Jackson type inequalities for averaged moduli of smoothness

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXVII (1987)

He l e n a Mu sie l a k (Poznan)

1. In [3], there were given converse type inequalities for averaged moduli of smoothness тк and best one-sided approximation £ „ (/) of the function / by means of trigonometric polynomials in generalized Orlicz spaces E2n generated by a convex «^-function </> depending on a parameter, satisfying some condition ( B J .. Analogous results in case of Orlicz spaces generated by a concave, s-convex ^-function q> independent of the parameter were obtained in [4]. Here we shall give direct type inequalities in both cases, thus generalizing some results of [5], Chapter 8, and of [6]. These results together with those of [3] and [4] enable us to conclude some saturation theorems for Lipschitz classes defined by averaged moduli of smoothness in H2n.

Let us recall that given an s-convex (0 < s ^ 1) ф-function depending on a parameter r, q>(t, u), 27t-periodic with respect to t, the modular

where / is measurable and 2^-periodic, defines the generalized Orlicz space E?2n = {/: gv (Af) < oo for some A > 0} with an s-homogeneous F-norm

Jackson type inequalities for averaged moduli of smoothness

M / ) = .f <р(*> l/(0 lM c 0

Il/||j = i n f '« > 0 : e „ ( i r I/!/ K 1!) ■ If s = 1, we shall write Ц/Ц^ in place of ||/|Ц .

Moreover, for any 2n-periodic f one defines

cok {/, x; Ô) = sup {\Akhf (f)|: t, t + k h e Ik0(x) \ ,

where I^(x) = (x — kô/2, xA-kô/2) for Ô > 0, к = 1, 2, ..., and arbitrary real x. If / i s measurable and bounded, then cok(f, •; <5) is also measurable ([5], p.

28). For such functions / the averaged modulus of smoothness xk of order к

in Iî2n is defined by

т * (/;< 5 )= .|К (/, -; а д , <5>о

(see also [4]). The usual modulus of smoothness œk of order к in lf2lt is defined by

ojk(f; Ô) = sup \\\Akhf\%\ 0 < h ^ Ô], S > 0. '

Let Я„ be the set of all trigonometric polynomials of degree ^ n, and let for a measurable, bounded, 2^-periodic function f H + ( /) be the class of all PeH„ satisfying the inequality f (x) ^ P(x) in <0, 2л:), and Hn ( /) be the class of all Q e H n satisfying the inequality f (x) ^ <2(x) in <0, 2л>. Then we denote, as usually,

En( f) = inf III/- Щ : T e H n),

£„(/) = in f 111 P-QWV- Р е Я „+ ( / ) ,0б Я ; ( Л ) .

It is easily seen that for any / е Я 2л and arbitrary n there exists a T *eH„

such that En( f ) = \\f— T*||^. Also, for any bounded and arbitrary n, there exist P * e H + ( /) and Q *eH ~ ( /) such that £ „ ( / ) = \\P* — Q*||* (see [3], Proposition 4 and [4], 2.1).

2. In this section we shall investigate the case of convex <p. We shall assume that (p is an iV-function depending on a parameter (see [1]). Let us recall that the function q> satisfies the condition (В rj > 0, if there exist a set A c <0, 2л) of measure zero, a number c > 0 and a family of non-negative, measurable, 2л-

2 n

periodic functions / ( • , h), satisfying the inequality Sv = sup ( f (t, h)dt < oo IM <0 ô

such that for every и ^ 0, t e <0, 2 л )\А and \h\ ^ ц there holds the inequality (p(t — h, и) ^ <p(t, cu)+ f(t, h) (see [3]). It is easily seen that if q> satisfies (Bn), then it satisfies (ВJ for all ц ^ л with the same constant c > 0 and = Sn. If (p satisfies (BJ, then for / е Я 2л there holds the inequality ||/(• + /î)ll<p

^ 2c max (1, S J H /I^ for any h > 0 (see [2], Theorem 1).

