On the rate of convergence of the Durrmeyer polynomialsAbstract. For functions of one and two real variables, the rate of pointwise and uniform

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXX (1990)

G. A niol and P. P ych -T aberska (Poznan)

On the rate of convergence of the Durrmeyer polynomials

Abstract. For functions of one and two real variables, the rate of pointwise and uniform convergence of some Bernstein type algebraic polynomials is investigated. Estimates presented here generalize and extend the corresponding results announced in [2, 3].

1. Preliminaries. Let / be the interval <0, 1) and let N be the set of all positive integers. Denote by L q the class of all complex-valued functions bounded and measurable on I. Assuming thatf e L q and m e N , let us consider the modified Bernstein polynomials M m[f] introduced by Durrmeyer and defined for x e l by

m

1

= X Pm.j(x)$ f(s)p„j(s)ds,

where 1=0 0

\uJ{ l - u ) m~J (uel).

In this paper we present some quantitative estimates of the difference

М т [ Я ( * ) - Н / ( * +

0

) + / ( х -

0

)}

at those points xe(0, 1) at which the one-sided limits f ( x ± 0 ) exist. In the special case when / is of bounded variation in the Jordan sense on I, we obtain the recent result of Guo [3]. From our general estimate it also follows that the degree of approximation of any continuous function / by M m [f] , in terms of the modulus of continuity of /, is co(\/y/m; f ) (see [2]). Finally, similar problems for the two-dimensional Durrmeyer operators are considered.

Analogous results for the classical Bernstein polynomials of one and two variables can be found in [5] and [6].

In what follows, the symbols ct (ieiV) will mean some positive absolute constants. The integral part of a real number q will be denoted by [q].

2. Basic estimates. Let us start with some known facts concerning the Durrmeyer polynomials. Writing

m

K m(x, s) = (m + 1) X Pm.j(x)Pm,j(s), j= 0

M m[f]{x) = $ f (s)К m(x, s)ds 0

( 1 ⁾

we have

10 G. A n i o l and P. P y c h -T a b e r s k a

and

(2) j K m(x, s)ds = 1 (x e I, m + le N ) . о

It is easy to verify that, for every positive integer m,

(3) 1 K m{x, s)ds< 2 2 + i f 0 ^ £ < x ^ l ,

о ( x - £ ) 2\ m m2 )

(4) JK m(x, s)ds < 2 ^ + if 0 ^ x < ц ^ 1

^ (rj — x ) \ m m j

(cf. Lemmas 5 and 6 in [3]). Moreover, if 0 < x < 1, then Pmj(x) ^ 2(mx(l — x))~1/2 (meiV, j = 0, 1, 2, ..., m) (see the proof of Lemma 2 in [4]) and

(5) И т (х)| = | j K m(x, s ) d s - ] K m(x, s)ds\ ^ 7(m x(l-x))~1/2

X

0

for all positive integers m. The last inequality was stated in [3] for sufficiently large m; however, more precise calculations show that it remains valid for all meiV.

In the estimates which will be presented below we shall use the modulus of variation of a bounded function g on some intervals J = <a, /?>. Given any positive integer r, this modulus is denoted by vr(g; J) or vr(g; a, f$) and is defined as the upper bound of the numbers

r - l

Z \e(X2j+l)-g{X2j)\

j = 0

over all systems IJr of r non-overlapping open intervals (x2y, x2j + 1) c (a, P), j = 0, 1, ..., r — 1 (see e.g. [1]); in this definition we assume, ad­

ditionally, that the length of each interval of any system Пг does not exceed (P-a)/r.

L

e m m a

1. Suppose that a function g of class L q vanishes at a fixed point xe(0, 1). Let a and b be arbitrary positive numbers satisfying x ^ a ^ l , 1 — x < b ^ 1. I f mx2 ^ 4a2, then

(6) i b w ^ o c , s)ds\ < - i j s t

0 “ l j = l J

where p = ['x^Jm/a]. I f m( 1—x)2 ^ 4b2, then

(7) Ifg (s )K J x , s)ds| ^ j ~2 ¹ л 8 Z ^L

^v

vjte> x ’ x + j b' f rn) . 2vAd> x >

! ) + ■

j = i

where о = [(1 —x)sJ~mfb'\.

