Volume 33, Issue 4

In this paper, we give a reformulation of the Riemann hypothesis for automorphic L -functions in terms of orthogonal polynomials which extends the Cardon and Roberts criterion for the classical case of the Riemann zeta function. Actually, for any principal L -function L ( s ,  π ) attached to irreducible cuspidal unitary automorphic representation π of GL ( N ), we show that the generalized Riemann hypothesis with simple zeros is equivalent to the existence of a certain family of orthogonal polynomials { P n ( z )} such that where ξ ( s ,  π ) is the completed automorphic L -function.

We list multiplier and convolutor spaces of the spaces occurring in L. Schwartz' “Théorie des distributions”. Furthermore we clarify whether the multiplications and convolutions are continuous or not.

The aim of this work is to study the coexistence of oscillatory and nonoscillatory solutions of a certain scalar delay difference equation.

Given two k -forms α and β on a compact Riemannian manifold M with boundary ∂ M , we derive an identity relating to an integral on the boundary ∂ M . Herein F k is a bundle endomorphism depending only on the Riemannian curvature tensor. Essential is how the tangential and normal parts of α and β , respectively their derivatives, appear in the integrand of the boundary integral. The identity gives a very simple proof for many classical results, which require that α  =  β and also that the tangential part α T (or the normal part α N ) vanishes on ∂ M . Moreover we generalize some of the boundary conditions in Gaffney inequality and in nonexistence theorems for harmonic fields.

In the paper, the authors introduce a notion “log - h -convex functions” and establish several Hermite–Hadamard type integral inequalities for this kind of functions.

Consider an integral I ( s ): =∫ 0 b e − s ( x 2 − icx ) x 2 dx , where c  > 0 and b  > 0 are arbitrary positive constants. It is proved that as s  →+∞. Possible applications of this result to the Pompeiu problem are outlined.

In this paper, we derive some interesting argument results of symmetric analytic functions which will be useful in obtaining applications of geometric function theory.

In the paper, the authors construct three new sequences converging fast to Euler–Mascheroni constant γ and obtain a sharp double inequality for bounding the harmonic number.

The aim of this article is to generalize the theory of functions “universal” with respect to interpolation operators from the univariate case introduced by Vogt [6,7] to the multivariate situation. Therefore, new arguments and different function spaces beyond mere polynomials are investigated to obtain functions f which have the following property: If one interpolates them in a certain sequence of nodal sets under certain conditions, the sequence of interpolating functions or some multiples of them allows uniform (or even stricter) approximation of any continuous function, or from a certain substantial subset of those, by a suitable subsequence.