Convergence factors for double series

Convergence Factors for Double Series Summable by Nörlund Means.

In a note' published in volume 21 of these PROCEEDINGS I gave necessary and sufficient conditions for convergence factors in simply infinite series summable by N6rlund means. The purpose of the present note is to furnish analogous results for double series. We are given a doubly infinite set of complex constants cij (i, j = 0, 1, 2, 3, .. .), where coo 0 0 and Zc,ix'y' is convergent for I x I <...

L1-convergence of complex double Fourier series

It is proved that the complex double Fourier series of an integrable function f (x,y) with coefficients {c jk} satisfying certain conditions, will converge in L 1norm. The conditions used here are the combinations of Tauberian condition of Hardy– Karamata kind and its limiting case. This paper extends the result of Bray [1] to complex double Fourier series.

Uniform Convergence of Double Trigonometric Series

It is a classical problem in Fourier analysis to give conditions for a single sine or cosine series to be uniformly convergent. Several authors gave conditions for this problem supposing that the coefficients are monotone, non-negative or more recently, general monotone. There are also results for the regular convergence of double sine series to be uniform in case the coefficients are monotone ...

Convergence of Double Fourier Series and W -classes

The double Fourier series of functions of the generalized bounded variation class {n/ ln(n + 1)}∗BV are shown to be Pringsheim convergent everywhere. In a certain sense, this result cannot be improved. In general, functions of class Λ∗BV, defined here, have quadrant limits at every point and, for f ∈ Λ∗BV, there exist at most countable sets P and Q such that, for x / ∈ P and y / ∈ Q, f is conti...

Uniform convergence of single and double sine series and integrals