P ro o f. In order to obtain (6), set Sj = x — jaf+Jm (j = 1, 2, g), sn+i = 0 and write

X X f l S j

f g(s)Km(x, s)ds = j g{s)Km(x, s)ds+ Z g{sj) J K m{x, s)ds

0 si ^{j =} 1 ^{s j + 1}

+

1

? (

9

( s ) -

9

( s j ) ) K » ( x . s ) < i s = У . + П + П . s a y -

j = l S j + l

It is easy to see that

X

l^il ^ 1 \g{s)-g(x)\Km(x, s)ds < v\{g\ st , x).

( 8 )

By virtue of (3),

«

1

f K m(x, s)ds ^ — ——

2

if 0 ^ £ < x < 1, m ^ 4.

о myx ç)

This inequality and the Abel transformation lead to

l^2l = b (S l)Z

T K m ( x ,

s)ds+ Z

( g ( S j + l ) - g ( S j ) )

Z J Km(x, s)rfs|

. fc=lsjc+l

j=

1 fc=j+lSk+l

I f

^_1 1

Z \g(Sj+ t ) - 9( s j ) \ j p ^

1

** ]

Z l^(s*+i)— flr(s*)| >

№ k =

1 J

< L L , a - S x ) + 2 Y 2 v i + A e ’ S j * l ' - - h Vl‘ i e ; s > ' x )

^ a 2 \ l ( 9 ’ i ’ ^’ Д Ü + 1 ) 3 /<2

1 f-, V 1 pito ;5/’ *) »,(g; o. *)

< 3 2 1

Next,

j=i

< I

v ^ g ; s j + u Sj

]= 1

- ± \ f у

v i t e ;

sk+

1 ..2

U = 1 v P

\ v u ( g ;

0, x) t

l ? '

1

"

>?

Sj +1 "

^j

= 1

, ^

V j,

! ( # ; S / + 1 , x )

6Д

^{o- + d;}

1 f^ (g ; 0, x) ^ ^-(g; s,, x)]

a2 \ g2 j ^ 2 f

Collecting the results we get (6). The proof of (7) runs analogously.

12 G. A n i o l and P. P y c h -T a b e r s k a

R em ark 1. In case mx2 ^ a2 [m( 1 —x)2 ^ b2] the inequality (6) [resp. (7)]

still holds with the right-hand side multiplied by 5/2.

3. Main results. Estimates given in Section 2 enable us to deduce easily the following

T heorem 1. Suppose that f e L ^ and that, at a fixed point x e (0 , 1), the one-sided limits f { x ± 0) exist. Let a, b be arbitrary numbers such that x ^ a ^ 1,

1— x ^ b ^ 1. Then, if mx2 ^ 4a2 and m(1—x)2 ^ 4b2, we have

\Мт[ Л ( х ) - Н П х + 0)+ Я х-0)}\ ^ l(m x (l-x ))_1/2|/(x + 0 ) - /( x - 0 ) | + J_ L у Vjidx', x —ja/y/m, x) | 2у^{дх; 0, x)

a2 \ M f V2

, 1 f0 v vj (9 x > x > x + jb /^ m ) t 2va{gx; x, 1)

' 7 2 ) ® *3 ' 2

b2 l A Г . <72

where p — [x^/m/a], tr = [(1 — x)«Jm/b] and gx is defined by f f{s)—f{x + 0) when x < s ^ 1,

gx(s) — л 0 when s = x,

L/(s) —/(x —0) w h e n 0 ^ s < x . P ro o f. Identity (2) and inequality (5) lead to

|M „ [ /] ( x ) - i{ /( ^ + 0 )+ /(x -0 )} | = \MmlgJ(x) + i { f ( x + 0 ) - f ( x - 0 ) } A J x ) l

\M„ [g J(x)| + i(mx(l - x))~ 1,2 \ f ( x + 0) - f ( x - 0)|.

