Converse type inequalities for averaged moduli of smoothness in Orlicz spaces generated by concave functions

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXVI (1986)

H

M

(Poznan)

Converse type inequalities for averaged moduli

of smoothness in Orlicz spaces generated by concave functions In [1] there was obtained a converse type inequality in generalized Orlicz space If2n generated by a convex (^-function, estimating the averaged modulus of smoothness in lf2n of order к for a bounded, measurable, 2 k - periodic function / by means of best one-sided approximations of / with trigonometric polynomials. Here, we obtain an analogous inequality in case of a real Orlicz space lf2n generated by a concave strongly s-convex (p- function with 0 < s ^ 1.

1.1. A function (p: R+~>R, is called s-convex with an se(0, 1) if (p{ocu + l3v) ^ as <p(u) + (p(r) for all u, v ^ 0, a, /? ^ 0, such that cts + fis — 1.

A function (p: R + -» R+ will be called strongly s-convex with an se(0, 1) if the function ф(и) = (p(ul/s) is convex. It is easily seen that a strongly s- convex function is s-convex. Obviously, if q> is strongly s-convex with an se(0, 1) and 0 < p ^ s, then (p is also strongly p-convex. Let us remark that if a (^-function (see [3], 1.9) is convex, then it is superadditive, and if it is concave, then it is subadditive. If (p satisfies condition (d 2) for all и ^ 0, i.e.,

<p(2u)/<p(u) is bounded for и ^ 0, then we shall write ф{и) — sup [<p (uv)/(p (i;)] ;

v>

we have ф (м) < x for all и ^ 0. If q> is strongly s-convex (s-convex) and satisfies (Л2), then ф is strongly s-convex (s-convex), too.

An s-convex (^-function q> defines an s-convex modular M / ) = iv { \f(t)\)d t

0

in the space of measurable, 27r-periodic functions f generating the Orlicz space

Цт = {f- Q«,W) < 00 for some X > ° l ’ with an s-homogeneous F-norm

ll/IIJ = inf \u > 0: ^ ( u _1/s/ ) ^ l |

(see [3]).

1.2. For any 2я-periodic f one defines

£ ( - 1 r m( k) f { t + mh),

m= 0 W

cok(/, x; 5) = sup{|d{/(i)|: t, t + k h e lk0{x)}, where

/J(x) = ( x —jk ô , x + ^kô}, for ô > 0, к = 1 , 2 , . . . , and arbitrary real x.

In the case of measurable bounded f (ok{f, •; <$) is also measurable (see [5], p. 28). Thus, for such functions / we may introduce the averaged modulus of smoothness of order к in If2n:

S) = ||cok(f, •; Э Д

and a modular averaged modulus of smoothness of order к in If2n:

* * (/; <5) = e A ^ k if, •; <5))-

The modulus xk possesses the same properties as given in [1], Theorem 3, with obvious modifications sprung from the fact that in our case cp does not depend on the parameter.

13. P roposition . Let (p be a concave, strongly s-convex (p-function, 0 < s ^ 1, satisfying condition (A2) for all и ^ 0, and let

, , , <p(uv) ф(и) = sup——

t>> 0 <p(v) ф (u) = ф (u(2r 1)/2)

with an integer r ^ ( s + 2)/2s. Let 0 < kS^oc/n, where a > 0. Then the following inequalities hold for an arbitrary trigonometric polynomial T„ of

degree ^ n:

т1(тп-,& )< с.,геЛк»?т<к)) and

т,(Т„; Й) < max(C> r, 1)(W)**||7<%, where Ca r is a positive constant.

P ro o f. Following the notation of [2], Theorem 2, we shall write 2 m + 1

t = t(2r ln) 1 m

2rn

where m = 0, 1 ,..., 2r 1 n — 1. Then, applying Lemma 2 from [4], formula

(1.7), to the fcth derivative T^k) of the trigonometric polynomial Tn of degree

^ n, we obtain

\ n k)(t)\ ^

2rn~

m= 0 I \ n k)(tj\ sin j n ( f - r j n sin — rm)

2 r - 1

2rn — 1

m= 0 where

Dr,n(t) sin j nt \ 2r 1

n sin j t J (K Un(t))i2r~l)12 >

with K ln given in [2], Lemma 2.