Let us still remark that if К : (0, 2л) x (0, 2л) -> R is measurable in (0, 2л) x(0, 2л), K(-, t)eH 2n for almost every t e ( 0, 2л) and ||K(-, t ) ^ is integrable in (0, 2л), then the following Minkowski inequality holds:

(U II j0 (f)|K (-, 01*11» « 2 J 9 (r)||K(-, Oil»*

0 Ô

for every g ^ 0 integrable in (0, 2л).

Th e o r e m 1. Let tp be an N -function depending on a parameter, satisfying

(B„) with constants c and Sn. Let f be 2n-periodic, absolutely continuous and such that the derivative f'e L ? 2n. Then there holds the inequality

£ „ ( /) ^ - 7 г с т а х ( 1 , Sn)E „ (f).

( 2 ) n+ 1

In case of tp independent of the parameter, we have

(3) n+ 1

P ro o f. Let q e H n. x be such that E „ _ i(/') = \\ f '- q \ \ v . By [5], p. 243- 247, there exist Tl n, t 1>n, R l , R 2eH „ _l such that

2n

i f 2

> t liH(t) in <0, 2tu>, - = —

71 J n

0 and

Я 2 (*) < / ( x ) < /?! (x) in <0, 2^tu>, 2 n

Rt (x) - R 2(x) = ~ J (TUn( x - t ) - t Un( x - t ) ) \ f ( t ) - q ( t ) \ d t 0

2n

- I \Ti,n(t) — t l n(t)\ \ f ' ( x + t) — q(x + t)\ dt.

Hence, by inequality (1) applied to g(t) = \T1<n{ t ) - t Un{t)\ and K{x, t)

\f '( x + t) — q(x + t)\, we obtain 2n

W R i - R i W ^ l^i,n(0 —n,n(0l \\f'{' + t ) ~ q { ' + t ) \ \ < p dt.

But, by condition (Bn) we have

IIpO + OII* < 2c m ax(1, SJWpW^ for р е П 2к.

Hence, taking p(u) — f'(u ) — q(u), we get 2n

\\R1 - R 2\ \ * < 2 ^ I |Г1.|1( 0 - П .и(0 |2 ст а х (1 ,5 11) ||/ ' - 9 || , Л

= — cm ax (l, Sn) £ „ _ !(/'). 871 n

This implies inequality (2). If (p does not depend on the parameter, then

\\p(' + t)\\v = IIpOILpand we obtain (3).

Co r o l l a r y 1. Let tp be an N -function depending on a parameter,

satisfying (Bn) with constants c and Sn. Let a In-periodic function f posses absolutely continuous derivative f (k~ X) o f order к — 1, k ^ l , such that f (k)e L 2n.

10 — Prace Matematyczne 27.1

Then

where С = 8л;стах(1, Sn) for (p depending on the parameter, and C = 4n in case of (p independent of the parameter.

P ro o f. If P, Q e H n, Q(x) ^ f{ x ) ^ P(x) for all x, then \\f-Q \\v < \\P - 6IIV- Hence

En{f) = inf II/- ^{T\\v ^} inf W f - Q W ^ inf ||P -G II, = Ё Л Л -

T e H n Q e H n ( / ) Q ^ f ^ P

Thus, applying Theorem 1, we get

Ёп( Л ^ С п - ' Ё „ ( Л , and so by easy induction,

Ë „ ( f ) ^ C k n - kËn( f (k))•

But Ë „ (fik)) ^ \\ f {k)\\v , which gives the corollary.

Th e o r e m 2. Let cp be an N -function depending on a parameter, satisfying

condition (Bn) and such that 1 еТ?2л- Let f be a In-periodic, measurable and bounded function. Then

E„(f) < Cxx {f; 1/w) for « = 1 , 2 , . . . ,

where C = 66(4rc + l)c₂ m ax(l, S2). In case of (p independent of the parameter, one may take C — 228.

P ro o f. By [5], p. 249, formulae (8.11H8-13), there exist step functions S„ and Tn such that

/ „ w < / ( * ) < s n{x), 0 ^ Sn{ x ) - I n(x) < col if, x; 3n/n) in <0, 2tt) and

\S'n(x)\ < 2nn~ 1 (Oi {f, x; 4n/n), |/^(x)| < 2nn~ 1 w 1 (/, x; 4n/n)

for all хе(0,2л) which are not points of discontinuity of functions S„ and /„.

