1. Introduction. On some approximation problems

ANN ALES SOCIETATIS MATHEMATICAE POLONAE Series I : COMMENTATIONES MATHEMATICAE XII|(1968)

Zo f i a Cy b e r t o w i c z (P o z n a ń ) -

On some approximation problems

1. Introduction. First, I shall introduce some notation. Given a par­

tition x0 = 0 < х г < ... < Xi_x < Xi < ... < xn. = 1 of the interval <0, 1>

we consider a continuous function ipn, linear in all subintervals <#*_!, xf), i.e. a polygonal line defined by the formula ipn{x) = ( &— #i_i)/

l(xi — Xi_1)-{-ii_1 for Жг-i < x < Xi, i = l , 2 , . . . , n . Furthermore let us set ipn(0) — y>„(1). Then ipn can be defined not only in the interval < 0 ,1>

but for all values of x by the condition of periodicity y>n(x-\-k) = tpn(x), Tc = ± 1 , ± 2 , . . . In this paper we shall deal with approximation by means of periodic polygonal lines. The points Xi will be called abscissae of the polygonal line. We shall call points (a?0, £0), (xn, £n), and each point (Xi, £i), which has the property that in its neighbourhood the polygonal line is not linear, vertices of the polygonal line. It is obvious that one and the same polygonal line may be obtained by means of various systems of abscissae; however, we must always take into consideration the ab­

scissae of all the vertices. We shall call tpn an approximating polygonal line of a function/, if ipn(Xi) — £* = f{xf) for abscissae Xi of all the vertices of ¹fn.

By co(3) we shall denote a non-decreasing function defined for <5^0 and such that co(0) = 0 , со (3) > 0 if 3 > 0, and limсо(3) = 0 as <3 -> 0+ .

G will mean the Banach space of continuous functions / of period 1 with usual definition of addition and multiplication by a real number;

moreover, ||/j|c = max|/(a?)|, where Л — < 0 ,1>.

л

By L*v we shall denote an Orlicz space of periodic functions /, i.e.

the Banach space of measurable functions / of period 1, for which there i

exists a constant X > 0 such that l^Xf) = jcp[X\f{x)\)dx < ^{o o ,} where cp о

is a continuous function defined in <0, ^{o o )} and satisfying the following conditions: (a) cp(0) = 0, (b) cp{u) > 0 for и > 0, (c) <p(u) -> ^{o o} if и -> ^{o o ,} (d) <p{u) is a convex function in <0, ^{o o ).} Let me remind that a non-de­

creasing function cp{u) continuous for u ^ 0 , vanishing at 0 only, and such that lim cp (u) = o o if и -> o o has been called a cp-f unction. Moreover,

if cpsatisfied the conditions: (0Х) <p(u)/u-> 0 if и -> 0+ , (ooj cp{u)/u-> oo if и ^—> с о , it is possible to define a function cp*complementary to cp: cp*(v)

— sup [uv — cp{u)). It is well-known that we can introduce in L*ę two о

mutually equivalent norms: 1) a homogeneous norm || ||((p) (Luxemburg norm) defined by the formula

||/||w = inf{e > 0: I <p(\f\/e) < 1}, and 2) a homogeneous Orlicz norm

i

ll/ll* = snpj I f(x )g (x )d x |, о

where the supremum is taken over the set of all measurable functions g satisfying the condition Iy*(g) < ¹.

These norms satisfy the following inequality (cf. [4], p. 97) ll/łlW < № < 2 | l/ | | (ł).

Let feC , and let pi = sup\Ahf(x)\l<x>(ó), where Ahf(x ) = f(x-\-h) — - № , \M ^ ój and the su.premu.rn is taken over all x e ^0 , 1), By /7 we shall understand the space of functions belonging to G, such that pi < ^oo. In this space we shall introduce the norm ||/Цш = \\f\\c + У- Let f e L * (p and let ptv = sup \\Ahf{x)\\(q>)lcjo{d), where \h\ < <5, and the supremum is taken over all же<0, 1>. By Л" we shall denote the space of all functions f e L * ę for which piy < ^oo. The modulus of continuity of the function fe C and the integral moduli of continuity in L*ę will be suitably de­

fined as follows: = sup \Ahf(x)\, cOyfS) = sup IH^a/ H I L^j, o)y(d)

|A|<a

= sup \\Ahf{x)\\y, where 0 < <5 < 1. As is easily seen, coJd) (d)

|A|<(5

< 2o)y(d).

