ( x )

( 5 ) L ( f ; x ) - f ( x ) = 2 x p ( x ) f ' ( x ) 4 - f " ( x ) ( x + c x ' ) ^ _^^1^

TIIEuREM III Let {9 (x) j be such that we have the special case

x(n) = n, 'i'(n,x) = n,

?(m(n),x) = m(n), m(n) = n+c,

(for all sufficiently large values of n ) , v/here c is an integer independent of n and a (x) are independent

K, n

of k, say, a (x). If the a (x) possess the property that at a fixed point x £ [o,b]

v/here T(n) has the properties: I) T(n) 4= 0; 2)

lim T(n) = °o; 3) x(n) = o(n), n -* 00 , then we have n - 00 , .

for f £ i4^^+(x)

(7) L„(f,., - f(x) = ^ ^ ^ i ^ ^ . o ( - ^ ) .

B e f o r e . o b t a i n i n g c o n v e r g e n c e t h e . . r e m s f o r c l a s s e s

(2)

larger than the clasoes K„ „x(x) and lu T)+{^) of functions, we note that no particular purpose is served by taking n to be an integer in the definition of the operators L . We assume, therefore, that n £ U where U is an unbounded set of positive real numbers. \/ith this stipulation the numbers m(n) also need not be integers. Further, it v/ould be sufficient to assume that the properties i)-iii) hold for all suffiently large values of n.

The follo\/ing theorem extends the result of tneorem 1 of Scnurer to the classes H p+(x) and H„ P+<a,b>, witiKiut any extra assumption. THE OREL: 1 Let f £ K ;+(x), x £ [o,b]. Then for the

operators L defined in (l), there holds L^(f;x) -* f(x) as n — oo ,

Also, if a, (x) - 0 uniformly in x £ [a,c] (O g a < c g b) and if in (iii)p, (2) and (3) hold

uniformly in x £ [a,c], then L (f;x), (n £ U ) , converges uniformly on [a,c] to f(x) as n -« 00 for each

^ e

\,R+<"'^>-Proof. For k g m, where k and m are positive integers, we have . m-i m-1 a , m , , m k = k! s J-. -T-yr 1 = 0 (k-i"+i)'

where Q ~ , i=0,1,,..,m-1 are the Stirling's numbers of second kind ([22], § 58, p. I68-I73). Hence for an arbitrary positive integer m \/e have for k g m the relation ( k ) / V k „ / ., \k (k)/ X k m-i r.^ . . ^ k ! L ™ L k" m-1 G ^ O ^ ( n ) x a„ (8) (-1) — Y i ; r — = . ^ ^ (k-m+i) X (n) 1 = 0 ^ , \ ^""^ / -\k-m (k-m)/ \ k-m m__m „ ^ (-1) m (x) X a X m-1 . ^ im m _{- _ n M m - ( n ) , x ) ( l + a ^ _ i^^)(x))} ^} ? (n). n) 1=0 » \ / X (n) m-1 m-1 2 + ^S 'n Mm^(n),x)(l+a, . u ^(x))} . x"'(n) i=0 k-i,mi(n)^ ^^I

/ .\k-m+1 (k-m+l)/ N k-m+1 (-1 ) m^ . (x)x

^ ^ ^ m-1/ N^ '

m (n)

• (k-m+1)! + ... ,

v/here m (n), 1 = 0,1,2,... are defined inductively by o

m (n)=n, m (n)=m(m (n)), m (n)=m(m (n)), and so on. In the following we will use the easily verifiable fact that a = 1 and a = m(m-l)/2. Omitted terms correspond

to the values of i < m - 1 .