Hence

l|S „-/„||v ^ I K ( / , •; 3702)11^ = t i ( / ; 3n/n), USX < 2nn~1xl (/; 4n/n), \\T„\\ ^ 2 w T₁ xx (/; 4л/п).

Since satisfy the assumptions of Theorem 1, we have

87Г 8 л

En(Sn) < 77^-7 c max (1, S„)£„(S;) ^ ^7-7с т а х (1, 5Я)||5;||^

/2+ 1 /2 + 1

^ 16cm ax(l, iSJi! (/; 4n/n).

Similarly,

Ë„(I„) ^ 16cm ax( 1, 4n/n).

Now, we shall prove the inequality

(4) En ( /) ^ Ên(Sn) + ||S„ - I X + En(/„).

Let h be a bounded, 27r-periodic, measurable function. Let P(h)eH// (h), Q{h)eH~ (h) be such that

Ê„(h) = \\P(h)-Q(h)\\v . Then

P (f)(x) < P(S„)(x), Q (f)(x) > Q(I„)(x) and

(2(S„)(x) < f ( x ) ^ I n(x) ^ P(/„)(x) for all x.

Hence .

P(f)(x)-QU)(x) ^ P(Sn)(x)-Q(Inm

^ (P (Sn) (x) - Q (Sn) (x)) + (Sn (x) - I n (X)) + (P(In) (X) - Q (/„) (x)).

Thus,

£ * (/) ^ W P i S J - Q i s ^ + WSn-inh + W P i U - Q i i n X

= Ên(Sn) + \\Sn- l X + Èn(In)-

Applying (4) and the previous inequalities, we obtain E „ (f) < (32cm ax(l, S J + l ) i i (f; 4n/n).

But we have

T it/; 4n/n) ^ 2c m ax(1, Sn) (4л + 1)xt (/; 1/n) . (see [3], Theorem 1 (4)). Hence

E „ (f) ^ C M /; 1/n) with the desired constant C.

If q> is independent of the parameter, then

Ën(Sn) ^ 8t! (/; 4n/n), Ё„(1п) ^ 8тх (/; 4тг/п), and so

En( f ) < 16ц (/; 4п/п) + т1 (/; 3n/n) ^ 228^ (/; 1/n).

Let us still remark that Theorem 2 is, up to the constant, a

generalization of [5], Theorem 8.2. Now, applying the method from [5], p.

46-48, we shall prove the following

Le m m a 1. Let <p be an N-function depending on the parameter\nd let

f e И2п, where к is a positive integer, 0 < h ^ 2n/к. Let

fk,h(x) + . . .

hГ ( - h ) ~ k

0J

I —f ( x + h + ■■■ +*к) + ( ^ / •••

о

- + ( - i)kC i ) 4 t \ + . . -+ tk

dt j ... dtk

(cf. [5], p. 47, 2.8). Then

(i) If ( x ) - f kth(x)\ ^ ojk(f, x; 2h), (ü) \\f-fk,h\\<p ^ 2mk(/; h),

(iii) fk,h has an absolutely continuous derivative fk kh~ 1] and / $ еТ?2п with ll/a il, ^ 2(2k)k h-'cor(f; h) for r = 1 ,2 , ..., k.

P ro o f, (i) is shown in [5], p. 47. In order to prove (ii), we first remark that applying the Minkowski inequality (1) with interval (0, 2n) replaced by (0, 2n)k c= R k, we obtain

и/-/* j „ « p - ‘ Î- ■ ■ î и?,,+...+,t„i/wi

о 0

« 2 r ‘ J . . J | l 4 1 + ...+Illrt/ ( - ) M r , .... dtk s: 2mk(f\ h).

0 0

In order to show (iii) it is sufficient to apply formula [5], 2.9, p. 47, Minkowski inequality (1) with <0, 2n}k~r c= ^{R k ~ r} and estimation as in [5], p.

