Narrow Results

In this paper, utilizing the theory of Watson transform and Watson convolution, we explore the Watson wavelet convolution product and its related properties. The relation between the Watson Wavelet convolution product and Watson convolution is also computed. Watson wavelet transform and its inversion formula are analyzed heuristically. Watson two-wavelet multipliers and its trace class are derived from Watson wavelet convolution product

We derive some characteristic properties of the convolution operator acting on white noise functions and prove that the convolution product of white noise distributions coincides with their Wick product. Moreover, we show that the S-transform and the Laplace transform coincide on Fock space.

We investigate the stochastic process defined as the square of the (integrated) symmetric telegraph process. Specifically, we obtain its probability law and a closed form expression of the moment generating function. Some results on the first-passage time through a fixed positive level are also provided. Moreover, we analyze some functionals $\mathrm{\Phi }(\cdot ,\cdot )$ of two independent squared telegraph processes, both in the case $\mathrm{\Phi }(u,v)=u+v$ and $\mathrm{\Phi }(u,v)=u\cdot v$. Starting from this study, we provide some results on the probability density functions of the two-dimensional radial telegraph process and of the product of two independent symmetric telegraph processes. Some of the expressions obtained are given in terms of new results about derivatives of hypergeometric functions with respect to parameters.

The paper deals with generalized truncated integral convolutions. We study the spectral properties of the operator family with the symbol defined on [0, 1] × [0, 1] × ℝ. The operator for τ > 0 is defined as

Based on nuclear algebra of entire functions, we extend some results about operator-parameter transforms involving the Fourier-Gauss and Fourier-Mehler transforms. We investigate the solution of a initial-value problem associated to infinitesimal generators of these transformations. In particular, by using convolution product, we show to what extent regularity properties can be performed on our setting.

Unitary integral transforms play an important role in mathematical physics. A primary example is the Fourier transform whose kernel is of the form k(x,y) = k(xy), i.e., of the product type. Here we consider the determination of spectrum for such unitary operators as the issue is important in the solvability of the corresponding inhomogeneous Fredholm integral equation of the second kind. A Main Theorem is proven that characterizes the spectral set. Properties of eigenfunctions and eigenspace dimensions are further derived as consequences of the Main Theorem. Concrete examples are also offered as applications.