# N.A. Il’yasov. On the equivalence of some inequalities in the theory of approximation of periodic functions in the spaces $L_p(\mathbb T), 1 < p < \infty$ ... P. 93-106

We propose a method for proving, in particular, the equivalence of M.F. Timan's known estimates for the $r$th-order $L_{p}$-moduli of smoothness $\omega_{r}(f;{\pi/n})_{p}$ and O.V. Besov's estimates for the $L_p$-norms $\|f^{(r)}\|_{p}$ of $r$th-order derivatives by using elements of the sequence $\{E_{n-1}(f)_{p}\}_{n=1}^{\infty}$ of the best approximations of a $2\pi$-periodic function $f\in L_{p}(\mathbb T)$ by trigonometric polynomials of order at most $n-1$, $n\in \mathbb N$, where $r\in \mathbb N$, $1<p<\infty$, and $\mathbb T=(-\pi,\pi]$.

Theorem 1.  Let $1<p<\infty$, $\theta=\min\{2,p\}$, $r\in \mathbb N$, $f\in L_{p}(\mathbb T)$, and $\sum_{n=1}^{\infty}n^{\theta r-1} E_{n-1}^{\theta}(f)_{p}<\infty$. Then the inequality $\omega_{r}(f;\pi/n)_{p}\le C_{1}(r,p)n^{-r}\Big(\sum_{\nu=1}^{n}\nu^{\theta r-1}E_{\nu-1}^{\theta}(f)_{p}\Big)^{1/\theta}$, $n\in \mathbb N$, is satisfied if and only if $f\in L_{p}^{(r)}(\mathbb T)$ and $\|f^{(r)}\|_{p} \le C_{2}(r,p) \Big(\sum_{n=1}^{\infty}n^{\theta r-1} E_{n-1}^{\theta}(f)_{p}\Big)^{1/\theta}$, where $L_{p}^{(r)}(\mathbb T)$ is the class of functions $f\in L_{p}(\mathbb T)$ with absolutely continuous derivative of the $(r-1)$th order and $f^{(r)} \in L_{p}(\mathbb T)$.

Theorem 2.  Suppose that $1<p<\infty$, $\beta=\max\{2,p\}$, $r\in \mathbb N$, and $f\in L_{p}^{(r)}(\mathbb T)$. Then the inequality  $n^{-r}\Big(\sum_{\nu=1}^{n}\nu^{\beta r-1} E_{\nu-1}^{\beta}(f)_{p}\Big)^{1/\beta}\le C_{3}(r,p)\omega_{r}(f;\pi/n)_{p}$ is satisfied for $n\in \mathbb N$ if and only if the inequality $\Big(\sum_{n=1}^{\infty}n^{\beta r-1}E_{n-1}^{\beta}(f)_{p}\Big)^{1/\beta}\le C_{4}(r,p)\|f^{(r)}\|_{p}$ is satisfied.

In view of the order identity $\sum_{\nu=1}^{n}\nu^{\alpha r-1}E_{\nu-1}^{\alpha}(f)_{p}\asymp\sum_{\nu=1}^{n}\nu^{\alpha r-1} \omega_{l}^{\alpha}(f;\pi/\nu)_{p}$, $n\in\mathbb N\cup\{+\infty\}$, where $1\le\alpha<\infty$, $l\in\mathbb N$, and $l>r$, the assertions of Theorems 1 and 2 remain valid if we replace the sequence $\{E_{n-1}(f)_{p}\}_{n=1}^{\infty}$ by the sequence $\{\omega_{l}(f;\pi/n)_{p}\}_{n=1}^{\infty}$ (Theorems 3 and 4). The method used in the proof of Theorems 1 and 2 can be applied to derive equivalent upper estimates and equivalent lower estimates for the values $E_{n-1}(f^{(r)})_{p}$ and $\omega_{k}(f^{(r)};\pi/n)_{p}$, $n\in \mathbb N$, by means of elements of the sequence $\{E_{n-1}(f)_{p}\}_{n=1}^{\infty}$, where $k,r\in \mathbb N$ and $1<p<\infty$.

Keywords: best approximation, modulus of smoothness, inequalities of approximation theory, equivalent inequalities, Timan's inequalities, Besov's inequalities.