Now, we apply the formula

t + h H'l+A wm - i + h

AkhT„(t) = A\~n J { J ...( J T}m)(wm)dwm)...dw 2}dw1, m ^ k

f W1 wm~ 1

(see [6], p. 150), in the case m = k, obtaining

t + h w i + h wk - i + h

Akh T„(t) = J { J ...( J 7l k)(wk)dwk)...d w 2}dwl .

t V V J Wfc-1

Hence we have for t, t + k h e lk6{x)

x + kô/2

\AkhT M < m k~i j \m k](u)\du

x - k ô / 2

2rn — 1 x + kô/2

<(kô)k~l X |7?>(tJ J \DrJ u - t m)\du.

m = 0 x — kô/2

Thus,

2rn — 1 x + kô/2

( o M , x ; 6 ) ^ ( k S r l X |îï‘’( t j f

m = 0 x - k ô / 2

Hence, by subadditivity of (p and the definition of ф, we have

2n 2rn— 1 x + kô/2

if ( Г, ; 5) < f v - 1 X 17?> (tJI f |Д,,„ (u - rJ|Ai} dx

0 m= 0 x — kô/2

2n 2 rn — 1 x + kô/2

X <»>!(^)‘ |7Α,( U I( ^ ) - 1 f I D , J u - l J \ d u ) d x

0 m= O x - k ô / 2

2rn - 1 2

x + kô/2

X «> ( \DrJv)\dv}dx,

m = О Ь x — kô/2

Prace Matematyczne 26.2

because Drn(u) is 2л-periodic. Denoting

A4 f ,n , Ч| /2л; kô 2n(j+ \) k ô \\

M j t k

= max< |Д.>и(м)|: u e ( - — y , ---+ y we obtain

2 tc

Ф 1

0 kô x —kô/2

2тт n ^

\ D r . n ( v ) \ d v } d x ^ —

X

./= «

But 0 ^

^ oc/n, and so

^ Ca/(/ + l)2r 1 for every

where Ca > 0 is independent of

(see [6], p. 151). By s-convexity of

we thus have

ф ( М и ) ^ ф ( С а

1 1

ü + i p - n ^ ' ^ ^ ÿ + u 2’

and so

2 r"~1 2 tü 00 1

n j = i2

m= 0

я2 ___ 271 2Г"~1

= ~ т Ф { С а ) ~

£ <jo((^)k|7jfc,(OI}- 6 n ,„e0

Now, denoting = x,- = 2 tc //JV for j — 0, 1, 2, ..., iV—1, we have tj — x f r,l)

= n/2r n. Hence, defining

Q%(Tn) = ~ sup J] «pflr^x + jc,)!)

* 7 = 0

(see [2], 2), we get

2 k 2r”_1

62 ; n((kS)k T<k>)>— x <?{(/d5)w>(o)i}.

^ П 7 = 0 Hence

тГ(Т„; 5) ^ и 2 Ф { С , ) Т е ^ ( ( к 0 ) к V k>)- Consequently, applying [2], Theorem 2, we get

т ? (Т „ ;0 )а и 2Ф(С.)2'2г$(1 + 2жп/2' n) e„ ((kS)k 7?>)

= C „ e„((fa5)‘ T'k>), where Cxr = 5 it2

(CJ 22r ф (1 + 2n/2').

Let us write Car = max(Ca ,, 1); then

Qq> ) < - J - t U T . ; S) « e„((fa5)‘ 72**).

Taking in this inequality TJulls in place of Tn, where и > 0, we obtain

2.1. Now, we shall give an estimation of the modulus xk by means of one-sided approximations.

The best one-sided approximation in lf2n of a bounded, real-valued measurable, 27i-periodic function / by means of trigonometric polynomials of degree ^ n is defined as

where H + (/) is the set of trigonometric polynomials P of degree < n for which P { x ) ^ f{ x ) in <0, 2тг> and H~ (/) is the set of trigonometric polynomials Q of degree ^ n for which f (x) ^ Q(x) in <0, 2тг) (see [5], p.

242). Arguing as in [1], Theorem 7, it is easily shown that for every n there exist trigonometric polynomials P * e H ^ (f) and Q * e H ~ (f) such that

£„(/) = IIP*'-<2*11;.

2.2. T heorem . Let tp be a concave, strongly s-convex (p-J'unction independent of the parameter, where 0 < s ^ 1, satisfying condition (A2) for all и ^ 0. Let f be a bounded, measurable, 2n-periodic function. Then there holds the inequality

for n — 1, 2, 3, ..., with a constant CkrS > 0 depending on к and s.

P roof. We choose P * e H f( f) and Q * e H ~ (f) so that £ „ (/) = ll^î —ÔÎIIJ, л = 1,2, ... Arguing as in the proof of [1], Theorems 3 and 1 (3), we obtain for <5 > 0

By Proposition 1.3 applied to к — 1 and T„ — P* — Q* with a = 4n, r being the least integer satisfying the inequality r ^ ( s + 2)/2s, we obtain

Hence

E n( f ) = inf{||P-Q||‘ : Q e H ~ (/)},

But, by [2], Theorem 3, there is a constant C > 0 such that ll(J>? - e ? ) #IIJ < Crf\\P*-Q*\\i = Crf Ën(f).

