Summation of Fourier Series with Respect to Walsh–like Systems and the Dyadic Derivative
نویسنده
چکیده
In this paper we present some results on summability of oneand multi-dimensional Walsh-, Walsh-Kaczmarzand Vilenkin-Fourier series and on the dyadic and Vilenkin derivative. The Fejér and Cesàro summability methods are investigated. We will prove that the maximal operator of the summability means is bounded from the martingale Hardy space Hp to Lp (p > p0). For p = 1 we obtain a weak type inequality by interpolation, which ensures the a.e. convergence of the summability means. Similar results are formulated for the oneand multi-dimensional dyadic and Vilenkin derivative. The dyadic version of the classical theorem of Lebesgue is proved, more exactly, the dyadic derivative of the dyadic integral of a function f is a.e. f . This research was supported by the Hungarian Scientific Research Funds (OTKA) K67642. Mathematics Subject Classifications: Primary 42C10, 43A75; Secondary 60G42, 42B30.
منابع مشابه
Summation of Fourier Series
A general summability method of different orthogonal series is given with the help of an integrable function θ. As special cases the trigonometric Fourier, Walsh-, Walsh-Kaczmarz-, Vilenkinand Ciesielski-Fourier series and the Fourier transforms are considered. For each orthonormal system a different Hardy space is introduced and the atomic decomposition of these Hardy spaces are presented. A s...
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