The paper was received by the Editorial Office on March 13, 2018.

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

REFERENCES

1.   Timan M.F. Inverse theorems of the constructive theory of functions in $L_p$ spaces $(1\le p\le \infty)$. Mat. Sb. (N.S.), 1958, vol. 46 (88), no. 1, pp. 125–132 (in Russian).

2.   Timan M.F. On the Jackson theorem in $L_p$ spaces. Ukr. Mat. Zhurn., 1966, vol. 18, no. 1, pp. 134–137 (in Russian). doi: 10.1007/BF02537726 .

3.   Besov O.V. On some conditions for derivatives of periodic functions to belong to $L_p$. Nauch. Dokl. Vyssh. Shkoly. Fiz.-Mat.Nauki, 1959, no. 1, pp. 13–17 (in Russian).

4.   Timan A.F., Timan M.F. Generalized modulus of continuity and best approximation in the mean. Dokl. Akad. Nauk SSSR, 1950, vol. 71, no. 1, pp. 17–20 (in Russian).

5.   Zygmund A. A remark on the integral modulus of continuity. Univ. Nac. Tucuman Revista, 1950, ser. A, vol. 7, pp. 259–269.

6.   Zygmund A. Smooth functions. Duke Math. J., 1945, vol. 12, no. 1, pp. 47–76. doi: 10.1215/S0012-7094-45-01206-3 .

7.   Stechkin S.B. On the Kolmogorov — Seliverstov theorem. Izv. Akad. Nauk SSSR. Ser. Mat., 1953, vol. 17, no. 6, pp. 499–512 (in Russian).

8.   Stechkin S.B. On the order of the best approximations of continuous functions. Izv. Akad. Nauk SSSR. Ser. Mat., 1951, vol. 15, no. 3, pp. 219–242 (in Russian).

9.   Timan A.F. Theory of approximation of functions of a real variable. Oxford; London; N Y: Pergamon Press, 1963, 655 p. ISBN: 048667830X . Original Russian text published in Timan A.F. Teoriya priblizheniya funktsii deystvitel’nogo peremennogo, Moscow, Fizmatgiz Publ., 1960, 624 p.

10.   Hardy G.H., Littlewood J.E. Some properties of fractional integrals. I. Math. Zeit., 1928, vol. 27, no. 4, pp. 565–606. doi: 10.1007/BF01171116 .

11.   Marcinkiewicz J. Sur quelques integrals du type de Dini. Ann. Soc. Polon. Math., 1938, vol. 17, pp. 42–50.

12.   Zygmund A. On certain integrals. Trans. Amer. Math. Soc., 1944, vol. 55, no. 2, pp. 170–204.

13.   Riesz M. Sur les fonctions conjuguees. Math. Zeit., 1927, vol. 27, no. 2, pp. 218–244. doi: 10.1007/BF01171098 .

14.   Zygmund A. Trigonometric series, vol. I, II. Cambridge: Cambridge Univ. Press, 1959, vol. I. 383 p.; vol. II. 354 p. ISBN(3rd ed.): 0-521-89053-5 . Translated to Russian under the title Trigonometricheskie ryady, Moscow, Mir Publ., 1965, vol. I, 616 p; vol. II. 538 p.

15.   Quade E.S. Trigonometric approximation in the mean. Duke Math. J., 1937, vol. 3, no. 3, pp. 529–543. doi: 10.1215/S0012-7094-37-00342-9 .

16.   Brudnyi Yu.A. Criteria for the existence of derivatives in $L_p$. Math. USSR-Sb., 1967, vol. 2, no. 1, pp. 35–55. doi: 10.1070/SM1967v002n01ABEH002323 .

17.   Zhuk V.V. Approksimatsiya periodicheskikh funktsii [Approximation of periodic functions]. Leningrad: Leningrad Univ. Publ., 1982, 368 p. (in Russian).

18.   Hardy G.H., Littlewood J.E., Polya G. Inequalities. London: Cambridge Univ. Press, 1934, 314 p. ISBN(2nd ed.): 978-0-521-35880-4 .

19.    Timan M.F. Best approximation and modulus of smoothness of functions defined on the entire real axis. Izv. Vyssh. Ucheb. Zaved. Mat., 1961, no. 6 (25), pp. 108–120 (in Russian).