In this paper we shall consider some approximation problems by means of polygonal lines. We shall deal with some theorems on approxi­

mation of the type of D. Jackson (Theorems I and II), and S. Bernstein (Theorems I' and II'). The Theorems I and I' relate to functions belong­

ing to the space Я® (applying approximation in the sense of the metric in C) and the remaining theorems refer to the functions from (applying approximation in the sense of the metric in L*v).

I should like to express my great appreciation to Professor Orlicz for his kind help and assistance during the preparation of this paper.

Let me remark that some results obtained here are generalization of those obtained by Z. Ciesielski [2]. Most of the results of this paper were an­

nounced without proofs in [3].

2. Lemma 1. For a given convex ⁹⁹-function <p(u) let us denote y(u) U

= j <p(t)dt. Let a0 be the greatest positive value of the constant a, for which о

the inequality

(n) ucp{au) < y(u) for и ^ 0

is satisfied. Then -J < aQ < 1.

Proof. Let ns put x(t) = t, a = 0, b — и in the integral Jensen’s inequality

ь b

' a 9 a

We get

■w \ U \ U

— I tdt) < — I (p(t)dt, <p(iu) < — I (p(t)dt, ua>{\u) ^ y ( u ) .

и J o и J o и Jо

Hence we have a0 > On the other hand, if the inequality (n) is satisfied, then the inequalities ucp(au) < у (и) < u<p{u) imply a0 < 1.

R em ark. The constant a0 cannot be equal 1 for any 99-function.

U

Putting a — 1 in the last inequality we get ucp(u) — fcp{t)dt. Hence 9Ąu)

0

is derivable for every и > 0. Differentiating we see that 9o' (u) = 0 for и > 0. Hence <p(u) = c for и > 0; but 9o{u) — c is not a 99-function.

Ex a m ples. 1. Let <p(u) = up, 1 ^ p < 00. Then y(u) = uv+l}{p-\-l).

Hence a0 = (p -f 1)~1IP.

2. Let 9o(u) = eu—u —1, и > 0. Then y{u) — eu—\u2— и —1, and the inequality (n) obtains the form ueau—au2 < eu — \u2—1. Developing eau in a power series, we can write this inequality for и > 0 in the form

(n') or

2T a 3u

3!

an+2un {n-\- 2)!

X u un

-)-••• ^ — -j--- -f-... -j---

3! 4! (w + 3)! +

Suppose the above inequality is satisfied for a given a and for each и > 0.

Let и -» 0. Then a2/2! < 1/3!, a < l/l/з. In order to show that the in­

equality (n') holds for a ~ l/ł/з it suffices to verify that аи/п! < l /(n + l)!

for a ==l/ł/3, n — 2 , 3 , ... This follows immediately from the fact that an = (n-\-l)~1,n is an increasing sequence. Hence for the function 99(u)

= eu—u —1 there is a0 = I /1/3.

Lemma 2. Lei be a polygonal line defined as in the introduction and let yj'n (x) denote its derivative at all points, at which this derivative exists.

Moreover, let us define the function y/n(x) at the abscissae of all vertices, talcing the right-hand side derivative as the value of the function ipn(x) at the point x, i.e.

Vn(x) = (Vn(xi) — y>n(0i-i))l(Xi — Xi_-f) for Xi_x <ж < xt,

i = 1, 2 , . . . , n. We shall suppose that y>'n{x) is extended to the whole x-axis by the condition of periodicity of the function ipn{x). Then the following in­

equalities hold for n = 1 , 2 , . . .