Applying Lemma 1 to

x 1

Mm[0 j(x ) = J gx{s)Km{x, s)ds + $gx(s)Km(x, s)ds

0

x

we obtain the desired assertion.

R em ark . 2. In view of the continuity of gx at x, the right-hand side of the inequality in Theorem 1 converges to zero as n tends to infinity (cf. Remark 2 in [5]).

R em ark 3. If the fun ctio n / of class L q is such that vn(f; I) — o(n) as n — > oo, then the conclusion of Theorem 1 holds at every x e (0, 1) (cf. Theorem 4 in [1]).

Passing to some special estimates we first give the simple

C orollary 1. I f f is continuous on the interval I and if (o(0;f) denotes its modulus of continuity, then

(9) max \Mm[f]{x)-f(x)\ ^ c ^ i l / ^ / m i f ) (meiV).

Indeed, in this case, for every interval <a, /?> c I, Vj{gx; a, jS) = Vj i f ’, a, f t ^ j(o f j (/' e N).

Hence, by Theorem 1 (with a = b = 1 and m > 16),

| M m[ / ] ( * ) - / M ! <

32

(

0

(

1

/ ^ ; / )

whenever 2Д/m ^ x ^ 1—2/yJm (cf. [5], pp. 70-71). If 0 ^ x ^ 2Д/m , we have

|Mm[/] (x )-/(x )| < j) + \\(f(s)-f{x))K m{x, s)ds| ^ 1 8 ^ ( 1 /^ ,- /) ,

X

by (7). Analogously, in view of (6), the same estimate is true if 1 — 2Д/m ^ x ^ 1.

Thus (9) follows for m > 16. Under the assumption m ^ 16, this inequality is obvious.

Next, let us denote by ВУФ the class of all complex-valued functions of bounded Ф-variation on / and by Уф(д; a, /?) the total Ф-variation of a given function g on the interval <a, /?> c= I (for the definition see e.g. [1] or [5]). We assume that the function Ф is continuous, convex, strictly increasing on the interval <0, oo) and that Ф(0) = 0.

If / е ВУФ, then the function gx introduced in Theorem 1 is of bounded Ф-variation on <0, x ) and <x, 1). Moreover, for every positive integer j and every subinterval <a, /?> of <0, x ) or <x, 1),

Vj(gx; a, Д) ^ ;Ф 1 \ j v 0 {gx; a, £ )j, where Ф-1 denotes the function inverse to Ф.

Taking in Theorem l a = x ,b = l —x, recalling Remark 1 and repeating the argument of [5] (p. 72), we obtain

C

o r o l l a r y

2. I f f e ВУФ, then, for every xe(0, 1) and for all positive integers m,

\Mm [ / ] (x) - H /(* + °) + f(x - 0)}K 1 (mx (1 - x)) " 1/2 |/(x + 0) - / (x - 0)|

1

^Л

1

+

w™ ⁴ 9*;x ^{} f x}

+ --- У — Ф ^{- 1} _{- У}

ф

I gx; * , x +

Finally, observe that if mx ^ 4 ad m(l — x) ^ 4, then (3) and (4) become

? 5x ^ 5fl — vt

l K ” (X> S )d S ^ 2 ^ P and

l K J X ’ S ) d $ i 2 M ^ x f

provided 0 < £ < х < 1 о г 0 < х < » / < 1 , respectively. Hence, the factors 1 /a2

and 1/b2 on the right-hand sides of the inequalities in Lemma 1 and Theorem 1

14 G. A n i o l and P. P y c h -T a b e r s k a

can be replaced by 5/2a and 5/2b, respectively. Consequently, the suitable results for functions of class ВУФ can be formulated for sufficiently large m. In particular, we have

