On Bary–Stechkin theorem

A. I. Rubinshtein

Ufa mathematical journal, Tome 11 (2019) no. 1, pp. 70-74

Résumé

In the beginning of the past century, N.N. Luzin proved almost everywhere convergence of an improper integral representing the function $\bar f$ conjugated to a $2\pi$-periodic summable with a square function $f(x)$. A few years later I.I. Privalov proved a similar fact for a summable function. V.I. Smirnov showed that if $\bar f$ is summable, then its Fourier series is conjugate to the Fourier series for $f(x)$. It is easy to see that if $f(x)\in\mathrm{Lip}\,\alpha$, $0\alpha1$, then $\bar f(x)\in\mathrm{Lip}\,\alpha$. The Hilbert transformation for $f(x)$ differs from $\bar f(x)$ by a bounded function and has a simpler kernel. It is easy to show that the Hilbert transformation of $f(x)\in\mathrm{Lip}\,\alpha$, $0\alpha1$, also belongs to $\mathrm{Lip}\,\alpha$. In 1956 N.K. Bari and S.B. Stechkin found the necessary and sufficient condition on the modulus of continuity $f(x)$ for the function $\bar f(x)$ to have the same modulus of continuity. In 2016, the author introduced the concept of conjugate function as Hilbert transformation for functions defined on a dyadic group. In the present paper we show an analogue of the Bari–Stechkin (and Privalov) theorem fails that for a conjugated in this sense function.

Détail

Keywords: conjugate function, modulus of continuity, Bari–Stechkin theorem.
Mots-clés : dyadic group

Bibliographie

[1] N.K. Bari, Trigonometric series, GIFML, M., 1961 (in Russian)

[2] N.N. Luzin, Integral and trigonometric series, GIFML, M., 1951 (in Russian) | MR

[3] I.I. Privalov, Cauchy integral, Sovgrafija 13 otd., Saratov, 1919 (in Russian)

[4] N.K. Bari, “On best approximation of two conjugate functions by trigonometric polynomials”, Izv. Akad. Nauk SSSR Ser. Matem., 19:5 (1955), 285–302 (in Russian) | Zbl

[5] N.K. Bari, S.B. Stechkin, “Best approximations and differential properties of two conjugate functions”, Tr. Mosk. Mat. Obs., 5 (1956), 485–522 (in Russian)

[6] G.N. Agaev, N.Ya. Vilenkin, G.M. Dzhafarli, A.I. Rubinstein, Multiplicative systems of functions and harmonic analysis on zero-dimensional groups, ELM, Baku, 1981 (in Russian) | MR

[7] B.I. Golubov, A.V. Efimov, V.A. Skvortsov, Walsh series and transforms, Nauka, M., 1981 (in Russian) | MR

[8] A.I. Rubinstein, “Moduli of continuity of functions, defined on a zero-dimensional group”, Math. Notes, 23:3 (1978), 205–211 | DOI | MR

[9] S. Fridli, “On modulus of continuity with respect to functions defined on Vilenkin groups”, Acta Math. Hungar., 45 (1985), 393–396 | DOI | MR | Zbl

[10] A.I. Rubinstein, “About functions on the dyadic group and Walsh series”, Analysis Mathematica, 41:1–2 (2015), 73–81 | DOI | MR | Zbl

[11] A.I. Rubinstein, “On a certain integral operator acting on functions defined on the dyadic group”, Eurasian Mathematical Journal, 7:1 (2016), 68–73 | MR

[12] M.F. Timan, A.I. Rubinshtein, “Imbedding of classes of functions that are defined on zero-dimensional groups”, Soviet Math. (Iz. VUZ), 24:8 (1980), 74–85 | MR | Zbl | Zbl

[13] N.Ya. Vilenkin, A.I. Rubinshtein, “A certain theorem of S.B. Stečkin on absolute convergence, and series in systems of characters of zero-dimensional abelian groups”, Soviet Math. (Iz. VUZ), 19:9 (1975), 1–7 | MR | Zbl

[14] N.I. Chernykh, “The approximation of functions by polynomials with constraints”, Proc. Steklov Inst. Math., 88, 79–138 | MR | Zbl

[15] A.I. Rubinstein, “On the Analog of the Bary–Stechkin Theorem of the Conjugate Functions for the Dyadic Group”, Sixth international conference “The problems of Mathematical Physics and Mathematical modelling”, Book of abstracts (Moscow, Russia, 2017), 210–211