For the values of k g m - 1 , the only important case is when k=m-1. In tnis case there holds

m-1 o"^-^

(m-i)'" = (m-1)! E

jf-jy ,

1 = 1 ^ ^' and as in (8) we have / .Nm-1 ( m - l ) / \ m-1

^^^ (^iryi m. ^

^ X (n) m-1 m-1 _ (-1)% (°) (x) x° a X m - 2 . ^ ' Tiii_-i\ / _m n {?(m^(n),x)(l+a . . ±. Jx))] ^ ._„' ^ \ /» /\ m-1-i,m-^(nj^ "' 0! x"^(n) i=0 "T* • • • •

Using" this and (S) we have

( 1 0 ) 2 ( - 1 ) ' ( k ) / N k -^ ( x j X ,,m i n k ! ^f \ k=0 X ( n ) m m a X oo m - i = — 2 [ n W ( m ^ ( n ) , x ) , x ) ( l + ( x , . ±f s ( x ) ) i , m, s , n . r^ \ /» /» / \ m + k - i , m ^ ( n ) ^ ^ ' ' X ( n ) k=0 1=0 ' ^ ^ / . \ k ( k ) / N k ( - 1 ) 9 ^ ^ ( x ) x ^m-1 ^m-1 ^ ^_^ . ^ ] + 2 [ n ^' x'^Cn) k=0 1 = 0 { ? ( m ^ ( n ) , x ) ( l + a . , . ±f N ( X ) ) J , ' \ ' » / \ m - 1 + k - i , m - ^ ( n j ' ' * / . s k ( k ) / N k

^-'^

^

m-V ,^^^

^

k! ] + • • • !

where the omitted terms contribute a o ( l / x(n)) quantity,

m ( n ) — oo a s n -* oo) t h a t f o r e a c h f i x e d 1 = 0 , 1 , 2 , . . .

( 1 1 )

n — oo

, i , K 4 ^ = 1 .

xTnT

From this and (iii) it follows that given an arbitrary £ > 0 we can choose a positive number N such that for all n > N we have and m-2

n

i=0

(13) 1-£ < ""ii j('^("^VM))(i+a , ^ . u ^(x))j < 1+£,

^ ' •-n x(^) m-1+k-i,m-'-(n)^ '' We can choose N so large that if n > N then the contribution of omitted terms in (lO) does not exceed e/x(n) in absolute value. Then it follows from i ) , (10), (12) and (l3) that if n > N

m-1 m-1 a X

(14) L (t ;x)-x < £x + j—T— (1+e) + —7—r .

^ ^ ' n^ • ^ I x(n) x(n) Since e > 0 is arbitrary, we have

(15) lim L^(t"';x) = x'". n -* 00

Noting that m in (14) is an arbitrary positive integer and that L (l;x) = 1 for all n, from theorem 1.1.2,2 we have

(16) lim L^(f;x) = f(x) , (x £ [o,b]) n -* 00

for each f £ H p+(x) where m is any positive integer g 2. This is equivalent to the first assertion of

theorem 1.

If the conditions in the second assertion of theorem 1 are satisfied then for x £ [a,c], I' in (14) c m be chosen to be independent of x. It follows from (14) that (15) then holds uniformly in x £ [a,cl and the second assertion of theorem 1 follows from the sec md assertion of theorem 1.1.2,2. This completes the proof of theorem 1,

In the next two results we improve upon theorems Il-lll of Schurer, Again we note that no extra

assumptions are involved,

THEOREM 2 Let the operators L defined in (1) be such that v/e have the special case

x(n) = n+p, m(n) = n+c, il'(n,x) = n,

where for all sufficiently large values of n, p and c are constants independent of n. Further if for all sufficiently large n, a, (x) are independent of k, say

a (x), and satisfy

(17) a (x) = - e ^ + o(l) , n ^ 00 ,

(2) at a fixed point x £ [o,b], then for all f £ H); p+(x)

(18) L (f;x)-f(x) = 2(p(x)-p)xf'(x)+x(cx+l)f"(x).,^(l^ 2n

as n

Further, if (17) holds uniformly in x £ [a,c] (0 g a < c g b ) , where p(x) is bounded on [a,c], then

(2)

(18) holds uniformly in x £ [a,c] for each f £ Hl^ R+<S-,C^

Q, R Proof. If ollows from (1) that for an arbitrary positive integer m, we have

m m , o x m-1 L (t"';x) = -^S n [(n+ic)!l+ ^ ^ 4 ^ + o(l)j] n^ ' ^ / sm . „ '-^ '' n+ic ^n''-' (n+p) 1=0 m-1 m-1 a X m-2 m (n+p)'" 1=0 n [(n+ic)ji + - a ^ + o(^)j]+o(-i) „ '-^ ^ ' n+ic °^n^ 1J ^n' m m ^ fn+ic+p(x)+o(l), a X n ^—^ ^—'-] m . r. n+p ' 1 = 0 m-1 m-1 „ a X m - 2 . /• \ ^ ^ \ y, S n ,n+ic+p(x)+o(l), ^ ^^U n+p ._„ ' n+p ' ^n^

