Consider generalized adapted stochastic integrals with respect to independently scattered random measures with second moments. We use a decoupling technique, known as the “principle of conditioning”, to study their stable convergence towards mixtures of infinitely divisible distributions. Our results apply, in particular, to multiple integrals with respect to independently scattered and square ...

After reviewing the theory of triangular (causal) and rectangular quantum stochastic double product integrals, we consider examples when these consist of unitary operators. We find an explicit form for all such rectangular product integrals which can be described as second quantizations. Causal products are proposed as paradigm limits of large random matrices in which the randomness is explicit...

In this paper, we derive a new class of finite-dimensional filters for integrals and stochastic integrals of moments of the state for continuous-time linear Gaussian systems. Apart from being of significant mathematical interest, these new filters can be used with the expectation maximization (EM) algorithm to yield maximum likelihood estimates of the model parameters.

For a family of weight functions, hκ, invariant under a finite reflection group on R, analysis related to the Dunkl transform is carried out for the weighted L spaces. The generalized translation operator and the weighted convolution are studied in detail in L(R, h2κ) and the result is used to study the summability of the inverse Dunkl transform, including the Poisson integrals and the Bochner-...

For a family of weight functions, hκ, invariant under a finite reflection group on R, analysis related to the Dunkl transform is carried out for the weighted L spaces. Making use of the generalized translation operator and the weighted convolution, we study the summability of the inverse Dunkl transform, including as examples the Poisson integrals and the Bochner-Riesz means. We also define a m...

Based on a mathematical model involving Radon measure explicit computations on convolution integrals defining continuous (integral) wavelet transformations are carried out. The study shows that the truncated Morlet wavelet significantly depends on a rotation parameter and thus lay a foundation of edge detection in pattern recognition and image processing using rotational (directional) wavelets....

The algebraic theory of double product integrals and particularly its role in the quantisation of Lie bialgebras is described. When the underlying associative algebra is that of the Itô differentials of quantum stochastic calculus such product integrals are formally represented as operators which are infinite sums of iterated integrals in Fock space. In this paper we describe some of the analyt...

An extension of the general fractional calculus (GFC) is proposed as a generalization Riesz calculus, which was suggested by Marsel in 1949. The form GFC can be considered an from positive real line and Laplace convolution to m-dimensional Euclidean space Fourier convolution. To formulate form, Luchko approach construction GFC, Yuri 2021, used. integrals derivatives are defined convolution-type...

A solution to the more than 300-years old problem of geometric and physical interpretation of fractional integration and differentiation (i.e., integration and differentiation of an arbitrary real order) is suggested for the Riemann-Liouville fractional integration and differentiation, the Caputo fractional differentiation, the Riesz potential, and the Feller potential. It is also generalized f...

Using a special case of the Efros theorem which was derived by Wlodarski, and operational calculus, it possible to derive many infinite integrals, finite integrals integral identities for function represented inverse Laplace transform. The are mainly in terms convolution with Mittag–Leffler Volterra functions. integrands determined include elementary functions (power, exponential, logarithmic, ...