Shannon's sampling theorem, incongruent residue classes and Plancherel's theorem
Sampling theory for multi-band signals is shown to have a logical structure similar to that of Fourier analysis.
Sidon sets and Riesz sets for some measure algebras on the disk
Sidon sets for the disk polynomial measure algebra (the continuous disk polynomial hypergroup) are described completely in terms of classical Sidon sets for the circle; an analogue of the F. and M. Riesz theorem is also proved.
Solving dual integral equations on Lebesgue spaces
We study dual integral equations associated with Hankel transforms, that is, dual integral equations of Titchmarsh’s type. We reformulate these equations giving a better description in terms of continuous operators on spaces, and we solve them in these spaces. The solution is given both as an operator described in terms of integrals and as a series which converges in the -norm and almost everywhere, where denotes the Bessel function of order ν. Finally, we study the uniqueness of the solution....
Some classical function systems in separable Orlicz spaces
The boundedness of (sub)sequences of partial Fourier and Fourier-Walsh sums in subspaces of separable Orlicz spaces is studied. The boundedness of the shift operator and Paley function with respect to the Haar system is also investigated. These results are applied to get the analogues of the classical theorems on basicness of the trigonometric and Walsh systems in nonreflexive separable Orlicz spaces.
Some footprints of Marcinkiewicz in summability theory
Four basic results of Marcinkiewicz are presented in summability theory. We show that setting out from these theorems many mathematicians have reached several nice results for trigonometric, Walsh- and Ciesielski-Fourier series.
Some properties for functions of VMO().
Some remarks on quasi-Cohen sets
We are interested in Banach space geometry characterizations of quasi-Cohen sets. For example, it turns out that they are exactly the subsets E of the dual of an abelian compact group G such that the canonical injection is a 2-summing operator. This easily yields an extension of a result due to S. Kwapień and A. Pełczyński. We also investigate some properties of translation invariant quotients of L¹ which are isomorphic to subspaces of L¹.
Spherical quadrature formulas with equally spaced nodes on latitudinal circles.
Statistical convergence of Walsh-Fourier series.
Strong convergence theorems for two-parameter Walsh-Fourier and trigonometric-Fourier series
The martingale Hardy space and the classical Hardy space are introduced. We prove that certain means of the partial sums of the two-parameter Walsh-Fourier and trigonometric-Fourier series are uniformly bounded operators from to (0 < p ≤ 1). As a consequence we obtain strong convergence theorems for the partial sums. The classical Hardy-Littlewood inequality is extended to two-parameter Walsh-Fourier and trigonometric-Fourier coefficients. The dual inequalities are also verified and a...
Strong summability of Ciesielski-Fourier series
A strong summability result is proved for the Ciesielski-Fourier series of integrable functions. It is also shown that the strong maximal operator is of weak type (1,1).
Summability and integrability of Vilenkin series.
Summability estimates of double Vilenkin-Fourier series.
Summation of Fourier series.
Sur les séries dont le terme général dépend de deux angles et qui servent à exprimer des fonctions arbitraires entre des limites données.