T heorem 2. Suppose that f еВУф with Ф satisfying the additional condition Ф(2и) ^ хФ(и)/ог и ^ 0 (x = const). Then, for every x e(0, 1) and for all integers m ^ 4 (x (l-x ))_1,

|Mm[ /] ( x ) - è { /( x + 0 )+ /(x -0 )} |

c m 1

+ --- ^{S i} --- £ _ L

х (

1

- х ) У т

7

=

1

ч / /

<

Ф

"1

(m x(l-x)) 1/21/(x + 0) —/ (x — 0)|

R em ark 4. In case Ф(и) = и (и ^ 0), the result of [3] follows im­

mediately.

4. Two-dimensional analogues. Let Q be the square I x l . Given any complex-valued function / of two variables, bounded and measurable on Q, let us consider the polynomials

A*m,« [/](* , jO = t)Km(x, s)Kn(y, t)dsdt ((x, y)eQ, m, neJV),

Q

with K m defined by (1). Suppose that, at a fixed point (x, y)elnt Q, all the limits f ( x , y±0), f ( x ± 0 , y), f ( x ± 0, y + 0) exist and are finite. Write

Sf (x, y) = i { /( x + 0, y + 0 )+ /(x + 0, y - 0 ) + f ( x - 0 , y + 0)+f(x — 0, y-0)}.

Introduce the function q> = (px<y defined for (s, t)eQ, with (p(s, t) equal to f(s, t)— /(x + 0, y + 0) [/(s, t)—f(x + 0, y — 0)] when s > x, t > у [s > x, t < y], f{s, t ) - f ( x - 0, y -0 ) [/(s, t ) - f ( x - 0, y + 0)] when s < x, t < y [s < x, t > y], /(x, t) f (x, y -0 ) [/(x, t) f (x, y + 0)] when s = x , t < y [s = x, t > y], f{s, y ) - /( x - 0 , y) [/(s, y)—/(x + 0, y)] when s < x , t = y [s > x, t = y],

0 when s — x, t = y.

Then

Mm,„[/](x , y ) - S f (x, y) = Mm,„ M (x , y)

+ 4 {/(* + 0, у + 0) - / ( x - 0, у + 0) - / ( x + 0, у - 0) + /(x - 0, у - 0)} Лт (х) Л„(у) + i{ /( * - ° > У- 0)—/( x + 0, y + 0 ) } ( jf - J J ) X m(x, s)K„(y, t)dsdt

0 0 X J7

+ i{ /(x + 0, y - 0 ) - / ( x - 0 , y + 0 ) } ( J J - jj) X m(x, s)K„(y, t)dsdt.

x 0 0 у

Consequently, in view of (2) and (5), for all positive integers m, n,

IMm>„ [/](x , y ) - S f (x, у)I < |Mm,„ M (x , y)| + £w,„ ( /; x, y),

where

(10) x, y) = (|/(x + 0, y + 0 ) - /( x - 0 , y + 0)—/(x + 0, y — 0)+f(x — 0, y-0)|

+ \f(x ~ 0, y -0 ) -/(x + 0, y + 0)| + |/(x + 0, y —0)—/(x —0, y+ 0)|}

( 4 9 7 7 1

(2 (mnx(l—x)y(l—y))1/2 + 2(mx(l—x))1/2 + 2(ny(l—y))1/2J Moreover, putting

A(q>; x', x ÿ , y") = (p(xf, ÿ )-(p (x', y")-<p(x", y') + <p(x", y"), we can write

x 1 У 1

(11) Мм,„ M (

x

, y) = (J + f)<p(s, y)Km(x, s)ds + (J + J)<p(x, t)K„(y, t)dt

O x 0 y

x y x 1 1 y 1 1

+ (f f + f f + f f + f f)d (<PÎ S‘>y> t)Km(x, s)Kn(y, t)dsdt.

