N.A. Il’yasov. On the order of decrease of uniform moduli of smoothness for the classes of periodic functions $H_{p}^{l}[\omega],\ l\in \mathbb N,\ 1\le p<\infty$ ... P. 162-175

S.B. Stechkin posed the following problem: for given $1\le p<q\le\infty$, $r\in\mathbb Z_{+}$, $l,k\in\mathbb N$, 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 $1\le p<q<\infty$. In the present paper, we give a solution in the case $1\le p<q=\infty$; 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 bound 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 bound 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 bound in Theorem 1 employs the inequality $n^{-l}\|T_{n,1}^{(l)}(f;\cdot)\|_{\infty} \le C_{2}(l)n\omega_{l+1}(f;\pi/n)_{l}$, 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 $\|t_{n}^{(l)}(\cdot)\|_{\infty}\le 2^{-1}\pi\|t_{n}^{(l+1)}(\cdot)\|_{1}$, $n\in\mathbb N$.

Keywords: modulus of smoothness, best approximation, inequalities between moduli of smoothness of various orders in different metrics, sharp order of decreases of uniform moduli of smoothness on a class.

The paper was received by Editorial Office on August 10, 2017.

N.A. Il’yasov, Cand. Sci. (Phys.-Math.), Baku State University, Baku, Azerbaijan,
e-mail: niyazi.ilyasov@gmail.com.

REFERENCES

1.   Il'yasov N.A. On the inequality between moduli of smoothness of various orders in different metrics. Math. Notes, 1991, vol. 50, no. 2, pp. 877-879.

2.   Timan A.F. Theory of approximation of functions of real variables. Oxford, London, NY, Pergamon Press, 1963, 655 p. This translation has been made from A.F. Timan's book entitled Teoriya priblizheniya funktsii deystvitel'nogo peremennogo,  Moscow, Fizmatgiz Publ., 1960, 624 p.

3.   Zygmund A. Trigonometric series, vol. I, II.  Cambridge: Cambridge Univ. Press, 1959; vol. I, 383 p.; vol. II, 354 p. Translated under the title Trigonometricheskie ryady.  M.: Mir Publ., 1965, vol. I, 616 p;  vol. II, 538 p.

4.   Il'yasov N.A. On the different-metrics inequality for derivatives of trigonometric polynomials in $L_{p}(\mathbb T)$. Approximations theory: abstracts of Intern. conf., dedicated to the 90-th anniversary of S.B. Stechkin. Moskow, Steklov Inst. Math. Press, 2010, pp. 36-37 (In Russian).

5.   Tikhonov S. Weak type inequalities for moduli of smoothness: the case of limit value parameters.  J. Fourier Anal. Appl., 2010, vol. 16, no. 4, pp. 590-608.

6.   Ul'yanov P.L. Absolute and uniform convergence of Fourier series. Math. USSR-Sb., 1967, vol. 1, no. 2, pp. 169-197.

7.   Il'yasov N.A. Embedding theorems for structural and constructive characteristics of functions: Cand. Sci. (Phys.-Math.) Dissertation, Baku, 1987, 150 p. (in Russian).

8.   Konyushkov A.A. Best approximations by trigonometric polynomials and Fourier coefficients. Mat. Sb. (N.S.), 1958,  vol. 44(86), no. 1, pp. 53-84 (in Russian).

9.   Tyrygin I.Ya. Turan-type inequalities in certain integral metrics. Ukrainian Math. J., 1988, vol. 40, iss. 2, pp. 256-260 (in Russian).

10.   Bari N.K. A treatise on trigonometric series. Vols. I, II.  Oxford, New York: Pergamon Press, 1964, vol. I, 533 p; vol. II, 508 p. Original Russian text published in Trigonometricheskie ryady, Moscow, Fiz.-Mat. Giz. Publ., 1961, 936 p.

11.   Stechkin S.B. On absolute convergence of Fourier series. Izv. Akad. Nauk SSSR. Ser. Mat., 1953, vol. 17, no. 2, pp. 87-98 (in Russian).

12.   Gheit V.E. Imbedding conditions for the classes $H_{k,R}^{\omega}$ and $\widetilde{H}_{k,R}^{\omega}$. Math. Notes, 1973, vol. 13, no. 2, pp. 101-106.

13.   Il'yasov N.A. On the order of uniform convergence of partial cubic sums of multiple trigonometric Fourier series on the function  classes $H_{1,m}^{l}[\omega]$. Tr. Inst. Mat. Mekh. UrO RAN, 2015, vol. 21, no. 4, pp. 161-177 (in Russian).

14.   Il'yasov N.A. On the direct theorem of approximation theory of periodic functions in different metrics. Proc. Steklov Inst. Math., 1997, vol. 219, pp. 215-230.

15.   Il'yasov N.A. An inverse theorem of approximation theory of periodic functions in various metrics.  Math. Notes, 1992, vol. 52, no. 2, pp. 791-798.

16.   Gheit V.E. On the exactness of certain inequalities in approximation theory.  Math. Notes, 1971,  vol. 10, no. 5, pp. 768-776.

17.   Gheit V.E. The structural and constructive properties of a function and its conjugate in $L$. Izv. Vyssh. Ucheb. Zaved. Mat., 1972, no. 7(122), pp. 19-30 (in Russian).

18.   Il'yasov N.A. On inequalities between best approximations and module of smoothness of different orders of periodic functions in $L_{p},\ 1\le p\le \infty$ . Singular integral operators, Baku, Baku State Univ. Press, 1991, pp. 40-52 (in Russian).