= o™ x"" (1 + ^i^-^)'^ + "i(p(^)-p) m ' 2n n m-1 m-1

m /1 \

n 'n'

Thus, putting the values of a and a , for

' ram' m=0,1,2,...

(19) L^(t'";x) = x'" + "^("'-1)(^^+I)x"'~^

+ I^(P(^)-P)^'" + o(l).

From (19) and the first part of corollary 1.1.3.1 the rel:ition (I8) is immediate.

To prove the second part of tiie^rem 2, it is easily verified tnat under the given uniformity

conditions (l9) holds uniformly in x £ [a,c]. Thus the second part follows from (19) and the second assertion of corollary

1.1.3.1.3-[1 ] 1.1.3.1.3-[1 ]

Let RL jj!(x) and Ht^ i[a,b] respectively denote the

Si«,A y , A T i l

totality of functions of the classes H L J ( X ) and •m,X'

[1 ]

H ^[a,b], m varying over all positive numbers.

m , A

V/e have the following generalization of theorem I H THEOREM 3. If in the statement of theorem 2 the condi condition (l7) is replaced by

(20) a (x) = - 4 4 + oi-r^) ,

^ ' n^ ' x(n) ^T(n)' '

where T(n) 4= 0, lim T(n) = <», T(n) = o(n), n -^ oo,

n - oo , ,

then \/e have for each f £ Hl^ v+^-^^

(21) L (f;x) - f(x) = ^P^^lf'l"^ + o{-K), n ^ oo,

R ~ ^ -"

^^TUT'

Further, if (2) holds uniformly in x £ [a,c] (O g a < c g b) where (x) is bounded on [a,G], then the convergence in (21) is uniform in x £ [a,c1 for

(2)

each f £ H^^+<a,c>,

Also, in tne first assertion above, the class n\ /}+{x) can be replaced by the larger class H|- j+(x)

U,n / s Q,K

and in the second asserti.,n the class R \ pj.<a,c> can be replaced by the class H^ J^[a,c].

The proof of theorem 3 proceeds on the lines similar to those of the proof of theorem 2. After pr. ving the first two assertions, we apply remark

1.1.3.1 made in connection with the relation (1.1.3.21) to obtain the third assertion.

1.1.5 C-eneralizations for functions of several variables In this section we obtain generalizations of some of the results of earlier secti ns, suitable for

studying the approximation of certain classes of real or complex valued functions defined on a subset of a

Euclidean m-space R by means of a seauence of linear m

positive operators. Proofs of the result of this section can be carried out on the ba,sis of the notions

intr-^duoed in earlier sections and along the lines of the proofs of results in ([62], chapter 2) and [59]. Definitions, conventions -.nd notations

Let the ra-tuples (n.,n^,...,n ) and (p. ,p,.,... ,p ) ,

1 ' 2 ' ' m ' \x--|Fx-2» '-^m '

where m is a natural number, be denoted by the symbols II and p respectively. This notation is consisitent with that of Schurer ([62], chapter 2 ) . 'v/e shall always assume that II,p £ R"^, the first hyperquadrant of R . The notation R — oo signifies that n. — oo, j = 1,2,...,m. Let U d R be such that the set! min n.: II £ U }

"•"•'" 1 g j g m -^ "' is unbounded. Let lL.,,N £ U \ denote a class of linear

' N' m'

operators mapping a linear space D ( X ) of real or complex valued functions defined on the set X d R

m — ra into a linear space D ( X ) of real or complex valued functions defined on the set X ^E_ R . V/e assume that

m m

if f £ D ( X ) then f, the complex conjugate -jf f, also belongs to L ( X ) . The symbol X in D(X ) represents an