(1 ) llAl(p) < ^{1 snp 2}

!o Oi l Wife)»

where a0 is the greatest constant satisfying the inequality (n) and ó; = Xi—x^ , i = 1 ,2 , . . . , n.

P roof. First let us assume that the function ярп{х) increases in the interval <xi_x, ж*>, and its graph cuts the ж-axis at a point x)-e(xi_1, xf).

Then \ipn{oci) — Vn(Xi-i)\ = №n{Xi)\ + |y»(a?<_i)|. We are looking for the smallest value of /?, for which the following inequality is satisfied:

(+) f <p(Wn{®)\)dx < J <p(p\y>n(x)\)dx,

xi- 1 xi- 1

/ = bi(p

xi—l

where и = |y>»(a?i_i)|, v — |v>»(®<)|.

Now, we calculate the integral at the right-hand side of the above in-

U

equality. Let us write, as in the Lemma 1, y{u) = j cp(t)dt, о

f <p(P\Vn{x)\)dx = —

xi-1

dj

p{u + v) (y(0) — y{pu)) + _{p {u + v )} [y(M — y( ⁰⁾⁾

p(u + v)(y(pu) + y(M)-

The inequality (+) may be written in the form {u-\-v)(p({uJrv)lbi)

< (y(jfftt)-f- y((3v))/p. With respect to the Lemma 1 and from the con­

vexity of the function у we obtain p{\a(5{u-{-v))lf}(u-\-v)

< 1у{Ри) + 1у(М у ł < a < l . Hence (u + v)<p($aP(u + v))^(y(Pu) + y(Pv))lp.

The inequality (+) would be satisfied if we take ajS/2 > 1 jbi. Therefore

1 2

we may choose /? = - sup—-. Similarly, we can prove with the same

a bi

constant /? the inequality (+), when щ (х) is a decreasing function in an interval <ж<_1? xf) and its graph cuts the ж-axis at a point xj €(xi_ 1, ж<).

Obviously in the case when ^(ж) is constant in <Жг_и ж*>, the inequality (+) holds, too.

Now, let ns suppose the line ipn{x) does not cut the ж-axis in <ж*_х, Xi>

and let xpn{Xi) > y>n(Xi_i) > 0. In this case we may take /3 = sup2/<V It follows from the Jensen’s inequality, when we apply this inequality to the function x(t) = t and put a = fiu, b = (iv. In the remaining cases, i.e. when 0 < Wni^i-i), and if both грп{Хг_f) and у)п(яц) are negative, it is possible to take ft = sup2 /^ , too. Thus we have proved

1 2

that ft = ■— sup — , where | < a0 < 1; here a0 denotes the constant

a0 di i

defined in Lemma 1. Therefore from (+) it follows that J cp[\'ipń(x)\lfte)dx

1 о

< / y>(\ipn(x)\le)dx for an arbitrary e > 0. Hence, in view of Theorem 9.5 in [4] (p. 97), we get \\грп/ft\\i<p) < IWIw At last we have \\rpń\\i<P) < P\\y>n\\w

E e m a rk 1. If cp{u) = up, 1 < p < oo, we have already found a0

= (р + 1 )_1/5Э. Hence/? = ( p + l)1/3,sup2/^. Forp = 1 we have ft = sup4/<V In [2] the inequality (1) was proved in the particular case (p{u)

— up,p > 1, but with a constant ft1 = sup 4 /<5*. Tending with p to in­

finity Z. Ciesielski gets (see [2]) the limit inequality in the sense of the metric in the space of bounded functions, but with the constant ftx. We see that in the limit case this constant may be taken equal sup2/<5i, and our inequality may be written in the form

(2) sup|t/40r)| < /3 sup \грп{х)\, where /? = sup2/<54,

and the supremum at the left-hand side is taken over the interval < 0 ,1>

minus the set of all points ж* of the partition corresponding to vertices of the polygonal line y>n(x).

