On the summability of power series and Fourier series

Logarithmic Summability of Fourier Series

A set of regular summations logarithmic methods is introduced. This set includes Riesz and Nörlund logarithmic methods as limit cases. The application to logarithmic summability of Fourier series of continuous and integrable functions are given. The kernels of these logarithmic methods for trigonometric system are estimated.

Wiener amalgams and summability of Fourier series∗

Some recent results on a general summability method, on the so-called θ-summability is summarized. New spaces, such as Wiener amalgams, Feichtinger’s algebra and modulation spaces are investigated in summability theory. Sufficient and necessary conditions are given for the norm and a.e. convergence of the θ-means.

Product Summability Transform of Conjugate Series of Fourier Series

A known theorem, Nigam 2010 dealing with the degree of approximation of conjugate of a signal belonging to Lipξ t -class by E, 1 C, 1 product summability means of conjugate series of Fourier series has been generalized for the weighted W Lr, ξ t , r ≥ 1 , t > 0 -class, where ξ t is nonnegative and increasing function of t, by ̃ E1 nC n which is in more general form of Theorem 2 of Nigam and Shar...

On the Summability of the Differentiated Fourier Series

l1-summability of higher-dimensional Fourier series

It is proved that the maximal operator of the l1-Fejér means of a d-dimensional Fourier series is bounded from the periodic Hardy space Hp(T ) to L p(T ) for all d/(d+1) < p ≤ ∞ and, consequently, is of weak type (1, 1). As a consequence we obtain that the l1-Fejér means of a function f ∈ L1(T ) converge a.e. to f . Moreover, we prove that the l1-Fejér means are uniformly bounded on the spaces ...