Numerical integration of Hölder continuous, absolutely convergent Fourier, Fourier cosine, and Walsh series

Nonanalytic Functions of Absolutely Convergent Fourier Series.

On Walsh-fourier Series^)

Every function f(x) which is of period 1 and Lebesgue integrable on [0, 1 ] may be expanded in a Walsh-Fourier series(3), f(x)~ ?.?=n ak\pk(x), where ak=fof(x)ypk(x)dx, k=0, 1, 2, • • • . Fine exhibited some of the basic similarities and differences between the trigonometric orthonormal system and the Walsh system. He identified the Walsh functions with the full set of characters of the dyadic ...

Convergence of Trigonometric and Walsh - Fourier Series

In this paper we present some results on convergence and summability of oneand multi-dimensional trigonometric andWalsh-Fourier series. The Fejér and Cesàro summability methods are investigated. We will prove that the maximal operator of the summability means is bounded from the corresponding classical or martingale Hardy space Hp to Lp for some p > p0. For p = 1 we obtain a weak type inequalit...

Statistical Convergence of Walsh-fourier Series

This is a brief and concise account of the basic concepts and results on statistical convergence, strong Cesàro summability and Walsh-Fourier series. To emphasize the significance of statistical convergence, for example we mention the fact that the one-dimensional Walsh-Fourier series of an integrable (in Lebesgue’s sense) function may be divergent almost everywhere, but it is statistically con...

Determination of a jump by Fourier and Fourier-Chebyshev series

‎By observing the equivalence of assertions on determining the jump of a‎ ‎function by its differentiated or integrated Fourier series‎, ‎we generalize a‎ ‎previous result of Kvernadze‎, ‎Hagstrom and Shapiro to the whole class of‎ ‎functions of harmonic bounded variation‎. ‎This is achieved without the finiteness assumption on‎ ‎the number of discontinuities‎. ‎Two results on determination of ...