E em ark 2. From the inequality ||/||((p) < Ц/Ц,, < 2 ||/||(ę)) it follows that HI A < WVnWw <£1Ы1(*) < P\\v>n\\v, i-e- for the equivalent Orlicz

, 1

norm we have ||wn|L < 2/?||y>n|L, where, as before, ft = — sup2/<5*.

aQ

Lemma 3. For a function f e L*ę satisfying the Lipschitz condition there holds the inequality \\Ahf(x)lh\\v < ||/, (^)ll93-

Proof. In order to prove this lemma we shall use the following remark, which we get applying the Lebesgue’s theorem on bounded con­

vergence. Let a function geL *<p*. The derivative J'{h ) of the function

i i

J(h ) = J /(ж-f h)g(x)dx exists for every h and equals j f (x-\-h)g(x)dx.

о о

For an arbitrary e > 0 let us choose a function ge{x) such that I<p*(ds) < 1, and

0

Yet a0 -> 1 as p -* oo.

0<ж<1

Y Ahf{®) h

Roczniki PTM — P race M atem atyczne X II ⁵

i

Denoting by J e(h) the integral f /(ж + h)ge(x)dx, we have

о

Ahf{x ) — s < i ( J e(fr)—'J*(0))

ф ib j'e(dh), where 0 < # < 1.

Now, we are going to apply the above formulated remark, taking fth instead of h

1 0Л+1

j'e(M ) = f f (ocJr'&h)ge{x)dx — J f (x)ge{x —dh)dx.

0

But I ęt(ge{x — M)) = I ę*{ge). Therefore Цд.Ц* = \\да(я—Щ\\9* and on the basis of generalizated Holder’s inequality (see [4], p. 91) we have J ’s{dh)

< т ы * » - As I 9*(ga) < 1, there is ||gre||v. < 1 , and finally, \\Ahf(x )/h ||„ —

— e < ||/'(a?)||9, for an arbitrary positive e.

R em ark 1. From the inequality for the norms || \\9, || ||(ę)) it follows that \\Ahf(x)IJi\\(<p) <21|/'(aOII(v).

R em ark 2. Let ipn(x) be a polygonal line of the above defined form.

Then the inequality (cf. [2])

4h 1

1ИаVfcfaO lta, < --- s u p — Ь п {х )\ \ {(р)

а0 Oi

holds, where a0 denotes the constant from Lemma 1. This inequality follows from the previous remark and from the inequality (1), imme­

diately.

Th e o r e m I. Let x0 = 0 < x x < ... < Xi_x < Xi < ... < xn = 1 Ъе а partition of the interval < 0 ,1 ), let fe C and let ipn denote an approxi­

mating polygonal line for n = 1 , 2 , . . . Suppose, there exists a constant e ^ 1 such that max <5i ^ c f n for n — 1 , 2 , . . . , where di = Xi—Xi_x. Under these hypotheses the inequalities \ f{x)~ у>п(х)\ < coc(eln) are satisfied for n = 1 , 2 , . . . , where coc is the modulus of continuity of the function f.

Proof. Suppose that y>n(x) is a non-decreasing function in <^_х, Let f(x ) ^ yn{x) for a given #€<#*_!, xf). Then \f(x) — yn{x)\ = f ( x ) —

— fn{%) < /(^ ) — Vn(Xi-l) = /( ® ) — f(®i-l) < ОсС^г)- Now, let us suppose there is f(x ) < yn(x) for a given ж е ^ _ „ xf). Then |/(a?) — yn{x)\ = ipn{%) —

—f(x ) < — f ix ) = —f ( x ) < We proceed similarly in the case when the function yn(x) is decreasing in the interval

Thus in each case we have \f(x) — f n{x)\ < coc(di) for i = 1 , 2 , . .. , n.

We have supposed that max <5* < c/n. Hence \f(x) — tpn(x)\ < coc(c[n).

1

R em ark. Evidently, in the above theorem the assumption that the function / is a periodic function is superfluous.