47 and 48, obtaining

h h

ll/S l ( - /) ) -k d h f ( X + t 1 + • • • + f i c - r )

+ 1 / \/c— 1 ^[k-Dhjkf ( * +к - 1

(t 1 + • • • + f i c - r ) +

•' + ( - 1^ ( k k_ l) kr Ah/kf(KX + tl + ' rj j ^ l ■■■dtk-

h h

<2 A -‘ j . . . j | p ; / ( x + t1+ . . . + f lt. r)||.

0 0

... +

^ 2h~r

^ f [ x + - ~ ' - ' k + t k~ r-

+ ...

àti ... Л*-,

« г ( / ; /l) + ( j ) ( ^ z j ) ® r f / ; - j ~ h ) + •

• + (fc*

^ 2 (2kf h~ 2 cor(f\ h).

Lemma 2. Let tp be an N -function depending on a parameter, satisfying condition (B K) and such that \ e E 2n. Let f be a 2n-periodic, measurable and bounded function and let gn(x) = cok(f, x; m/n), where m ^ 1. Then

E„{gn) ^ Czk(/; 2m/n) for n = 1 , 2 , . . . , where C is given in Theorem 2.

P ro o f. Let g(x) = cok( f x; h), h > 0. Then, estimating as in [5], p. 251, we obtain

Hence

a*! (g, a; <5) ^ cok(f, x; h + ô/k) for Ô > 0.

Ti (»*(/, •; h )\3) ^ zk(f; h + ô/k) ^ zk(f; h + 0).

Taking here h = 3 = m/n we obtain

*1 (0„; l/и) ^ Tj (бг„; m/n) ^ zk( f ; 2m/n).

Thus, applying Theorem 2, we get

En(gn) < C ii (#„; 1/w) ^ Czk{f; 2m/n).

Lemma 3. I f tp is a convex (p-function and f g, oteL?2n, |/(x ) —g(x)| ^ ot(x) for all x , then

£ „ ( /) ^ Ën(g) + 2Ën(x) + 2\\ot\\<p.

P ro o f. Keeping the notation from the proof of Theorem 2, we obtain

£ „ ( / ) < £„(gf)4-2||P(oc)||(?), and the desired result follows form the inequality 0 < P (a )(x )-a (x ) ^ P (a)(x )-()(a)(x ) for all x.

Theorem 3. Let q> be an N -function depending on a parameter, satisfying condition (Bn) and such that \ е П 2п. Then for every positive integer к there exists a positive constant c(k) such that for every 2n-periodic, measurable and

Ë»(f) ^ c{k)zk{ f ; 4/n).

P ro o f. We take functions / M from Lemma 1 with h = i/n. Applying Lemma 3 and property (i) from Lemma 1, we obtain

£ „ (/) « Ën(fl n ^ ) + 2Ë„{o)k(f, -, 2 n - ‘)) + 2 |H ( / , ■; 2n~l)\\v . By Lemma 2 with m = 2,

È„(o)k{ f •; 2n ~ 1) ) ^ Cxk(f; 4/n), whence

(5) Ën( f) ^ Èn(fktH- i ) + 2C4 {f; 4/n) + 2xk(f; 2/n).

bounded function f there holds the inequality

Applying Corollary 1 and Lemma 1, (iii), we obtain

£ „ (/M - i) ^ Ck n~k \\fkin-i\\q> < 2(2Ck)k cofc(/; \/n) ^ 2 (2Ck)k xk (/; 2/n).

Substituting this inequality into (5), we obtain the desired result with c(k)

= 2{(2Ck)k + C + \).

Theorem 3 together with [3], Theorem 3, give easily the following result, generalizing [5], Corollary 8.3, p. 257:

Theorem 4. Let <p be an N -function depending on a parameter, satisfying condition (Bn) and such that 1 e Tf2 and let f be a 2n-periodic, measurable and bounded function. Then

(a) if 0 < a < k, then the condition xk(f; <5) = 0(<5a) as 3 —> 0 is equivalent to the condition En( f) = 0(«~a) as n-* oo,

(b) if a = k, then xk(f; (5) = 0 (3 k) implies Ên( f) = 0 (n ~ k) as n —> oo, and E„{f) = 0 ( n ~ k) as n —>oо implies xk(f; <5) = 0 (3 k |log<5|) as <5-> 0,

(c) if a > k, then xk(f; Ô) = 0 (3 a) as Ô -* 0 implies Ê„(f) = 0 ( n ~a) as n

—>oo, and Ën( f ) = 0 (n ~ a) as n —> oo implies xk(f; 3) = 0{ôk) as 3 —> 0.