° ^ m' •' m ^ m' -^ assumption that for f £ D(X ) , II £ U and

^ ^ m ' m

X = X(x-,x„,...,x ) £ X we can construct L (f;X) \ -|» 2' ' m' n^ ' ' provided the values f(H), 3 = zi^^ ,1^, . ..l^) £ X^ .are given. An operator L,., II £ U , is said to be

_^N' m'

positive on a set JT cz. X if for each f £ D ( X ) the m — m' ^ m^

assumption f(H) g 0 for each E £X leads to L,.(f;X) > 0 ^^' m 1\^ ' ' ~

for each X £ X . The class (L,-, K £ U j is said to be m ' ir ^ m'

ultimately positive on a set X S: X , if to each m m '

f £ D ( X ) and satisfying f(E) s 0 for each S £ X , _{m ./ o \ / m} there exists a natural ntunber n such that L (f;X) g 0

for each X £ jt whenever min n. > n. The set 7.

may consist of a single point X £ X or may contain more than one point.

In the sequel we shall deal with the following classes of functions:

H : the class of all real or complex valued functions

P,A

f(H) defined on X for each of which there exist ^ ' m

positive constants C,D, say, such that there holds m p.

f ( H ) g C + D 2 | ^ . | ^ , for all S £ X .

j = 1 '^ ^

K „ (H): subclass of H ,. consisting of the functions

P , A P , A

' m •^' m

f(s) which, with an extension, are continuous at the point 3 = X.

(2)

H^ / (x): subclass of H consisting of the functions

p.A P«X

-^' m •^' m

f(3) which, with an extension, are twice differentiable at the point 3 = X, in the sense tliat at the point X all the first and second partial derivatives of an extension of f exist and there holds

m f(3) - f(X) = 2 (l.-x.)f' (X) j-1 ^ ^ ""j ., m m p + j 2 (^-x )(^ -X )f" (x)+o( 2 (^.-x )^) m p if 2 ( E . - X . ) tends to zero. j = 1 ' '

Let S c=-E , Denote by H „ (S ) the class of all m — m '' p,X ^ m'

•^' m

functions f £ H ^ which with an extension are p,X

"^ (2) continuous at all points X £ S . By H^ /. <S > we denote

m p , X ra

^' m

the class of all functions f £ H ^ which with an

p,X ^ ' m

extension a r e unif-^rmly twice d i f f e r e n t i a b l e on S , i . e . m' *

twice differentiable at all points X £ S such that m

the above twice differentiability relation holds uniformly in X £ S .

fl 1 ^

R^ 4 (X) : subclass of H ^ of functions f(H) which

P tA P f A

•^' m ' m

with an extension possess all first order partial derivatives at the point X and in addition satisfy

m m p f(H) - f(x) = 2 (^-x.)f' (X) + 0( 2 (^.-x )^)

^ a = i ^ ^ ""o j = 1 ^ ^

if 2 (t.-x.) tends to zero. The class of all functions f(3) £ H"- ^ (X) for each X £ S and for

^ ' p,X ^ ' m

' m

which this relation holds uniformly in X £ S is

ril " m

denoted by E^ 4 [S ] .

p.A ^ m-'

•^' m

If for some p , f £ H (X) we say that P , A

f £ E^^ ( X ) . The classes E^^^ ( x ) , H^^^ ( x ) , ' m ' m ' m

H „ ^ (S ) , E),^ <S > and H^''^ [s ] are defined in the

Q,A m fi.A m Q,X '- m-' ' m ' m ' m same way,

In the direction of approximation of functions of many variables Schurer [62] (also see [59]) gave the following two theorems.