Theorem I'. Let a junction f eC mid a sequence of polygonal lines {ipn}

be given. Suppose that the abscissae off of the polygonal line ipn(x) satisfy the inequalities min 6i ^ c/n, where <5* = 0 < c < 1 , and all the abscissae x™ are contained in set of all abscissae х$+1 of the polygonal line ipn+i(x). Let mico(d)/d = r > 0, where the infimum is taken over < 0 ,1>

and let \f(x) — грп(х)\ < со (c/n) for n — 1 , 2 , . . . , where c is a positive con­

stant. Moreover, we suppose that for a certain increasing sequence of natural numbers k0 = 1, kx, k2, ... the following conditions are satisfied for m = 1 , 2 , . . .

m ^

(**) ~r / kncoi— J ^ koo I— |,

кvi-1 \ kn_ i J \ km J

where к and к are positive constants.

Then f e l l 10; more precisely: the following inequality is satisfied coc(d) < Mco(d) for 0 < d < 1, where M — l k f - l k l c Jr 2k1\\f\\clcr.

Proof. The proof will be performed in the same way as the classical proof of Bernstein’s theorem on approximation. We introduce functions иг(х) = y)kl(x), un(x) = Щп(х) — Укп- г(х) for n = 2 , 3 , . . . Then %(#) +

-\-u2(x) + . . . = f ( x ), where the series at the left-hand side is uniformly convergent in the interval < 0 ,1>. Let us take x e ( 0 , 1>, 0 < h < <5 < 1, and for a given S let us choose a constant m such that 1 jkm < д < 1 lkm_ 1.

The following inequality

m o o oo

(i) \Ahf(x)\ ^ ^ \ A hun{x)\ + £ Iun{x + h)\+ \un(x)\

n = l n = m+ 1 n= m+ 1

is true. For n = 1 we have \u1(x)\^\y)kl{x )-f(x )\ -lr \f(x)\^.co(clk1)-lr + ll/llc. For n = 2 , 3 , ... there is \un{x)\ < \wn(x)-f{x)\ + \f {x) - i p ^ i x )]

< 2col—-—J. The condition (*) implies

\kn_ x/

(И)

00 n=w+l

^ 2hoo(<5), m = 1 , 2 , ...

We can apply the inequality (2) which is of Bernstein-Zygmund type to the functions un(x). In that inequality <5* denotes the distance between two successive abscissae of the polygonal line un(x) which is also a poly­

gonal line because it is a difference of two polygonal lines. From the assumption about the abscissae of polygonal lines щ п(х) it follows that all the abscissae i = 0 , 1 , . . . , кп_ г, belong to the set of abscissae

cĄn, i — o, 1, . .. , lcn, and therefore the difference щ п(х) —■ tpk (х) is a polygonal line with abscissae х\п, i = 0 , 1 , kn. Hence the minimum of the distance between two successive abscissae of un(x) satisfies the inequality min<5i..> cjhn. Thus we have max(l/<5i) <

But there is satisfied the inequality \Ahun(x)\ < sup \uń(x)\h, 0<Ж<1

where h < <5, and the supremum is taken over the interval <0, 1> minus the set of abscissae of all vertices of un(x). Next, we have for n = 2 , 3 , ...

, 2кп д 4:Тсп ё

sup \un{x)\h < —-— sup \un(x)\ < —-— со

0г£Жг£Х С 0<ж<1 C f c ) ’

/ , 21cx 6 21cxd ( ( c \ \

(a?)I <sup \ux(x)\h < —r— sup \ux(x)\ < —— о — + ||/||c ,

c c \ \k11 I

Using the condition (**) we get

m m

A _ / e \ 2Tcx6

i K a \ i d \ + -

4<5 V"

(in) 2 j \AhUn(v)l < y 2 j

n = l n =

llfllc

4(5

C M O l

2Tcxd , /4&

- 4 - ll/lic < —c \ c ^c ‘ -ll/lic)r )

Here we have taken into account that co(<5)/<5 > r if 0 < <5 < 1. From (i),

( ^4fc 21c 1 \

4&+ — + — ||/||c)

c c r I

3. Now we are going to formulate analogous theorems on approxi­

mation in the space L*v and in the classes Щ . Lemma 4. Let f be a measurable function. Then

b b b—a b—h

J ( f <p(\f(®)—f (*)№*)dx = 2 / ( / <p[\Ahf(t)\)di>)dh,

a a о a

where 0 < ft < & — a. Since f is measurable, the integral at the left-hand side exists.