3. In this section we shall consider the case of s-convex (^-functions, 0

< s ^ 1, limiting ourselves to (p independent of the parameter. Let /„ be a Ire- periodic step function with discontinuities at Xj = nj/n, j = 0, ± 1 , ± 2 , ..., i.e.,

2 n - 1

Ц х ) = у0+ X ( y j - y j- 1

j= 1 0 X j ( x ) for x e <0, 2re>,

where 9y is 2re-periodic and

0y(x) = j j for 0 ^ x < y, for у ^ x < 2re;

obviously, ln(x) = y j for X E ( x j , x j+ l), ^{j =} 0, 1, ..., ^{2 n —} 1. The method

applied here follows the same lines as in [6] (case (p{u) = |u|s). We shall define for ÿ = (y0, yi, y2n- i) the pseudomodular

e l<pn]( y ) = Z < p ( \ y j - y j - i \ )

j= 1

and the respective s-homogeneous F-pseudonorm ilÿllS,"1 = inf {м > 0: $ }{7/ulls) < !}•

We shall also assume that cp satisfies the condition (A2) for all и > 0, and apply the function ф(и) = sup[(p(uv)/(p(v)f Let us remark that from s-

v > О

convexity of the ^-function tp follows strict monotonicity of cp and, supposing (A2), also s-convexity and strict monotonicity of ф. Let cp_x and ф ^ г be the inverse functions to (p and ф, respectively.

Lemma 4. Let cp be a concave, s-convex tp-function without parameter, 0

< s ^ 1, satisfying the condition (A2). Then

M n ) ^

CAs)

Ф~Лс2А) n)ll)C[n]

where C1 (s) and C2(s) are positive constants independent of n and y.

P ro o f. The idea of the proof is the same as in [6]. We write

K m(v) 1 /sin j n v A r

am \ sin it; ) where a„ sin^nw 42r

sin^w ^dw

for m —{n — \)r + \, n, r are positive integers; then K m is a non-negative trigonometric polynomial of degree < m. Let

u mu m i я

Я

f ( z ) K m{ x - z ) d z ,

ъ

— Я

x eR ;

then Um'[ f ] e H m_ l (see [6]). Let us write

/s in in (x —у + я /2 п )\2(г_1) /sin-|rc(x — у — я/2и)\2(г-1) y,m \u s in i( x — у + Tt/2n)J \n s in i( x — у — n/2n)J

^ п ^ п ( х + я/2п )\2(г_1) /sin^n(x —я/2п) \ 2(r_1)

\n s in i( x + n/2n)J \rcsini(x — n/2n) J U i [0,](x) = Uml9A(x) + ASy>m(x),

U~ [0 J(x ) = Um [9y] (x) — A S 2n_ ym (2k — x).

R. Taberski [6] proved that t/ + [0У] е Я + _ , (ву) and U~ [ 0 J e x (0y) for

sufficiently large A = A(r), depending on r. Hence

Ëm- i ( 9 y) < III/; W - U - [0 J||„ = M (S ,>mO + S2, - , . m(2*--))||,.

