On the order of decrease of uniform moduli of smoothness for the classes of periodic functions $H_{p}^{l}[\omega],\ l\in \mathbb N,\ 1\le p \infty$

N. A. Il'yasov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 23 (2017) no. 4, pp. 162-175

Résumé

S. B. Stechkin posed the following problem: for given $1\le p$, $r\in\mathbb Z_{+}$, $l,k\in\mathbb N$, and $\omega \in\Omega_{l}(0,\pi]$, find the exact order of decrease of the $L_{q}(\mathbb T)$-modulus of smoothness of the $k$th order $\omega_{k}(f^{(r)};\delta)_{q}$ on the classes of $2\pi$-periodic functions $H_{p}^{l}[\omega]=\{f\in L_{p}(\mathbb T):$ $\omega_{l}(f;\delta)_{p}\le\omega(\delta),\,\delta\in(0,\pi]\}$, where $\mathbb T=(-\pi,\pi]$, $L_{\infty}(\mathbb T)\equiv C(\mathbb T)$, and $\Omega_{l}(0,\pi]$ is the class of functions $\omega=\omega(\delta)$ defined on $(0,\pi]$ and satisfying the conditions $0\omega(\delta)\downarrow 0\ (\delta\downarrow 0)$ and $\delta^{-l}\omega(\delta)\downarrow (\delta\uparrow)$. Earlier the author solved this problem in the case $1\le p$. In the present paper, we give a solution in the case $1\le p$; more exactly, we prove the following theorems. Theorem 1. Suppose that $1\le p\infty$, $f\in L_{p}(\mathbb T)$, $r\in\mathbb Z_{+}$, $l,k\in\mathbb N$, $l>\sigma=r+1/p$, $\rho=l-(k+\sigma)$, and $\sum_{n=1}^{\infty}n^{\sigma-1}\omega_{l}(f;\pi/n)_p\infty$. Then $f$ is equivalent to some function $\psi\in C^{r}(\mathbb T)$and the following estimate holds:$\omega_{k}(\psi^{(r)};\pi/n)_{\infty} \le C_{1}(l,k,r,p)\Big\{\sum_{\nu=n+1}^{\infty}\nu^{\sigma-1}\omega_{l}(f;\pi/\nu)_{p}+ \chi(\rho)n^{-k}\sum_{\nu=1}^{n}\nu^{k+\sigma-1}\omega_{l}(f;\pi/\nu)_{p}\Big\}$, $n\in\mathbb N$, where $\chi(t)=0$ for $t\le 0$, $\chi(t)=1$ for $t>0$, and $C^{r}(\mathbb T)$ is the class of functions $\psi \in C(\mathbb T)$ that have the usual $r$th-order derivative $\psi^{(r)}\in C(\mathbb T)$ $($we assume that $\psi^{(0)}=\psi$ and $C^{(0)}(\mathbb T)=C(\mathbb T))$. Note that this estimate covers all possible cases of relations between $l$ and $k+r$. Theorem 2. Suppose that $1\le p\infty$, $r\in\mathbb Z_{+}$, $l,k\in\mathbb N$, $l>\sigma=r+1/p$, $\rho=l-(k+\sigma)$, $\omega \in\Omega_{l}(0,\pi]$, and $\sum_{n=1}^{\infty}n^{\sigma-1}\omega(\pi/n)\infty$. Then $\sup\{\omega_{k}(\psi^{(r)};\pi/n)_{\infty}:$ $f\in H_{p}^{l}[\omega]\}\asymp\sum_{\nu=n+1}^{\infty}\nu^{\sigma-1}\omega(\pi/\nu)+\chi(\rho)n^{-k} \times \sum_{\nu=1}^{n}\nu^{k+\sigma-1}\omega(\pi/\nu)$, $n\in\mathbb N$, where $\psi$ denotes the corresponding function from $C^{r}(\mathbb T)$ equivalent to $f\in H_{p}^{l}[\omega]$. In Theorems $1$ and $2$, the case $l=k+\sigma=k+r+1/p$ $(\Rightarrow \chi(\rho)=0)$ is of the most interest. This case is possible only for $p=1$, since $r\in\mathbb Z_{+}$ and $l,k\in\mathbb N$. In this case, the proof of the estimate in Theorem $1$ employs the inequality $n^{-l}\|T_{n,1}^{(l)}(f;\cdot)\|_{\infty} \le C_{2}(l)n\omega_{l+1}(f;\pi/n)_{1}$, where $T_{n,1}(f;\cdot)$ is a best approximation polynomial for the function $f\in L_{1}(\mathbb T)$. The latter inequality is derived from the strengthened version of the inequality of different metrics for derivatives of arbitrary trigonometric polynomials $\|t_{n}^{(l)}(\cdot)\|_{\infty}\le 2^{-1}\pi\|t_{n}^{(l+1)}(\cdot)\|_{1}$, $n\in\mathbb N$.

