Analysis Mathematica

, Volume 14, Issue 1, pp 49–63

• Т. КОВАЛЬСКИ
Article

# On some properties of partial sums of Fourier—Walsh—Paley series

## Abstract

In this paper we consider the behaviour of partial sums of Fourier—Walsh—Paley series on the group62-01. We prove the following theorems: Theorem 1. Let {n k } k =1/∞ be some increasing convex sequence of natural numbers such that
$$\mathop {\lim sup}\limits_m m^{ - 1/2} \log n_m< \infty$$
. Then for anyfL (G)
$$\left( {\frac{1}{m}\sum\limits_{j = 1}^m {|Sn_j (f;0)|^2 } } \right)^{1/2} \leqq C \cdot \left\| f \right\|_\infty$$
. Theorem 2. Let {n k } k =1/∞ be a lacunary sequence of natural numbers,n k+1/n kq>1. Then for anyfεL (G)
$$\sum\limits_{j = 1}^m {|Sn_j (f;0)| \leqq C_q \cdot m^{1/2} \cdot \log n_m \cdot \left\| f \right\|_\infty }$$
. Theorems. Let µ k =2 k +2 k-2+2 k-4+...+2α 0,α 0=0,1. Then
$$\begin{gathered} \{ \{ S_{\mu _k } (f:0\} _{k = 1}^\infty ;f \in L^\infty (G)\} = \{ \{ a_k \} _{k = 1}^\infty ;\sum\limits_{k = 1}^m {a_k^2 = 0(m)^2 \} .} \hfill \\ \{ \{ S_{\mu _k } (f:0\} _{k = 1}^\infty ;f \in C(G)\} = \{ \{ a_k \} _{k = 1}^\infty ;\sum\limits_{k = 1}^m {a_k^2 = o(m)^2 \} = } \hfill \\ = \{ \{ S_{\mu _k } (f:0\} _{k = 1}^\infty ;f \in C(G),f(0) = 0\} \hfill \\ \end{gathered}$$
. Theorem 4. {{S 2 k(f: 0)} k =1/∞ ,fL (G)}=m.
$$\{ \{ S_{2_k } (f:0\} _{k = 1}^\infty ;f \in C(G)\} = c. \{ \{ S_{2_k } (f:0\} _{k = 1}^\infty ;f \in C(G),f(0) = 0\} = c_0$$
.

## Preview

### Литература

1. [1]
A. B. Alexandrov,Essays on non locally convex Hardy classes, Lect. Notes Math.,864 (1981), 1–89.
2. [2]
L.Carleson, Appendix to the Paper of J.-P. Kahane and Y. Katznelson,Stud. Pure Math. Mem. Paul Turán, (Budapest, 1983), 411–413.Google Scholar
3. [3]
В. А. Глухов, О сумми ровании кратных рядо в Фурье по мультиплик ативным системамМа тем. заметки 39 (1986), 665–673.Google Scholar
4. [4]
J. P.Kahane, Y.Katznelson, Séries de Fourier des fonctions bornées,Stud. Pure Math. Mem. Paul Turán (Budapest, 1983), 395–410.Google Scholar
5. [5]
Jui-Lin Long, Sommes partielles de Fourier des fonctions bornées,C. R. Acad. Sc. Paris,288 (1979), 1009–1011.
6. [6]
F. Schipp, Über die Grössenordnung der Partialsummen der Entwicklung integrierbarer Funktionen nach W-System,Acta Sci. Math.,28 (1967), 123–134.
7. [7]
Н. А. Загородный, Р. М. Тригуб, Об одном воп росе Салема,сб. Теоре ма функций и отображе ний, Наукова думка (Ки ев, 1979), 97–101.Google Scholar