On the logarithmic summability of Fourier series.(English)Zbl 1221.42049

The authors dicuss the logarithmic summability, defined by G. Tkebuchava [Acta Math. Acad. Paedagog. Nyházi. (N. S.) 21, 161–167 (2005; Zbl 1094.42002)], for Walsh-Paley and Walsh-Kaczmarz Fourier series. Tkebuchava defined his logarithmic means $$l_{m,n}$$ by $$l_{m,n}(f)=(l(m,n))^{-1}(\sum_{k=0}^{m-1}S_k(f)/(m-k+1)+S_m(f)+ \sum_{k=m+1}^{n}S_k(f)/(k-m+1))$$, where $$S_k(f)$$ is the $$k$$-th partial sum of a Fourier series, and $$l(m,n)=\sum_{k=0}^{m-1}1/(m-k+1)+1+ \sum_{k=m+1}^{n}1/(k-m+1)$$. This summation method includes the Riesz (in case $$m=0$$) and Nörlund (in case $$m=n$$) logarithmic methods.
Let $$\mathcal F_{m,n}$$ be the kernel function of Walsh-Paley or Walsh-Kaczmarz Fourier series. Then the authors’ main result says that there are $$C_2>C_1>0$$ such that $$C_1\leq \sup_{m,n\in\mathbb Z,\,0\leq m\leq n}\|\mathcal F_{m,n}\|_{1}\leq C_2$$. From this, they deduce that, for $$m_m=O(\exp \sqrt{\log n})$$ (as $$n\to\infty)$$, $$\|t_{m_n,n}(f)-f\|_1\to 0$$ for $$f\in L^1(0,1)$$, and $$\|t_{m_n,n}(f)-f\|_\infty\to 0$$ for $$f\in C(0,1)$$. These results correspond to those in the case of trigonometrical Fourier series by Tkebuchava.
The authors also discuss quadratical partial sums of two-dimensional Fourier series (Walsh-Paley, Walsh-Kaczmarz and trigonometrical).

MSC:

 42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) 42B08 Summability in several variables

Zbl 1094.42002
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