Vanishing Fourier Transforms and Generalized Differences in $L^{2}(\mathbb{R})$

Abstract

Let $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}\in \mathbb{R}$ and $s\in \mathbb{N}$ be given. Let $\unicode[STIX]{x1D6FF}_{x}$ denote the Dirac measure at $x\in \mathbb{R}$, and let $\ast$ denote convolution. If $\unicode[STIX]{x1D707}$ is a measure, $\unicode[STIX]{x1D707}^{\star }$ is the measure that assigns to each Borel set $A$ the value $\overline{\unicode[STIX]{x1D707}(-A)}$. If $u\in \mathbb{R}$, we put $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},u}=e^{iu(\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D6FD})/2}\unicode[STIX]{x1D6FF}_{0}-e^{iu(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD})/2}\unicode[STIX]{x1D6FF}_{u}$. Then we call a function $g\in L^{2}(\mathbb{R})$ a generalized$(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})$-difference of order$2s$ if for some $u\in \mathbb{R}$ and $h\in L^{2}(\mathbb{R})$ we have $g=[\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},u}+\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},u}^{\star }]^{s}\ast h$. We denote by ${\mathcal{D}}_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},s}(\mathbb{R})$ the vector space of all functions $f$ in $L^{2}(\mathbb{R})$ such that $f$ is a finite sum of generalized $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})$-differences of order $2s$. It is shown that every function in ${\mathcal{D}}_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},s}(\mathbb{R})$ is a sum of $4s+1$ generalized $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})$-differences of order $2s$. Letting $\widehat{f}$ denote the Fourier transform of a function $f\in L^{2}(\mathbb{R})$, it is shown that $f\in {\mathcal{D}}_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},s}(\mathbb{R})$ if and only if $\widehat{f}$  “vanishes” near $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ at a rate comparable with $(x-\unicode[STIX]{x1D6FC})^{2s}(x-\unicode[STIX]{x1D6FD})^{2s}$. In fact, ${\mathcal{D}}_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},s}(\mathbb{R})$ is a Hilbert space where the inner product of functions $f$ and $g$ is $\int _{-\infty }^{\infty }(1+(x-\unicode[STIX]{x1D6FC})^{-2s}(x-\unicode[STIX]{x1D6FD})^{-2s})\widehat{f}(x)\overline{\widehat{g}(x)}\,dx$. Letting $D$ denote differentiation, and letting $I$ denote the identity operator, the operator $(D^{2}-i(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD})D-\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}I)^{s}$ is bounded with multiplier $(-1)^{s}(x-\unicode[STIX]{x1D6FC})^{s}(x-\unicode[STIX]{x1D6FD})^{s}$, and the Sobolev subspace of $L^{2}(\mathbb{R})$ of order $2s$ can be given a norm equivalent to the usual one so that $(D^{2}-i(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD})D-\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}I)^{s}$ becomes an isometry onto the Hilbert space ${\mathcal{D}}_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},s}(\mathbb{R})$. So a space ${\mathcal{D}}_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},s}(\mathbb{R})$ may be regarded as a type of Sobolev space having a negative index.