Détail

Keywords: modulus of smoothness, best approximation, inequality between moduli of smoothness of different orders in different metrics, exact order of decrease for uniform moduli of smoothness on a class.

Bibliographie

[1] Ilyasov N.A., “K neravenstvu mezhdu modulyami gladkosti razlichnykh poryadkov v raznykh metrikakh”, Mat. zametki, 50:2 (1991), 153–155 | Zbl

[2] Timan A.F., Teoriya priblizheniya funktsii deistvitelnogo peremennogo, Fizmatgiz, M., 1960, 624 pp.

[3] Zigmund A., Trigonometricheskie ryady, v 2 t., v. 1, Mir, M., 1965, 616 pp. ; т. 2, 538 с. | MR

[4] Ilyasov N.A., “K neravenstvu raznykh metrik dlya proizvodnykh trigonometricheskikh polinomov v $L_{p}(\mathbb T)$”, Teoriya priblizhenii, tez. dokl. Mezhdunar. konf., posvyasch. 90-letiyu S.B. Stechkina, Izd-vo MIRAN, M., 2010, 36–37

[5] Tikhonov S., “Weak type inequalities for moduli of smoothness: the case of limit value parameters”, J. Fourier Anal. Appl., 16:4 (2010), 590–608 | DOI | MR | Zbl

[6] Ulyanov P.L., “Ob absolyutnoi i ravnomernoi skhodimosti ryadov Fure”, Mat. sb., 72(114):2 (1967), 193–225

[7] Ilyasov N.A., Teoremy vlozheniya dlya strukturnykh i konstruktivnykh kharakteristik funktsii: dis.. kand. fiz.-mat. nauk, Baku, 1987, 150 pp.

[8] Konyushkov A.A., “Nailuchshie priblizheniya trigonometricheskimi polinomami i koeffitsienty Fure”, Mat. sb., 44(86):1 (1958), 53–84 | MR

[9] Tyrygin I.Ya., “O neravenstvakh tipa Turana v nekotorykh integralnykh metrikakh”, Ukr. mat. zhurn., 40:2 (1988), 256–260 | MR

[10] Bari N.K., Trigonometricheskie ryady, Fizmatgiz, M., 1961, 936 pp. | MR

[11] Stechkin S.B., “Ob absolyutnoi skhodimosti ryadov Fure”, Izv. AN SSSR, Ser. mat., 17:2 (1953), 87–98 | Zbl

[12] Geit V.E., “Ob usloviyakh vlozheniya klassov $H_{k,R}^{\omega}$ i $\widetilde{H}_{k,R}^{\omega}$”, Mat. zametki, 13:2 (1973), 169–178 | MR

[13] Ilyasov N.A., “O poryadke ravnomernoi skhodimosti chastnykh kubicheskikh summ kratnykh trigonometricheskikh ryadov Fure na klassakh funktsii $H_{1,m}^{l}[\omega]$”, Tr. In-ta matematiki i mekhaniki UrO RAN, 21:4 (2015), 161–177 | MR

[14] Ilyasov N.A., “K pryamoi teoreme teorii priblizhenii periodicheskikh funktsii v raznykh metrikakh”, Tr. MIRAN, 219, 1997, 220–234 | Zbl

[15] Ilyasov N.A., “K obratnoi teoreme teorii priblizhenii periodicheskikh funktsii v raznykh metrikakh”, Mat. zametki, 52:2 (1992), 53–61 | Zbl

[16] Geit V.E., “O tochnosti nekotorykh neravenstv v teorii priblizhenii”, Mat. zametki, 10:5 (1971), 571–582 | MR

[17] Geit V.E., “O strukturnykh i konstruktivnykh svoistvakh funktsii i ee sopryazhennoi v $L$”, Izv. vuzov. Matematika, 1972, no. 7(122), 19–30 | MR | Zbl

[18] Ilyasov N.A., “K neravenstvam mezhdu nailuchshimi priblizheniyami i modulyami gladkosti raznykh poryadkov periodicheskikh funktsii v $L_{p},\ 1\le p\le \infty$”, Singulyarnye integralnye operatory, sb. statei, Izd-vo Bakinskogo gos. un-ta, Baku, 1991, 40–52