Related papers: Hardy's theorem for the q-Bessel Fourier transform

An analytical result is given for the exact evaluation of an integral which arises in the analysis of acoustic radiation from wave packet sources: $ I_{mn}(\beta,q) = \int_{-\infty}^{\infty} e^{-\beta^{2}x^{2}-i q x}x^{m+1/2}J_{n+1/2}(x)…

In this paper we prove a noncommutative version of Hardy-Littlewood inequalities relating a function and its Fourier coefficients on the group $SU(2)$. As a consequence, we use it to obtain lower bounds for the $L^p-L^q$ norms of Fourier…

A q-type Holder condition on a function f is given in order to establish (uniform) convergence of the corresponding basic Fourier series S_q[f] to the function itself, on the set of points of the q-linear grid. Furthermore, by adding others…

New proofs of the classical Hermite-Hadamard inequality are presented and several applications are given, including Hadamard-type inequalities for the functions, whose derivatives have inflection points or whose derivatives are convex.…

It is proved that given $-1/2<s<1/2$, for any $f\in L^2(\mathbb{R})$, there is a unique $u\in \widehat{H}^{|s|}(\mathbb{R})$ such that $$ f=\boldsymbol{D}^{-s}u+\boldsymbol{D}^{s*}u\,, $$ where $\boldsymbol{D}^{-s}, \boldsymbol{D}^{s*}$ are…

For functions $f(x_{1},x_{2})=f_{0}\big(\max\{|x_{1}|,|x_{2}|\}\big)$ from $L_{1}(\mathbb{R}^{2})$, sufficient and necessary conditions for the belonging of their Fourier transform $\widehat{f}$ to $L_{1}(\mathbb{R}^{2})$ as well as of a…

We show that many classical results in Hardy space theory have exact analogues when the Fourier coefficients are allowed only to be real.

A q-version of the Fourier transformation and some of its properties are discussed.

This is primarily an overview article on some results and problems involving the classical Hardy function $$ Z(t) := \zeta(1/2+it){\bigl(\chi(1/2+it)\bigr)}^{-1/2}, \quad \zeta(s) = \chi(s)\zeta(1-s). $$ In particular, we discuss the first…

Let $C$ be a smooth projective curve defined over the finite field $\mathbb{F}_q$ ($q$ is odd) and let $K=\mathbb{F}_q(C)$ be its function field. Removing one closed point $C^\text{af} = C-\{\infty\}$ results in an integral domain…

An addition and product formula for the Hahn-Exton $q$-Bessel function, previously obtained by use of a quantum group theoretic interpretation, are proved analytically. A (formal) limit transition to the Graf addition formula and…

Recently, many surveys are devoted to study the Clifford Fourier transform. Dealing with the real Clifford Fourier transform introduced by Hitzer [10], we establish analogues of the classical Heisenberg's inequality and Hardy's theorem in…

We generalise a result of Hedenmalm to show that if a function $f$ on $\mathbb{R}$ is such that $\int_{\mathbb{R}^2} \bigl|f(x) \, \hat f(y)\bigr| \,e^{\lambda \left|xy\right|} \,dx\,dy = O( (1-\lambda)^{-N} )$ as $\lambda \to 1-$, then $f$…

Let T_t=e^{-tL} be a semigroup of self-adjoint linear operators acting on L^2(X,mu), where (X,d mu) is a space of homogeneous type. We assume that T_t has an integral kernel T_t(x,y) which satisfies the upper and lower Gaussian bounds:…

A Hardy inequality of the form \[\int_{\tilde{\Omega}} |\nabla f({\bf{x}})|^p d {\bf{x}} \ge (\frac{p-1}{p})^p \int_{\tilde{\Omega}} \{1 + a(\delta, \partial \tilde{\Omega})(\x)\}\frac{|f({\bf{x}})|^p}{\delta({\bf{x}})^p} d{\bf{x}}, \] for…

For each $f\!:\!\mathbb{R}\to\mathbb{C}$ that is Henstock--Kurzweil integrable on the real line, or is a distribution in the completion of the space of Henstock--Kurzweil integrable functions in the Alexiewicz norm, it is shown that the…

Let $M$ denote the centered Hardy--Littlewood operator on $\mathbb{R}$. We prove that \[ {\rm Var} (Mf)\le {\rm Var} (f) - \frac12\big| |f(\infty)|-|f(-\infty)|\big| \] for piecewise constant functions $f$ with nonzero and zero values…

Motivated by problems in control theory concerning decay rates for the damped wave equation $$w_{tt}(x,t) + \gamma(x) w_t(x,t) + (-\Delta + 1)^{s/2} w(x,t) = 0,$$ we consider an analogue of the classical Paneah-Logvinenko-Sereda theorem for…

In the paper the question - Is the q-Fourier transform of a q-Gaussian a q'-Gaussian (with some q') up to a constant factor? - is studied for the whole range of $q\in (-\infty, 3)$. This question is connected with applicability of the…

We prove a Hardy inequality on convex sets, for fractional Sobolev-Slobodecki\u{\i} spaces of order $(s,p)$. The proof is based on the fact that in a convex set the distance from the boundary is a superharmonic function, in a suitable…