The proof runs the same lines as the proof of Lemma 2 in [6].

Lemma 5. Let us suppose that f(x) is a measurable periodic function x

of period 1, and f (p{\f(x)\)dx < oo. Let x0 = 0 < x x < ... < Xi_x < Xi < ...

о

... < xn = 1 be a given partition of the interval <0, 1>, and let <5* = Xi — Xi_x, i — 1, 2 , . . . , n. We define a step function s(x) in < 0 ,1>, talcing 1

s(x) = mi = —1

Oi

/

and extend it to a function of period 1. Then

f

sup ( (p[\Ahf(x)\)dx,

where d0 = max di, 6 — mindi for i — 1 , 2 , . . . , n.

Proof. We have

xi

v { \ f ( x ) - s ( x ) \ ) J \ f { x ) - f ( t ) \ d t y

xi- 1 We apply now the Jensen’s inequality

xi xi

J J

xi— 1 x-' ’

According to Lemma 4 we have xi xi

f ( / <p(\f(%)—f{t)\ )d t)dx^ 2 f ( f (p(\Ahf(t)\)dtjdh,

xi _ 1 x{ _ ! о x%_ x

where 0 < h < <5*. Hence we obtain for i — 1 , 2 , . . . , n 4- 1

Xj—h

J ( J < ^р (И ^л /(^)|)^) ^

:~1 г о жг—1

_ йо Ч

J ( J 9?(| dh.

< —

d -0 4 —1

Adding the above inequality over all intervals <#г_и^г> we obtain

j <p(\f{%) — s (o c )\ )d x ^ ^ j ( J (p(\Ahf{t)\)dt} dh sup j p(\Ahf(x)\)dx.

0 0 0 1лКйо 0

Le m m a 6 . Let f e L * 9 and let x0 — ^{0 <} x ± ^< ... < X i_ x < Xi < ... ^< xn

— 1 be a partition of the interval < 0 ,1> into parts of equal length, i.e. di = Xi —

— Xi_x — l\n for i = 1 , 2 , . . . , n. Let s(x) be the step function defined in Lemma 5. Then \\f(x) — s(x)\\^) ^2(o,p(lln).

Proof. Given the function F (x ) = /(ж)/2со<р(1/'?г) we construct a step function in the following way

s (x) 1 rч

S{x) = —— 7тгт— ~Г F(t)dt for 2a)<p(l/n) di J

i — 1, 2, . . . , n, and we extend it to a function of period 1. Applying Lemma 5 we have

and \\Ahf{x)\\{tp) for \h\ < 1 jn, are equivalent. This remark and the previous inequality imply \\f(x) — s(x)\\^ <2ft),,(l/w).

In the next lemma we shall take some polygonal lines, which will be denoted by y>n{x). It is easily seen that it is of no use to apply the approximating polygonal lines ipn(x) here.

Lemma 7. L e tfe L * v and let s be a step function defined as in Lemma 5, where di = Xi—Xi_x = 1/n. Let us define a polygonal line tpn in the follow­

ing way

Vn{x) = n((a)i—x)ci_ 1 + (x — Xi_f)Ci) for Xi_x < x < xi, i = 1 , 2 , . . . , n , by the convexity of p. Notice, that the conditions

for \h\ < 1/n,

where

Then

\{тх-\-тп) , for i — 0 and for i = n

%(mi + mi+1), for i = 1, 2, ..., n —1.

Proof.

dx

by the convexity of cp. Notice that

'Mn); i = 1,

(j’l_J —“

\{mi — i = 2 , 3 , . . . , w , wi+1), ś = 1 , 2 , ..., w—1, i ( w n—ą ) , i = n.

