Narrow Results

A time and space inhomogeneous Markov process is a Feller evolution process, if the corresponding evolution system on the continuous functions vanishing at infinity is strongly continuous. We discuss generators of such systems and show that under mild conditions on the generators a Feller evolution can be approximated by Markov chains with Lévy increments.

The result is based on the approximation of the time homogeneous spacetime process corresponding to a Feller evolution process. In particular, we show that a d-dimensional Feller evolution corresponds to a (d + 1)-dimensional Feller process. It is remarkable that, in general, this Feller process has a generator with discontinuous symbol.

Refinable functions play an important role in the construction and properties of wavelets. Basically, most of the wavelets are generated from refinable functions. In this paper, a study on the approximation properties of refinable functions on $p$-adic fields is carried out with necessary theoretical background. Various equivalent forms of approximation order and the connection between the approximation order and the Strang–Fix condition are derived. Finally a characterization for the approximation order of a refinable function is given in terms of order of the sum rules associated with the refinement mask.

We establish several a priori estimates of local or global nature in Sobolev spaces with general exponent $s\le 0$ for a class of second-order hyperbolic operators with double characteristics in presence of a transition in a domain of the Euclidian space ${ℝ}^3$.

Pseudo-differential operator (p.d.o) associated with the symbol a(x, y) whose derivatives satisfy certain growth condition is defined and the Zemanian-type spaces H_μ(I) and S(I) are introduced. It is shown that the p.d.o is continuous linear map of the space H_μ(I) and S(I) into itself. An integral representation of p.d.o h_{1, μ, a} is obtained. Using the Hankel convolution it is shown that p.d.o h_{1, μ, a} satisfies a certain -norm inequality. Properties of Sobolev-type space are studied.

Using the inverse of Fourier–Jacobi transform a symbol is defined, and the pseudo-differential operator (p.d.o.) 𝒫_{α, β} (x,D) associated with Jacobi-differential operator in terms of this symbol is defined. It is shown that the p.d.o. is bounded in a certain Sobolev type space associated with the Fourier–Jacobi transform. Continuous Jacobi wavelet transform (JWT) and Fourier-cosine wavelet transform are defined and a reconstruction formula is obtained for Fourier-cosine wavelet transform. Properties of Fourier-cosine wavelet transform are investigated.

Pseudo-differential operators (p.d.os) involving generalized Hankel–Clifford transformation associated with the symbol $a(x,y)$ whose derivatives satisfy certain growth condition are defined and the Zemanian type function spaces ${ℍ}_{\beta }(I)$ and ${𝕊}_{\alpha }(I)$ are introduced. It is shown that p.d.o’s are continuous linear map of the space ${ℍ}_{\beta }(I)$ and ${𝕊}_{\alpha }(I)$ into itself. Also an Integral representation of p.d.o is obtained.

The main goal of this paper is to study properties of the linear canonical transform (LCT) on Schwartz-type space ${{𝒮}}_A(ℝ)$. The symbol class ${{𝒮}}_{\rho ,\sigma }^{m,A}$ is introduced. The pseudo-differential operator (p.d.o.) involving LCT is defined and also some of its properties including boundedness are investigated in Sobolev-type space. Kernel and integral representation of p.d.o. are obtained. Some applications of LCT to generalized partial differential equations and canonical convolution integral equation have been solved.