0 0 О y x 0 x y

In the next estimates the modulus of variation of the function (p on some rectangles R = < a ,/l> x < j,(5 > c Q will occur. For any pair of positive integers r, q , this modulus is defined by

r — 1 Q— 1

i +Д ф ; R) = sup X £ \A((p; x 2j, x 2j+1; y 2 k, y2k +i)l.

П г,е J = 0 k = 0

where the supremum is taken over all systems n rQ of rg rectangles ( x 2j, x 2j+l} x ( y 2k, y2k+1} such that a ^ x0 < Xj < x2 < ... < x2r_2

< * 2 r - i ^ j 3 , У < y 0 < У1 < У г < • • • < У2 е - 2 < y 2 e - i < ô a n d \ x 2 j + i - x 2j\

< (Д — a)/r (j = 0, 1 ,...,г - 1 ) , |у2к + 1-У2к1 < (à-y)/Q (к = 0, 1, £ -1 ).

Recalling the inequality (8) and arguing as in the proof of Lemma 2 in [6], we obtain

L emma 2. Let a, c be two positive numbers satisfying x ^ a < l , y ^ c ^ l . Set Sj = x - j a / J m for j = 1 , 2 , . . . , p; p = [x ^ m /a ], tk = y - k c / ^ f n for к = 1, 2 , . . . , v; v = [y-Jn/c]. Then, if mx2 ^ 4a2, ny2 ^ 4c2, we have

X у

If f A(q>; x, s; y, t)Km(x, s)Kn(y, t)dsdt\

о 0

1 (20 £ vjtV{(p;

S j ,

x; 0, y) 20 * iy fc(<p; 0, x; tk, y)

< a2с2 ) v2 I

j=

i

+ S X _t*

_{k =}

₁ k3

+ 112 Z I

j= 1

^{k =}

1

£ ^ VjA<P’ Sj, x; tfc, y) ty,v((p; 0, x; 0, y)

j 3k3 ju2v2

By symmetry, we deduce the corresponding estimates for the remaining

three double integrals on the right-hand side of (11). The four single integrals of

16 G. A n i o l and P. P y c h -T a b e r s k a

this identity can be estimated, via Lemma 1, in terms of the modulus of variation of the functions <p(1) or <p(2) of one variable, defined for

u gI

by (p(1)(u) = (p(u, y), (p{2)(u) — (p(x, u).

Thus, we have established the two-dimensional analogue of Theorem 1.

But it will not be formulated explicitly. This result together with the obvious inequality vjtk((p; R) < jkvltl((p; R) {R <= Q) yields

T heorem 3. Under the restrictions on f and (x, y) given at the beginning of this section, for all integers m ^ 4, n ^ 4,

c pb»*]

IM m>„[ / ] ( * ,

y ) ~ S f ( x , y

)I ^ ^

^{-f -}

Z Uj((p(1); /*)

(х (1 -х ))У ?и

j =

1

[Vn]

+ - ---=

71

-

7

= Z

+ -

( x { l - x ) y { l - y ) ) 2yfmn j =i k=i W l - y J V " k=l

[Vm] [Vn]

Z Z «1,1 O ^p ; Rj!k)+Qm,n(fi y), where I* = <x - x/;, x + (1 - x)//>, J* = <y- y/fc, у + (1 - y)/fc>, Я*,* = I* x Я (/ = 1 , 2 , . . . , [^ m ]; к = 1, 2, ..., [^ й ]) and x, y) is de/ined by (10).

Let Ф be a function as in Section 3, satisfying also the condition Ф(2и) < усФ(и) for и ^ 0 (x = const). Denote by V0((p; R) the total Ф-variation of the function (p on a rectangle R = <a, /?> x (y, Ô) c: <2, defined as the upper bound of the numbers

i

— 1

i - l

Z Z &(M<Pl Xj , Xj + 1 ; y k, y k + 1)\)

j

= 0 k = 0

corresponding to all partitions a ^ x0 < x 1 < ... < x { ^ fi, у ^ y0 < yj < ...

