This Memoir studies Weil’s well-known Explicit Formula in the theory of prime numbers and its associated quadratic functional, which is positive semidefinite if and only if the Riemann Hypothesis is true. We prove that this quadratic functional attains its minimum in the unit ball of the $L^2$-space of functions with support in a given interval $\left[-t,t\right]$, and prove again Yoshida’s theorem that it is positive definite if $t$ is sufficiently small. The Fourier transform of the functional gives rise to a quadratic...

It is well known that the condition “f ∈ L¹ and f̂ ∈ L¹” is not sufficient to ensure the validity of the Poisson summation formula ∑f(k) = ∑f̂(k). We discuss here a stronger condition "$x^{a}f\in L^p$ and ${\xi }^{b}f̂\in L^q$" and see for which values of a and b the condition is sufficient.

On donne une démonstration nouvelle (et un peu plus générale) d’un théorème de J. Delsarte sur les fonctions moyenne-périodiques de deux variables.

In this paper, we relate the notions of remote almost periodicity and quasi-asymptotical almost periodicity; in actual fact, we observe that a remotely almost periodic function is nothing else but a bounded, uniformly continuous quasi-asymptotically almost periodic function. We introduce and analyze several new classes of remotely $c$-almost periodic functions in ${ℝ}^n,$ slowly oscillating functions in ${ℝ}^n,$ and further analyze the recently introduced class of quasi-asymptotically $c$-almost periodic functions...

Some recent results on spline-Fourier and Ciesielski-Fourier series are summarized. The convergence of spline Fourier series and the basis properties of the spline systems are considered. Some classical topics, that are well known for trigonometric and Walsh-Fourier series, are investigated for Ciesielski-Fourier series, such as inequalities for the Fourier coefficients, convergence a.e. and in norm, Fejér and θ-summability, strong summability and multipliers. The connection between Fourier series...

Riemann’s memoir is devoted to the function π(x) defined as the number of prime numbers less or equal to the real and positive number x. This is really the fact, but the “main role” in it is played by the already mentioned zeta-function.

Elementary estimates for the Riesz kernel and for its derivative are given. Using these we show that the maximal operator of the Riesz means of a tempered distribution is bounded from $H_p\left(ℝ\right)$ to $L_p\left(ℝ\right)$ (1/(α+1) < p < ∞) and is of weak type (1,1), where $H_p\left(ℝ\right)$ is the classical Hardy space. As a consequence we deduce that the Riesz means of a function $⨍\in L_1\left(ℝ\right)$ converge a.e. to ⨍. Moreover, we prove that the Riesz means are uniformly bounded on $H_p\left(ℝ\right)$ whenever 1/(α+1) < p < ∞. Thus, in case $⨍\in H_p\left(ℝ\right)$, the Riesz means converge...

Given a set of positive measure on the circle and a set Λ of integers, one can ask whether $E\left(\Lambda \right):=e_{\lambda \in \Lambda }^{i\lambda t}$ is a Riesz sequence in L²(). We consider this question in connection with some arithmetic properties of the set Λ. Improving a result of Bownik and Speegle (2006), we construct a set such that E(Λ) is never a Riesz sequence if Λ contains an arithmetic progression of length N and step $\ell =O\left(N^{1-\epsilon }\right)$ with N arbitrarily large. On the other hand, we prove that every set admits a Riesz sequence E(Λ) such that Λ does contain...

In this paper we obtain the $L^p$-boundedness of Riesz transforms for the Dunkl transform for all $1\&lt;p\&lt;\infty$.

The efficient evaluation of a discrete convolution is usually carried out as a repated evaluation of a discrete convolution of a special type with the help of the fast Fourier transform. The paper is concerned with the analysis of the roundoff errors in the fast computation of this convolution. To obtain a comparison, the roundoff errors in the usual (direct) computation of this convolution are also considered. A stochastic model of the propagation of roundoff errors. is employed. The theoretical...