Dual of 2D fractional Fourier transform associated to Itô--Hermite polynomials

Wavelet frames, Bergman spaces and Fourier transforms of Laguerre functions

The Fourier transforms of Laguerre functions play the same canonical role in wavelet analysis as do the Hermite functions in Gabor analysis. We will use them as analyzing wavelets in a similar way the Hermite functions were recently by K. Groechenig and Y. Lyubarskii in "Gabor frames with Hermite functions, C. R. Acad. Sci. Paris, Ser. I 344 157-162 (2007)". Building on the work of K. Seip, "Beurling type density theorems in the unit disc, Invent. Math., 113, 21-39 (1993)", concerning sampling sequences on weighted Bergman spaces, we find a sufficient density condition for constructing frames …

具有加权Hardy类函数的Hermite展开

Hermite expansions of functions from the weighted Hardy class

本文研究了具有加权Hardy类函数的Hermite展开，通过分析Log型权重和指数型权重，证明了Hermite系数衰减估计及谐振子解的衰减率，并改进了亚临界Hardy不确定性原理。

关于 Hardy 类函数的 Hermite 系数的成对相关性

On pair correlation of Hermite coefficients of functions from the Hardy class

本文研究 Hardy 类函数的 Hermite 系数成对相关性，在函数及其 Fourier 变换受 Gauss 函数控制的条件下，证明特定 Hermite 系数组合可获得优于已知精确估计的衰减率。

On bicomplex Fourier--Wigner transforms

We consider the $1$- and $2$-d bicomplex analogs of the classical Fourier--Wigner transform. Their basic properties, including Moyal's identity and characterization of their ranges giving rise to new bicomplex--polyanalytic functional spaces are discussed. Particular case of special window is also considered. An orthogonal basis for the space of bicomplex--valued square integrable functions on the bicomplex numbers is constructed by means of the polyanalytic complex Hermite functions.

Bivariate poly-analytic Hermite polynomials

A new class of bivariate poly-analytic Hermite polynomials is considered. We show that they are realizable as the Fourier-Wigner transform of the univariate complex Hermite functions and form a nontrivial orthogonal basis of the classical Hilbert space on the two-complex space with respect to the Gaussian measure. Their basic properties are discussed, such as their three term recurrence relations, operational realizations and differential equations (Bochner's property) they obey. Different generating functions of exponential type are obtained. Integral and exponential operational representatio…

Bicomplex analogs of Segal-Bargmann and fractional Fourier transforms

We consider and discuss some basic properties of the bicomplex analog of the classical Bargmann space. The explicit expression of the integral operator connecting the complex and bicomplex Bargmann spaces is also given. The corresponding bicomplex Segal--Bargmann transform is introduced and studied as well. Its explicit expression as well as the one of its inverse are then used to introduce a class of two--parameter bicomplex Fourier transforms (bicomplex fractional Fourier transform). This approach is convenient in exploring some useful properties of this bicomplex fractional Fourier transfor…