< y t ^ S , i , l e N . Introduce the class Нф of those functions which are of bounded Ф-variation on the square Q and which, for each value of one variable, are of bounded Ф-variation with respect to the other variable on the interval /.

The following analogue of Theorem 2 can be deduced easily (see also Theorem 2 in [6]).

T heorem 4. Suppose that f e H 0 and that (x, y)elnt<2. Then, for all integers m ^ 4(x(l — x))-1 and n ^ 4(y(l — y))-1 , we have

|MM,n[ /] ( x , y ) - S f (x, y)| ^ ¹

+

x(l —x)y/m

^{j =}

1 J~j

+

m n j

—

CjL—r i Г уф(<р,2>; Л )

y ( i - y ) % A

k = i л А W "

Z Z ^{ф -1}

x(l — x)y(l — у)Лу/Йш j=l к =1 У /ife « Уф(<р; Qtl)) + Qm,n{f ; x» у),

where J°j = ( x - x / y / j , x + ( 1-х)Д //> , Я = <У~У/^Д, У + О ~У )/^к), Qff = Я Х Я (j = 1> 2 , т ;к = 1, 2 ,..., п).

R em ark 5. The function <р is continuous at (x, y);<p(1) and cp{2) are continuous at x and y, respectively. Consequently, the right-hand sides of the inequalities in Theorems 3 and 4 converge to zero as m, n->oo.

Finally, suppose that / is continuous on the square Q and denote by o (f; À, 0), (o(f; 0, z) its partial moduli of continuity (defined as in [7], p. 124).

Clearly, for arbitrary positive integers j, к and for every rectangle R = (a, p) x <j, <5> c= g ,

vj,k(<p>R

^{) =}

vj,k(f ; R )

^<

i k

(3 — 0L

j

^,

⁰

^{/ ;}

⁰

^,

Consequently, the inequality given in Lemma 2 becomes

* у

и - r

|J $A(cp; x, s; y, t)Km(x, s)Kn(y, t)dsdt\ < c8 {co(f; l/yjm, 0) + co(f; 0, 1Д /n)}

о о

for arbitrary m, и g IV and all (x, y)eQ (cf. Corollary 1). Analogously, all remaining terms of the right side of (11) can be estimated. Thus, we get

T heorem 5. I f f is continuous on Q, then

max |Мм>и[/] (х , y ) - f ( x , y)| ^ c9{co{f; 1 /У т, 0) + co(/; 0, 1 Д/й)}

(x,y)eQ

for all positive integers m, n.

References

[1] Z. A. C a n tu r ija , On the continuity o f functions of classes V[v(nf\, in: Constructive Function Theory ’77, Sofia 1980, 179-189 (in Russian).

[2] M. M. D e r r ie n n ic , Sur l’approximation de fonctions intégrables sur <0, 1) par des polynômes de Bernstein modifiés, J. Approx. Theory 31 (1981), 325-343.

[3] S. G u o , On the rate o f convergence of the Durrmeyer operator for functions of bounded variation, ibid. 51 (1987), 183-192.

Г 41 F. H e r z o g and J. D. H ill, The Bernstein polynomials of discontinuous functions, Amer. J. Math.

68 (1946), 109-124.

[5] P. P y c h -T a b e r s k a , On the rate of pointwise convergence o f Bernstein and Kantorovic polynomials, Funct. Approx. Comment. Math. 16 (1988), 63-76.

[6] —, On Bernstein polynomials of two variables, in: Function Spaces, Proc. Intern. Conf. Poznan 1986, Teubner-Texte Math. 103, Leipzig 1988, 123-131.

[7] A. F. T im a n, Theory o f Approximation o f Functions o f a Real Variable, Moskva 1960 (in Russian).

INSTYTUT MATEMATYKI, UNIWERSYTET IM. A. MICKIEWICZA INSTITUTE OF MATHEMATICS

A. MICKIEWICZ UNIVERSITY

MATEJKI 48/49, 60-769 POZNAN, POLAND

2 — Commentationes Math. 30.1