We give a new version of Ivasev-Musatov’s construction of a measure whose support has Lebesgue measure zero but whose Fourier transform drops away extremely rapidly.

As in Part I [Annales de l’Inst. Fourier, 27-3 (1997), 97-113], our object is to construct a measure whose support has Lebesgue measure zero, but whose Fourier transform drops away extremely fast.

We investigate when the trigonometric conjugate to the periodic general Franklin system is a basis in C(𝕋). For this, we find some necessary and some sufficient conditions.

We give $A_p$ type conditions which are sufficient for two-weight, strong $(p,p)$ inequalities for Calderón-Zygmund operators, commutators, and the Littlewood-Paley square function $g_{\lambda }^{*}$. Our results extend earlier work on weak $(p,p)$ inequalities in [13].

In 1970 C. W. Onneweer formulated a sufficient condition for a periodic W-continuous function to have a Walsh-Fourier series which converges uniformly to the function. In this paper we extend his results from single to double Walsh-Fourier series in a more general setting. We study the convergence of rectangular partial sums in $L^p$-norm for some 1 ≤ p ≤ ∞ over the unit square [0,1) × [0,1). In case p = ∞, by $L^p$ we mean $C_W$, the collection of uniformly W-continuous functions f(x, y), endowed with the...

Let a single sine series (*) ${\sum }_{k=1}^{\infty }{a}_{k}sinkx$ be given with nonnegative coefficients $a_k$. If $a_k$ is a “mean value bounded variation sequence” (briefly, MVBVS), then a necessary and sufficient condition for the uniform convergence of series (*) is that $k{a}_k\to 0$ as k → ∞. The class MVBVS includes all sequences monotonically decreasing to zero. These results are due to S. P. Zhou, P. Zhou and D. S. Yu. In this paper we extend them from single to double sine series (**) ${\sum }_{k=1}^{\infty }{\sum }_{l=1}^{\infty }{c}_{kl}sinkxsinly$, even with complex coefficients $c_{kl}$. We also give a uniform...

In the present paper we consider a new class of sequences called GM(β,r), which is the generalization of a class defined by Tikhonov in [15]. We obtain sufficient and necessary conditions for uniform convergence of weighted trigonometric series with (β,r)-general monotone coefficients.

In [5], we characterized the uniform convexity with respect to the Luxemburg norm of the Besicovitch-Orlicz space of almost periodic functions. Here we give an analogous result when this space is endowed with the Orlicz norm.

The uniqueness theorem for the ergodic maximal operator is proved in the continous case.

Let $a₁,...,a_k$ be arbitrary nonzero real numbers. An $(a₁,...,a_k)$-decomposition of a function f:ℝ → ℝ is a sum $f₁+\cdots +f_k=f$ where $f_i:ℝ\to ℝ$ is an $a_i$-periodic function. Such a decomposition is not unique because there are several solutions of the equation $h₁+\cdots +h_k=0$ with $h_i:ℝ\to ℝa_i$-periodic. We will give solutions of this equation with a certain simple structure (trivial solutions) and study whether there exist other solutions or not. If not, we say that the $(a₁,...,a_k)$-decomposition is essentially unique. We characterize those periods for which essential uniqueness...

It is proved that the ergodic maximal operator is one-to-one.

It is shown that if two functions share the same uncentered (two-sided) ergodic maximal function, then they are equal almost everywhere.