We have for i = 2 , 3 , . . . , n, Wli—Ci

ЖМ+ 1/rl

■1—Co = £w( /

j f(t)dt

f(t)dt—

f(t)dt)

*0 xn — 1 ж/¹—1

xn xn xn

= J f { r + l l n ) d r —

j

ln J

X M— l / n Xj

cn-l

x i —l 4 - 1

wii—Gi_i = \п I f f(t)d t— j f(t)dtj = \n f A1/nf(x )d x .

x i —l жг ~ 2 xi —2

Similarly,

x i

mi—C i = — %n f A1/nf(x )d x for % — 1 , 2 , . . . , n —1,

жг—1

m,

i

~

J

Now,

/ 2 1 m

*P I' -— — \=ę>(----—j— f Aljnf(x )d x \

( l / « ) j J '<nn I

<<P

, xn

( n

/

4-1

Hi/n/(a?)l

<o»(l/w) ^{< n}

f” / M W W L U J

Collin) j

xn—1

Г J ^ m \ dx (or £ = 2 , 3 , . , . ,

\ ^ ( im

I J

ojyii/n)!

x l —2

n

Analogously, we obtain

■) 4- 1 ‘ - 1 ’ 8 ...* - 1 ’

/2И» <^1L r“y/MWMLb*;

\ <*>*№) / a J 0>v(l/w) /

\s{x)—y>n{x)\

(Ov{lln)

2\m1—y>n{x)\\

Mę {Hn ) ] j dx

dx-\-

+

n- 1 4 ,

Z SĄ

г=2 x, Oię{lln) I

xn ,

I Ą

2\mn—y>n(x)\\

^ № ) / dx 1

2w

/ / 2 Im, — c0 V l 0)v{lln)

n n — 1

V I / 2 |wtj ! | \ V I I 2 \ m i — d\

^ "»>№) ) ^ \ М В Д ⁺

Hence ||2(s(a?) —yw(aj))||(v) < ct)v(l/n) or ||е(ж) — у»п(ж)||(?)) < ВДСВД.

Theorem II. Leż feL *4* and let yn{x) be the polygonal line defined in Lemma 7. Then ||/(ж) — ^(ж)||(^ < |cop(1/r).

P r o o f. IIf(x) — y>n (x) ||(ł) < II/(ж) — 8 (ж) ||(v) + ||в (ж) - ysn(ж) ||(v) < 2соф (1 /п) + +£ft>v(l/w) =|ft)v(l/w).

Theorem II'. .Leż a measurable function f(x) of period p = 1 be given and let {ipn(x)} be a sequence of polygonal lines, abscissae of which satisfy the conditions of Theorem I'. Let ||/(ж) — ysn(x)\\{cp) < со (c/n) for n = 1 , 2 , . . . , where c is a positive constant. Finally, let inf со (d)Jd = r > 0, where the infimum is taken over < 0 ,1 ) . Suppose that k0 = 1 , kx, k2, . . . is an increasing sequence of natural numbers satisfying the conditions (*) and (**) of Theorem I '.

Then fell™, or more precisely: co^d) <Aco(<5) for 0 < d < 1, where N — 4& + (8&/a0c) + {Fk1\\f\\(<p)la0cr). The constant a0 was defined in Lemma 1.

P roof. As in the proof of the Theorem I', we take the functions un(x). Let \h\ < <5. From the inequality (i) it follows that

Ш oo oo

ш { х ) \ \ {(р)

< 2;1И^(Ж)Н«+

n = 1 n —rriĄ-1 n = m +1

Now, the proof becomes analogous to that of Theorem I', replacing the absolute value by || Instead of the inequality (2) of Bernstein-Zyg- mund type we apply the inequality (1) from the Lemma 2. Besides, from the remark to Lemma 3 it follows that ^ 2\h\\\un{x)\\{<p). Now we proceed as in Theorem I'. Finally, we get

00

£ \\un{oo)\\((p) < 2ксо(д) for m = l , 2 , . . . , n=m-f 1

m _

г — ĄżJc \

Therefore

8k 4Jc1 .

