Hankel Transform of the Type 2 (p,q)-Analogue of r-Dowling Numbers

In this paper, type 2 (p,q)-analogues of the r-Whitney numbers of the second kind is defined and a combinatorial interpretation in the context of the A-tableaux is given. Moreover, some convolution-type identities, which are useful in deriving the Hankel transform of the type 2 (p,q)-analogue of the r-Whitney numbers of the second kind are obtained. Finally, the Hankel transform of the type 2 (p,q)-analogue of the r-Dowling numbers are established.

On the Generalization of the Random Projection Method for Problems of the Recovery of Object Signal Described by Models of Convolution Type

Introduction. In technical systems, there is a common situation when transformation input-output is described by the integral equation of convolution type. This situation accurses if the object signal is recovered by the results of remote measurements. For example, in spectrometric tasks, for an image deblurring, etc. Matrices of the discrete representation for the output signal and the kernel of convolution are known. We need to find a matrix of the discrete representation of a signal of the object. The well known approach for solving this problem includes the next steps. First, the kernel matrix has to be represented as the Kroneker product. Second, the input-output transformation has to be presented with the usage of Kroneker product matrices. Third, the matrix of the discrete representation of the object has to be found. The object signal matrix estimation obtained with the help of pseudo inverting of Kroneker decomposition matrices is unstable. The instability of the object signal estimation in the case of usage of Kroneker decomposition matrices is caused by their discrete ill posed matrix properties (condition number is big and the series of the singular numbers smoothly decrease to zero). To find solutions of discrete ill-posed problems we developed methods based on the random projection and the random projection with an averaging by the random matrices. These methods provide a stable solutions with a small computational complexity. We consider the problem of object signals recovering in the systems where an input-output transformation is described by the integral equation of a convolution. To find a solution for these problems we need to build a generalization for two-dimensional signals case of the random projection method. Purpose. To develop a stable method of the recovery of object signal for the case in which an input-output transformation is described by the integral equation of a convolution. Results and conclusions. We developed the method of a stable recovery of object signal for the case in which an input-output transformation is described by the integral equation of a convolution. The stable estimation of the object signal is provided by Kroneker decomposition of the kernel matrix of convolution, computation of random projections for Kroneker factorization matrices, and a selection of the optimal dimension of a projector matrix. The method is illustrated by its application in technical problems.

A note on convolution operators on Riesz Bounded variation spaces

We show some estimates and approximation results of operators of convolution type defined on Riesz Bounded variation spaces in Rn. We also state some embedding results that involve the collection of generalized absolutely continuous functions.

Semi-Hyers–Ulam–Rassias Stability of a Volterra Integro-Differential Equation of Order I with a Convolution Type Kernel via Laplace Transform

In this paper, we investigate the semi-Hyers–Ulam–Rassias stability of a Volterra integro-differential equation of order I with a convolution type kernel. To this purpose the Laplace transform is used. The results obtained show that the stability holds for problems formulated with various functions: exponential and polynomial functions. An important aspect that appears in the form of the studied equation is the symmetry of the convolution product.

Two Forms of An Inverse Operator to the Generalized Bessel Potential

In this paper, we treat a convolution-type operator called the generalized Bessel potential. Our main result is the derivation of two different forms of its inversion. The first inversion is provided in terms of an approximative inverse operator using the method of an improving multiplier. The second one employs the regularization technique for the divergent integrals in the form of the appropriate segments of the Taylor–Delsarte series.