On absolute summation of Fourier series by subsequences

В работе изучается сл едующая задача. Пусть заданы числа 0<α≦1 и β<α. При каки х условиях на строго во зрастающую последов ательность натуральных чисел {n _k } _k ^t8 =1 для всех 2π-периодических функ ций \(f(x) \sim \sum\limits_{v = - \infty }^\infty {c_v e^{ivx} } \) , принадлежащих к лассу Lip α, равномерно пох будет выполнено неравенство $$\sum\limits_{k = 1}^\infty {|\sum\limits_{n_k \leqq |v|< n_{k + 1} } {c_v e^{ivx} } |n_k^\beta< \infty ?} $$ .     

On the convergence rate of Fourier series

The rate of convergence of a Fourier‐series representation of a given function depends on the nature of the function and of its derivatives. This is shown by using the graphical outputs of a desk computer for different cases. For full‐range series, the effects of continuity and discontinuity of the function and its first derivatives are shown first. The advantage of half‐range formulae due to the free choice of function in the second half‐range are demonstrated next, along with the importance of choice of sine and cosine series according to the function being represented. Finally, a method is given of modifying the given function in a simple manner whereby a dramatic increase in the rate of convergence of the Fourier series is obtained. Examples of the Gibbs overshoot and its elimination are included.     

On summability of subsequences of partial sums of trigonometric Fourier series

n- (n1) fL ^p ([–, ] ⁿ ),=1 = (L C) . , , f([–, ] ⁿ ).     

On convergence and summability of Fourier series

. , , –1<<0. .

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The present work was written on the basis of two earlier works received byAnalysis Mathematica on January 16, 1979, and July 20, 1979.     

On the convergence in mean of trigonometric Fourier series

We prove the sharpness of Zygmund’s theorem, which asserts that if a 2π-periodic function f belongs to L ln⁺ L, then its Fourier series is convergent in mean.     

On necessary conditions for the convergence of Fourier series

We establish necessary conditions for the convergence of multiple Fourier series of integrable functions in the mean.     

On strong convergence of trigonometric and Fourier series

Acta Mathematica Hungarica -     

On the convergence of double Fourier series of discontinuous functions

In the paper we obtain a formula for the computation of the rectangular partial sums of a double Fourier series of a function of two variables at the point (0, 0), having a discontinuity along any ray at this point. We also give an estimate of the remainder in the abovementioned formula.     

On the convergence of greedy approximants of trigonometric Fourier series

Efficient versions of the Cauchy criterion for the convergence in L _p of greedy approximants of trigonometric Fourier series are discussed.