+-^-||/||(ł) co(ó),

«0с a0cr '

E e m a rk 1. In Theorems I' and II' we have supposed that inf со (<5)/<5 = r > 0, where the infimum is taken over the interval < 0 ,1>.

This becomes clear in view of the following known remark: Suppose that

\Ahf(oc)I < co(ó) if \7i\ < <S; if liminfco(<5)/<5 = 0, then f(oc) = c.

<5-M)

Obviously, under the above hypothesis we have <oc (<5) < co((5). Hence liminf <oc (<5)/d = 0 because of liminf ^со (д)/д = 0. Let us choose a sequence

V->0

{<5n} such that со(дп)/дп -> 0 and дп < 6, 0 < ó < 1. Then [d/dn] — m —l f where m is an integer. Therefore we have m—1 < djdn < m. Hence a>c(d) < (oc(dnm) < mcoc(dn) < (1 + dJdn)coc(dn) = а>с(дп) + да)с{дп)1дп. Since сос(дп) 0 and moreover, (ос(дп)1дп -> 0 as dn -> 0, we have coc(<5) = 0 for an arbitrary <5 > 0, i.e. f(x ) — c (see [1], p. 159, where the proof is made under the hypothesis that limco(<5) <5-1 = 0 as <5 0).

E e m a rk 2. Let us suppose that a function co(u) satisfies the (m)- condition (cf. [5]), i.e.

(+) (jo{u v) < cQoo(u)co(v), (++) lim co(u)/u — 0.

ге— >-oo

Let us take a sequence kn = an, n = 0 , 1 , 2 , . . . , where the constant a > 1 satisfies the following conditions:

(a) to (a)

a (b) < 1

co

Under these hypotheses the function co(u) satisfies the conditions (*) and (**) from Theorem I'.

Proof. We get from (+) a){cjkn_ x) < c0co(c) со(Цап~1).

1. Let n > w = 1 , 2 , . . . We apply w —m—1 times the inequality (+):

We applied the condition (b) with к = c0co(c)/(l — c0co(l/a)).

2. Now, let w < m +1, m = 1, 2 , . . . If we apply the condition (+) ш — nĄ-l times, we have

by condition (a), where к = a2c\a){a)(o{c)j[a—c0co(a)), since a/c0a)(a) > 1.

For instance if we put co(u) = ua, 0 < a < 1, e = d1/a (d > 0), = an, w = 0 , 1 , 2 , . . . , in Theorem II', we get: If ||/(а?) — щ{х)\\(<р) < d/na, then

<л>Д<5) <Nco(ó), 0 < <5 < 1, where W = 4aad/(h“—l) + 8 « a+1d/(l — aa_1)a0c + + 4a||/||(?,)/a0cr. In this particular case the conditions (*) and (**) are satisfied with constants к = (ac)a/(aa—l) and к — aa+lcal{l — aa~l).

[1] N. I. A e h ie z e r, Teoria aproksymacji, Warszawa 1957.

[2] Z. C ie sie lsk i, Properties of the orthogonal Franklin system (I), Studia Math.

23 (1963), pp. 14 1 -1 5 7 ; - (II), Studia Math. 27 (1966), pp. 289-323.

[3] Z. C y b e rto w ic z , On some approximation problems, Bull. Acad. Polon.

Sci. 7 (1967), pp. 497-501.

[4] M. А. К р а с н о се л ь ск и й и Я. Б. Р ути ц ки й , Выпуклые функции и про­

странства Орлича, Москва 1958.

[5] W. O rlicz , On a class of operations over the space of continuous vector valued functions, Studia Math. 14 (1954), pp. 285-297.

[6] П. Л. У л ья н ов, О рядах no системе Хаара, Матем. сборник 63 (1964), рр. 356-392.

R eferen ces

DEPARTM ENT OF MATHEMATICS I., A. MICKIEWICZ U N IV ERSITY, POZNAN