We choose

Then

3e(*) =

sin^n(x + a) \ 2(r X) n s in |(x + a) J

f , l/s in in x \ 2(r_1)l , „ f , |/s in i n x \ 2(r" 1)| ,

^ (ej- 2J н и э д r x+2J*{Un3ï) r x-

0 K/n

Hence, taking r > 1 + l/2s, we have for an arbitrary и > 0:

4 5 0 ^21* ( i ) dx+2 M 4 ) '

0 n/n

n \U 1 Is

with a constant Cv > 0. Hence Pall* <

1 where C' = 1/CS

Г - l ( C > )

Hence the best one-sided approximation of 0y in 1%n, Em- 1 (0у)ф ^ Н З Д ^ {j/ s^ l (Cs n)

with some fy eR , for m = (n— l ) r + 1, for sufficiently large integer r. Now, let us choose an arbitrary n and taking r fixed as above, let us choose an integer q for which qr ^ n < (q + l)r. Then

Еп(@у)ф ^ Eqr(0у)ф ^ SAS 8 A s

for 0 < y e 271,

^ - . ( c ^ + i ) ) Г - Л с »

where C" = C’Jr. There exist trigonometric polynomials Pyn e H„ (ву) and Qy,ne H ~ (9 y) such that

E H( 6 y \ , = \ \ Р у , п ~ й у , п \ \ ф -

Hence, since 0 < Ру>п — ву ^ Py,„ — Qy,„, we have

(6⁾ Similarly, (7)

\\Py,n- e y\ \ i ^ \ \ p y,n- Q yj \ i < SAS Г - Л с ; п )

\\0y-QyJ\i <

8 A s Г - Л с ; п )

Writing (as in [6])

V

and

V

we may express l„(x) in the form

2n—1 2n—1

/„(*) = У0+ Z Z 4Г M X) for XGK.

J= 1 J=1

We take also (see [6]) trigonometric polynomials

2n— 1 2 n - l

Р„М = Уо +

I

j =

1

1

2n-l 2/1-1

ô,,(x) = yo + Z V 6 x ,,n W - Z p Xj.n(x), QHe H „ ( 0 -

j =

1 1=1

Applying the fact that cp is subadditive and the definition of ф, we obtain for arbitrary rjn > 0 and и > 0:

2 n

P „ ~ l 2/1— 1

I J

(

0

2n

"h Z ^ Qxj,n (-^))^ dx

0

< Z ч > ( ^ У * Ы Р хрп - о х) )

2/i— 1

• Z

|^l/s

))•

By inequalities (6) and (7) we have

\\>ln(PXj, n - 0 Xjm < 1 and -\\rin(Ox r QXjJ \ i ^ 1 for

(⁸) Hence

rjn = Ф- Л С » 81;M

^ fan (*%•.« - 0X) ) ^ 1 and Q+ fa„ (0Xj - QXjJ ) < 1

Thus, for цп given by (8) we have

where y = (y0, yu ..., y 2 n - i ) - Similarly,

for all и > 0 and tjn given by (8). By s-convexity of <p, we have

for every и > 0. Hence

R e m a rk 1. If (p(u) = \u\p, 0 < p ^ 1, then cp_ x (и) = t (w) = u1/p and taking s = p, we obtain ij/L l (C2(s)n) = C2(p)n, and Lemma 4 yields the lemma from [6] given in B2n, 0 < p < 1.

Theorem 5. Let (p be a concave, s-convex (p-function without parameter, 0

< s ^ 1, satisfying the condition (A2), and let f be a In-periodic, bounded and measurable function. Then there holds the inequality

Cj(s) and C2(s) being the constants from Lemma 4.

P ro o f. Taking as in [6] the 2rc-periodic step functions

g„(x) = inf {/(и): Xj ^ и ^ xj+ l}, Gn{x) = sup [/(и): Xj ^ u ^ x J+l) for x e ( Xj , xj+1), j = 0 , ± 1 , ..., we have gn{x) < / ( x ) ^ G „ ( x ) . By inequality (4), we have

E„ ( /) < E„ (G„)+ ||G. - gX + E„ Ы • By Lemma 4,

where

9n = ( в п ( хо ) , U n i xi), g n ( x 2 n - i ) ) , Gn = (G„(x0), •••, G J x ^ ! ) ) . But

sup {\f(u)-f(v)\: u, v e (X j, xj + l)}

< sup \ \ f ( u ) - f ( v ) \ : u, v e < x - x l5 x + x ^ ]

= w l { f , x - , 2 x l) for x e < x j , x j+1>.

