Related papers: The Minkowski ?(x) function and Salem's problem
The present article is devoted to the description of further investigations of the author of this article. These investigations (in terms of various representations of real numbers) include the generalized Salem functions and…
The asymptotic study of a time-dependent function $f$ as the solution of a differential equation often leads to the question of whether its derivative $\dot f$ vanishes at infinity. We show that a necessary and sufficient condition for this…
Fourier transforms of Lorentz invariant functions in Minkowski space, with support on both the timelike and the spacelike domains are performed by means of direct integration. The cases of 1+1 and 1+2 dimensions are worked out in detail,…
We extend the equivalence of the Salem type for the Riemann hypothesis by application of Titchmarsh's theorem. Other equivalences to the Riemann hypothesis and notes on related Fourier integrals are provided.
The purpose of this paper is to investigate the distribution of zeros of entire functions which can be represented as the Fourier transforms of certain admissible kernels. The principal results bring to light the intimate connection between…
We investigate an extension of an equilibrium-type result, conjectured by Ambrus, Ball and Erd\'elyi, and proved recently by Hardin, Kendall and Saff. These results were formulated on the torus, hence we also work on the torus, but one of…
The Cancellation Problem for Affine Spaces is settled affirmatively, that is, it is proved that : Let $ k $ be an algebraically closed field of characteristic zero and let $n, m \in \mathbb{N}$. If $R[Y_1,..., Y_m] \cong_k k[X_1,...,…
There have been many proposed forms of fractional calculus, which can be grouped into a few broad classes of operators. By replacing the kernel of the power function with another kernel function, the traditional Riemann-Liouville formula…
In this paper we investigate the existence of positive solution for a class of quasilinear problem on an Orlicz-Sobolev space that can be nonreflexive $$- \Delta_{\Phi} u +V(x)\phi(|u|)u= K(x)f(u)\mbox{ in } \mathbb{R}^{N}$$ where $N\geq2$,…
A generalization of the Zernike circle polynomials for expansion of functions vanishing outside the unit disk is given. These generalized Zernike functions have the form Zm,{\alpha} n ({\rho}, \vartheta) = Rm,{\alpha} n ({\rho})…
The Minkowski function is a crucial tool used in the study of balanced domains and, more generally, quasi-balanced domains in several complex variables. If a quasi-balanced domain is bounded and pseudoconvex then it is well-known that its…
We extend results of Andrews and Bressoud on the vanishing of coefficients in the series expansions of certain infinite products. These results have the form that if \begin{equation*} \frac{(q^{r-tk}, q^{mk-(r-tk)};…
The squared eigenfunctions of the spectral problem associated to the Camassa-Holm equation represent a complete basis of functions, which helps to describe the Inverse Scattering Transform for the Camassa-Holm hierarchy as a Generalised…
We revisit the Fourier transform of a Hankel function, of considerable importance in the theory of knife edge diffraction. Our approach is based directly upon the underlying Bessel equation, which admits manipulation into an alternate…
A negative solution of Problem 188 posed by Max Eidelheit in {\it the Scottish Book} concerning superpositions of separately absolutely continuous functions is presented. We discuss here his and some related problems which have also…
The classical Minkowski problem in Minkowski space asks, for a positive function $\phi$ on $\mathbb{H}^d$, for a convex set $K$ in Minkowski space with $C^2$ space-like boundary $S$, such that $\phi(\eta)^{-1}$ is the Gauss--Kronecker…
The subject of this paper is the history of the Minkowski-Funk Transform. After introducing the Minkowski-Funk Transform as well as its dual transform and a generalization of both, we will present an inversion formula of the Minkowski-Funk…
We study both analytically and numerically a coupled system of spherically symmetric SU(2) Yang-Mills-dilaton equation in 3+1 Minkowski space-time. It has been found that the system admits a hidden scale invariance which becomes transparent…
In the present paper by the Fourier transform we show that every linear differential equations of $n$-th order has a solution in $L^1(\Bbb{R})$ which is infinitely differentiable in $\Bbb{R} \setminus \{0\}$. Moreover the Hyers-Ulam…
In this paper, we show the existence and multiplicity of nontrivial, non-negative solutions of the fractional $p$-Kirchhoff problem \begin{equation*} \begin{array}{rllll}…