THEOREM I Let H ( X ) denote the class of all real functions f(x) which are defined in R and which have

^ ' m the properties

1) f(X) is continuous for X=X ,

2) f(x) = 0( 2 X.) when |x.| - oo (j = 1 ,2,... , m ) . j = 1 ^ ^

Let f(x) £ H(X ) and let jL j (n=1,2,...) be a sequence of linear and positive operators defined on H ( X ) . If we write

L^(1;X) = 1 + a^(x), L (t.;X) = x + p (X), (j=1 m; n=1,2,...) n J J _ 'i»J Ul p 1" p L ( 2 t;;X) = 2 x; + Y„(X), j = 1 '^ j = 1 "^

and if a (X), p .(x) and Y (l^) have the property that n '' n, J ^ ' n^ ^ ^ •'

lim a (X ) = lim P . (X ) = lim Y (X ) = 0 n^ o' n,j o' 'n^ o'

n-*oo n-'oo '" n->oo (j=1,...,m), then v/e have

lim L (f;X) = f(X ). n ' ^ o' n -* oo

TiIEOREM II Let H^^^(X ) denote the class of all real functions f(X) £ H(X ) of which all the second

° (2)

derivatives exist at the point X . Let f(x) £ H^ (X ) and let {IT^J te a sequence of linear and positive operators defined on H(X ) . If the operators L,. have the pr.iperty that in a fixed point X

L„(1;X ) = 1 + o( / 0 i N ^ ' o ' ^cp(n.)' Y. -(X )

+

'^

X

+ o(^^)

9(n.) > ( n . ) ' L„(t.;X ) = X . (k,j=1,...,m; k + j) L(t2;x ) = x 2 . , ! 2 i i ^ ^ o ( - ^ ) N' j' o' oj 9(n.) > ( n . ) ' J 0 L,,(t, t. ;X ) = X X . ir k j ' o' °k ''

-(XJ

^2-k i^^o) 9(n^) 2:i,k^ o' f 1 N t'\^^_^\

+

y \

+ o(—7 v) + o(~7 V) »

9(n^) > ( n j ) ^ >(j^k) where 9(n.) + 0 and 9(n.) -* oo when n. — oo (j = 1,...,m).

J J J If there exist positive integers p. (j=1,...,m) such

J that

then we have [?, .f +4('i'o • •-2X .?, .)f" ] m L i;j X. 2^ 2;j,j oj 1;j^ x.x.J

L,,(f;X J-f(Xj= 2 ^ ^

N^ ' 0 ' ^ o' ^^^ 9(n.

'

^ - L _

m 'I'o.v .-x„,^..v '^2;j,k~^o^^1;j •^i 2 p;k,.1 o.i-1;k, '-> k '-^^., ^ , , ( 1 ) 0 + k

where the values of all functions ¥, f and f" are taken in X , _o

Remark Theorem II of Schurer is not correct. It requires a modification. In fact a mere assumption on the existence of all the second derivatives at the point X is not enough. Following is a counter example.

In Rp, let an operator sequence JL j, (n=1,2,...), be defined as follows:

Vf;(x,y)) =lf(x+J, y + ^ ) -Jf(x-f. y - J ) .

n=1,2,..., where a and b are two positive numbers, We have \{M{x,y)) = 1, L (t;(x,y)) = X, and

\is',ix,y))

= y,

\ { i ;(x,y)) = X + (a/n) , L (s ;(x,y)) = y + (b/n) , 2 L^(ts;(x,y)) = xy + ab/n , V(t-x)2^-^2.(x,y)) = (a/n)2'"+^

L^((s-y)2^^S(x,y)) = (b/n)2^+2^

for an arbitrary positive integer m.

The asymptotic formula given by theorem II becomes L^(f;(x,y)) - f(x,y)

= iKg)^f;;,(x,y).(^)2f;,(x,y)j

Consider the follov/ing function

0 , (x,y) = (0,0) f(x,y) = \

(. 2 2

xy ^ 2^ 2 » otherwise, X +y

defined on the xy-plane. At the point (0,0) all the (2)

second derivatives of f exist and f £ H^ ((0,0)). We have

f(o,o) = f^(o,o) = f^(o,o) = f;^(o,o) = f^y(0,0)=0, f;;y(0,0) = -1 and f^^(0,0) = 1,

If theorem II would be correct we should have then

L^(f;("0,0)) = o ( ^ ) . n

However, an actual calculation shows that

n (a +b )

For an arbitrary choice of a and b the two results are clearly incompatible,

As we shall see in the sequel, a correction, to render theorem II applicable, would be to assume

further that f is twice differentiable at the point X , In the above counter example the function f is clearly not twice differentiable at the point (0,0).