Hence, we have for u > 0:

xj+1

0* ( J ' I ) <P y^T/s (/• ^ 2*l> = <?* ^ ^ ---

Since u > 0 is arbitrary, this implies

\\Gn- g n\L ^ Urn, (/, •; 2 х 1 ) | | ц) = x x(/; 2 X i ) .

Hence

£ „ (/) ^ M / ; 2xt)+ Ci(s)

Vs- 1 ( G 2 (s) nт Ж 1]+ ш л

But xj + l —Xj — n/n, and so

I < p { ^ s u p \ \ f { w ) - f ( v ) \ : D ,w e<xj . 1, x j+ 1>}

* j + i

n 2n_1 / 1 \

= 2^ I \ 4> ( y /ï ® i (/. xh 2*i) J dx Xj~ 1

n „

^ 271 !

2n— 1 *j+l/* / j

( TTT^^l ( / ’ 4 x l) )dx xj~ 1

^ JI V •; 4*i) ) < ^/ ( 4 / ^ 1 (/> ’i 4xi) )’\ W whence

1 \ , 1 _rnl / 0n

l/s for all и > 0. Hence

1

,1/s

[n]

< ll<^i if, •; 4*i)ll„ = Ji (/; 4 x j,

and so

Similarly,

Il9.lt”1 Ht, (/; 4x,) $ 2ht, (/; 2x,).

Il6.lt"1 « 2 n t,(/; 2x,).

Thus,

Èn( f К / 4C 1(s)n \ . _ , V + * U ( C 2(s)n

R e m a r k 2. In case of (p(u) = |u|s, 0 < s ^ 1, wè obtain 4C i(s)n _ C,(s)

Г - Л с2(*)п) Cz(s)’

whence Theorem 5 gives Theorem 1 of [6]. Let us still remark that in the general case, boundedness of

1 4Ci(s)n

Ф - i ( C 2 ( s ) n )

is equivalent to boundedness of -\j/(K n 1/s) with some К > 0, and this in turn n

is equivalent to the following condition: there exist positive constant К and C such that the inequality (p(Knl/su) ^ Crup(u) holds for every и > 0.

Theorem 5 together with [4], Theorem 2.2 give easily the following result:

Th e o r e m 6. Let cp be a concave, strongly s-convex (p-function without

parameter, satisfying the condition (d 2), 0 < s ^ 1, and let f be a 2n-periodic, measurable and bounded function. Then

(a) i/0 < a < s, then En( f) = 0 ( n ~ a) as n —*• oo implies i j ( /; <5) = 0 (S a) as Ô —> 0, and (/; <5) = 0 (ô a) as Ô -* 0 implies

£ „ (/) = o :+

Ф -1( c 2 (s) n) as n-+ оо ,

(b) if a = s, then En( f) = 0 (n s) as n-> oo implies (/; S) = 0((5s|log<5|) as Ô -* 0 and t i (/; <5) = 0(<5S) as Ô -* 0 implies

En{f) = o ( n ~ s + - П( г (л v) as n-* oo, V Ф- i (C2(s)n)J

(c) if a > s, then En( f ) = О (п~л) as n -> oo implies (/; <5) = 0(<$s) as Ô - 0 .

References

[1] H. H u d z ik , A. K a m in s k a , Equivalence o f the Orlicz and Luxemburg norms in generalized Orlicz spaces LM(T), Functiones et Approx. 9 (1980), 29-37.

[2] H. M u s ie la k , On some inequalities in spaces of integrable functions, Proc. Intern. Confer, on Constructive Theory of Functions, Sofia 1984, 629-633.

[3] —, On the x-modulus o f smoothness in generalized Orlicz spaces, Comment. Math. 25 (1985), 285-293.

[4] — , Converse type inequalities for averaged moduli o f smoothness in Orlicz spaces generated by concave functions, ibidem 26 (1986) (to appear).

[5] Bl. S e n d o v , V. A. P o p o v , Averaged moduli o f smoothness (in Bulgarian), Sofia 1983.

[6] R. T a b e r s k i, One-sided trigonometric approximation in metrics of the Fréchet spaces LP (0

< p < 1), Math. Nachr. 123 (1985), 36-46.