\(k\)-Hankel Gabor transform and its applications to the theory of localization operators
After the introduction, in the second section, the authors expose known definition of the \(k\)-Hankel transform \((\mathcal{F}_kf)(x)\) and its properties, including Parseval's and Plancherel's equalities. Next they define the shift operator \(\tau^k_yf(x)\). The shift operator is used to define the \(k\)-convolution operator \[(f\ast_k\varphi)(x):=\frac1{2\Gamma(2k)}\int_{\mathbb{R}}(\tau^k_xf)(y) \varphi(y)\vert x\vert ^{2k-1}\,dx,\] \[h_{y,t}(y)=\tau^k_yh_t(x)=\tau^k_y\mathcal{F}_k\left(\sqrt{\tau_t^k(\vert h(x)\vert ^2)}\right)\] which is commutative and bounded in \(L^p(\mathbb{R},\vert x\vert ^{2k-1}dx)\), \(1\leqslant p\leqslant\infty\), provided that the kernel \(f\in L^1(\mathbb{R},\vert x\vert ^{2k-1}dx)\). The \(k\)-Hankel transform maps the convolution into a product \(\mathcal{F}_k(f\ast_kg)(x)=(\mathcal{F}_kf)(x)(\mathcal{F}_kg)(x)\).\par In the third section, the authors define the \(k\)-Hankel-Gabor transform \[(\mathcal{G}^k_hf)(y,t):=\frac1{2\Gamma(2k)}\int_{\mathbb{R}} f(x)h_{y,t}(x) \vert x\vert ^{2k-1}\,dx=(f\ast_kh_t)(y),\] where \(h_t:=\mathcal{F}_k\left(\sqrt{\tau_t^k(\vert h\vert ^2)}\right)\) is the modulation of \(h\) by \(t\) for a function \(h(x,y)\) from the space \(L^2(\mathbb{R}^2,\vert x\vert ^{2k-1}dx\vert y\vert ^{2k-1}dy)\) and any \(t\in\mathbb{R}\). Results similar to those derived in the foregoing section for the \(k\)-Hankel transform, are proved for the \(k\)-Hankel-Gabor transform.\par In Section~4, for measurable even functions on \(\mathbb{R}\) and a measurable function \(\sigma\) on \(\mathbb{R}^2\), the authors define the two-Gabor localization operator on \(L^2_k(\mathbb{R})\) by \[ (\mathcal{L}_{u,v}(\sigma)f)(y):=\frac1{2\Gamma(2k)}\int_{\mathbb{R}}\sigma(x,t) (\mathcal{G}^k_uf)(x,t)v_{x,t}(y)\vert x\vert ^{2k-1}\,dx\vert t\vert ^{2k-1}\,dt, \quad y\in\mathbb{R},\] where \(v_{x,t}(y)\) is defined as \(h_{y,t}(y)\) above.\par Further, the authors investigate the localization operators. In particular, they study trace class properties and prove that they belong to the Schatten-von Neumann class. They also study the eigenvalues and eigenfunctions of the time-frequency localization operator. Finally, some results on the spectrograms for the \(k\)-Hankel Gabor transform are exposed. Finally, some applications of these operators are given.
scientific article; zbMATH DE number 7393073
scientific article