N.A. Il’yasov. On the equivalence of some relations in different metrics between norms, best approximations, and moduli of smoothness of periodic functions and their derivatives ... P. 176-188

Full text (in Russian)

We propose a method capable, in particular, of establishing the equivalence of known upper estimates for the $L_q(\mathbb T)$-norm $\|f^{(r)}\|_q$, the best approximation $E_{n-1}(f^{(r)})_q$, and the $k$th-order modulus of smoothness $\omega_k(f^{(r)};\pi/n)_q$ in terms of elements of the sequence $\{E_{n-1}(f)_p\}_{n=1}^\infty$ of 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 $r\in \mathbb Z_+$ ($f^{(0)}=f)$, \mbox{$1<p<q<\infty$,} and $\mathbb T=(-\pi,\pi]$. The principal result of the paper is the following statement. Let \mbox{$1<p<q<\infty$,} $r\in \mathbb Z_+$, $k\in \mathbb N$, $\sigma=r+1/p-1/q$, $f\in L_p(\mathbb T)$, and $E(f;p;\sigma;q)\equiv\Big(\sum_{\nu=1}^{\infty}\nu^{q\sigma-1}E_{\nu-1}^{q}(f)_{p}\Big)^{1/q}<\infty$. Then the following inequalities are equivalent in the sense that each of them implies the other two:

(a) \ $\|f^{(r)}\|_q\le C_1(r,p,q)\left\{(1-\chi (r))\|f\|_p+E(f;p;\sigma;q)\right\}$;

(b) \ $E_{n-1}(f^{(r)})_q\le C_2(r,p,q)\left\{n^\sigma E_{n-1}(f)_p +\Big(\sum\nolimits_{\nu =n+1}^\infty \nu ^{q\sigma -1}E_{\nu -1}^q (f)_p\Big)^{1/q}\right\}$, $n\in\mathbb{N}$;

(c) \ $\omega _k (f^{(r)};\pi/n)_q \le C_3 (k,r,p,q)\Big\{\Big(\sum\nolimits_{\nu =n+1}^\infty \nu^{q\sigma -1}E_{\nu -1}^q (f)_p\Big)^{1/q}+n^{-k}\Big(\sum\nolimits_{\nu =1}^n \nu ^{q(k+\sigma )-1}E_{\nu -1}^q (f)_p \Big)^{1/q}\Big\}$, $n\in \mathbb{N}$.

\noindent Inequalities (a), (b), and (c) depend on the key estimate
$$
\big\| S_m^{(l)} (f;\cdot )\big\|_q \le C_4(l,p,q)\Big\{(1-\chi (l))\|f\|_p +\Big(\sum\nolimits_{\nu =1}^m \nu ^{q\lambda -1} E_{\nu -1}^q (f)_p \Big)^{1/q}\Big\},\ \ m\in \mathbb{N},
$$
where $S_m (f;x)$ is the partial sum of order $m\in \mathbb{N}$ of the Fourier series of a function $f\in L_p(\mathbb T)$, $l\in \mathbb Z_+ $, $\lambda =l+
1/p-1/q$, $\chi (t)=0$ for $t\le 0$, and $\chi (t)=1$ for $t>0$, $t\in \mathbb{R}$. The latter estimate in the case $l=r$ and $\lambda =\sigma $ provides a necessary and sufficient condition for the fulfillment of inequality (a) under the condition $E(f;p;\sigma ;q)<\infty$, which guarantees that $f\in L_q^{(r)}(\mathbb T)$, where $L_q^{(r)} (\mathbb T)$ is the class of functions $f\in L_q (\mathbb T)$ with absolutely continuous $(r-1)$th derivative and $f^{(r)}\in L_q (\mathbb T)$. Necessary and sufficient conditions for the validity of inequalities (b) and (c) are also provided in terms of the behavior of elements of the sequence $\{\|S_m^{(l)} (f;\cdot )\|_q\}_{m=1}^\infty$.

Keywords: best approximation, modulus of smoothness, inequalities in different metrics, equivalent inequalities

Received September 10, 2018

Revised November 13, 2018

Accepted November 19, 2018

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

REFERENCES

1.   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).

2.   Timan A.F. Theory of approximation of functions of real variables. N Y: Macmillan, Pergamon Press, 1963, 631 p. Original Russian text published in Timan A.F. Teoriya priblizheniya funktsii deistvitel’nogo peremennogo. Moscow: Fizmatgiz Publ., 1960, 624 p.

3.   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).

4.   Kokilashvili V.M. On the estimate of best approximations and modulus of smoothness in the various Lebesgue spaces of periodic functions with transformed Fourier series. Bulletin of the Academy of Sciences of the Georgian SSR, 1964, vol. 35, no. 1, pp. 3–8 (in Russian).

5.   Ul’yanov P.L. Imbedding theorems and relations between best approximations (moduli of continuity) in different metrics. Math. USSR-Sb., 1970, vol. 10, no. 1, pp. 103–126. doi: 10.1070/SM1970v010n01ABEH001589

6.   Ul’yanov P.L. Embedding of certain classes of functions $H_{p}^{\omega}$. Math. USSR–Izv., 1968, vol. 2, no. 3, pp. 601–637. doi: 10.1070/IM1968v002n03ABEH000650

7.   Ul’yanov P.L. The imbedding theorems and best approximations. Dokl. Akad. Nauk SSSR, 1969, vol. 184, no. 5, pp. 1044–1047 (in Russian).

8.   Timan M.F. Some embedding theorems for $L_{p}$-classes of functions. Dokl. Akad. Nauk SSSR, 1970, vol. 193, no. 6, pp. 1251–1254 (in Russian).

9.   Timan M.F. The imbedding of the $L_{p}^{(k)}$ classes of functions. Izv. Vyssh. Ucheb. Zaved. Mat., 1974, no. 10 (149), pp. 61–74 (in Russian).

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

11.   Il’yasov N.A. Approximation of periodic functions by Fejer — Zygmund means in various metrics. Math. Notes, 1990, vol. 48, no. 4, pp. 1004–1010. doi: 10.1007/BF01139600

12.   Il’yasov N.A. An inverse approximation theorem in various metrics. Math. Notes, 1991, vol. 50, no. 6, pp. 1253–1260. doi: 10.1007/BF01158266

13.   Zygmund A. Trigonometric series. Warszawa: Instytut Matematyczny PAN, 1935, 331 p. Translated to Russian under the title Trigonometricheskie ryady. Moscow; Leningrad: Gostexizdat Publ., 1939, 324 p.

14.   Riesz M. Sur les fonctions conjuguees. Math. Zeit., 1927, bd. 27, no. 2, s. 218–244. doi: 10.1007/BF01171098

15.   Zygmund A. Trigonometric series, vol. I, II. Cambridge: Cambridge Univ. Press, 1959; vol. I, 383 p.; vol. II, 354 p. ISBN (3rd ed.): 0521890535 . Translated to Russian under the title Trigonometricheskie ryady. Moscow: Mir Publ., 1965, vol. I, 616 p.; vol. II, 538 p.

16.   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).