Hence

T i ( P * - Q * ; ô ) ^ C maix(Catrt l)ôsnsË„(f) for 0 < S ^ 4 n / n . Consequently,

4 ( / ; <5K тк (P* ; (5) + Tk (e* ; Ô) + Cl En(/) for 0 < <5 ^ 4n/n, where Cl = [2skC max(Car, 1)(4л)*+1] 2b .

Now, adopting the same notation as in the proof of [1], Theorem 3:

U0(x) = P f(x )-P $ (x ), Uv(x) = PJv(x )-P J v -iW , v = 1, 2, ...

we have

m

^ < Z т* ( ^ ; <5).

v = 0

Applying Proposition 1.3 with a' = 4кк to the polynomial Uv of degree ^ 2V and then [2], Theorem 3, we obtain

Tk(U v; ô) ^ max(C«,„ l)(k<5)kl t / < %

^ C2 (k<5)ks 2ksv || 17V||* for 0 < <5 sc 4 tt / 2v,

where C2 = Ckmax(Catr, 1). Hence, arguing as in the proof of [1], Theorem 3, we get

m

ч(Р*2„;б) * z c 2{ k s r £ 2tovi|(7v||j,

v = 0

m

s:2C2(fa5)“ (£0(/) + I 2‘" £ ,,_ ,(/))

v — 1

for 0 < <5 ^ 4n/2m. Taking into account an analogous inequality for T*(@2m;<5)> we thus obtain

xk(f;S)^4C2(kSr(Ë0(n+ I 2*"Ê m , (/)) + C, £ ,„ (/) v= 1

for 0 < 5 ^ 4n/2m.

Now, it is easily seen that

> V - 1

2favÊ2,_ 1( / ) 2 ' 2ta 2‘" Ё 2>- , ( / ) 2 - ‘* -‘

Hence, taking C, = max(22ftï, 2*s+1), we get 2b ’ £ 2,_ 1( / K C J

for ks ^ 1, for ks < 1 .

Л***-1 £,(/)■

This implies

2 m — 1

тк( / ; й ) ^ 4 С 2(И)ь !Ёо(Л + 2‘*Ё ,(Л + Сз X мь _ г £»(Л} + с , £ 2, ( Л Д=2

2

< 4 C 2C3(fa5f !£0(Л + 2“ £ ,( Л + X ( v + i f - ‘ £ ,+ i ( f } +

V — I

+ Ct Ë2„ (f) for 0 < S < 4 7 i/2 " . Since Ê,,+1 (Л < £,(Л> we thus obtain

2m" i - l t k( / ; Ô K 12C2C3( ^ ) A s X (v+1

for 0 < <5 ^ 4 к / 2 т.

Now, let n be an arbitrary positive integer and let 2 m~ 1 ^ n < 2m; then тк ( / ; 2тс/и) ^ тк( / ; 4 n / 2 m)

i 2m -1 - l

« 1 2 C J C,(4tik)‘>-iS; X (.+ 1|‘- £ >( Л + С , £ г, Ш

2 V = 0

< 12C3 C3(4itfcf Jg"X (v + I f 1 ЕЛ Л + С, £„(/)•

r r

v= 0

It is easily calculated that

£ . ( Л < с * Т х (V+ i f - ‘ £ vm .

" v = 0

where

Thus,

C4

m ax(l, 2fcs) if Acs < 1,

1 if ks = 1,

ks if ks > 1.

4 \ f ; 2n

« 12C2 C3 (4 n * f 4

(V + i f - 1 ЁЛЛ ^{+ C, c 4 4}

^{(V + i f - 1} ЁАЛ

n

v = 0

v = 0

= % " Z ( v + if- ‘ £,a) _П

у

— о

for w = 1, 2, 3 , . where C5 is a positive constant depending on Ic and on s.

Let us remark that Theorem 2.2 in the special case of (p(u) = |u|p, 0 < p

< 1, gives Theorem 4.1 of [6].

References

[1] H. M u s ie la k , On the т-modulus o f smoothness in generalized Orlicz spaces, Comment.

Math. 25 (1986).

[2] —, On some Nikolskii and Oswald-type inequalities, J. Approx. Theory (in print).

[3] J. M u s ie la k , Orlicz spaces and modular spaces, Lecture Notes in Math. 1034, Springer Verlag, 1983.

[4 ] P. O s w a ld , Some inequalities for trigonometric polynomials in L p-metric, 0 < p < 1, Izv.

Vys. Uc. Zav. No. 7 (1976), 65-75 (in Russian).

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

[6] R. T a b e r sk i, On modified integral moduli of smoothness and one-sided approximation o f

periodic functions, Funct. Approx. 10 (1980), 147-156.