Analysis Mathematica

, Volume 14, Issue 1, pp 49–63 | Cite as

О некоторых свойства х частичных сумм рядо в Фурье—Уолша—Пэли

  • Т. КОВАЛЬСКИ
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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.
$$\{ \{ S_{2_k } (f:0\} _{k = 1}^\infty ;f \in C(G)\} = c. \{ \{ S_{2_k } (f:0\} _{k = 1}^\infty ;f \in C(G),f(0) = 0\} = c_0 $$
.

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Литература

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Copyright information

© Akadémiai Kiadó 1988

Authors and Affiliations

  • Т. КОВАЛЬСКИ
    • 1
  1. 1.Technical UniversityKoszalinPoland

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