Two-dimensional dirac delta reconsidered

Distribution theory continues to be of significant importance in many branches of applied mathematics and especially within the research activities of theoretical and applied physicists. It is the belief of the author that the Dirac delta functional offers enormous impact in fostering advances within distribution theory together with its applications. Whenever one requires an example of a singular, ultra or new generalized function, a version of the Dirac delta satisfies that need. In this paper we have collected several very recent and important results for the Dirac delta and formulated them within a two-dimensional domain. We then go on and graph a three-dimensional version of the result implementing the software, Pro-Matlab. Within many branches of signal analysis the geometrical aspects of a particular mathematical concept are of paramount importance to the user. For example, when one implements a transform as a filter, the geometrical considerations give strong evidence of the utility of the filter for the particular application. We have also included a preliminary beginning for considering wavelet transforms applied to distributions.     

On best rational approximation of analytic functions

This paper contains some theorems related to the best approximation ρ_n(f;E) to a function f in the uniform metric on a compact set by rational functions of degree at most n. We obtain results characterizing the relationship between ρ_n(f;K) and ρ_n(f;E) in the case when complements of compact sets K and E are connected, K is a subset of the interior Ω of E, and f is analytic in Ω and continuous on E.     

The topology of the spaces of regular and absolutely regular Hankel‐transformable distributions

New topologies on the spaces of regular and absolutely regular Hankel‐transformable distributions are proposed and compared. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Scattering Matrix Method for Modeling Defect Modes in 2-D Photonic Crystals

A novel and numerically efficient treatment of electromagnetic modes localized at defects in two-dimension (2D) photonic crystals is presented in this paper. The method represents the fields in terms of scattered fields by each column of the photonic crystals. With the method, the field distributions in two photonic crystal structures are calculated with satisfying results.     

A Note on the Operator Equation Generalizing the Notion of Slant Hankel Operators

The operator equation λM_zX = XM_(zk), for k ≥ 2, λ∈ C, is completely solved.Further, some algebraic and spectral properties of the solutions of the equation are discussed.     

Integrals of Lipschitz–Hankel type,Legendre functions and table errata

The complete Lipschitz–Hankel integrals (LHIs) include the Laplace transforms of the Bessel functions, multiplied by powers. Such Laplace transforms can be evaluated using associated Legendre functions. It is noted that there are errors in published versions of these evaluations, and a merged and emended list of seven transforms is given. Errata for standard reference works, such as the table of Gradshteyn and Ryzhik, are also given. Most of the errors are attributable to inconsistent normalization of the Legendre functions. These transforms can be viewed as limits of incomplete LHIs, which find application in communication theory.     

Hankel transforms of generalized Motzkin numbers

We study Hankel transform of the sequences (u,l,d),t, and the classical Motzkin numbers. Using the method based on orthogonal polynomials, we give closed‐form evaluations of the Hankel transform of the aforementioned sequences, sums of two consecutive, and shifted sequences. We also show that these sequences satisfy some interesting convolutional properties. Finally, we partially consider the Hankel transform evaluation of the sums of two consecutive shifted (u,l,d)‐Motzkin numbers. Copyright © 2017 John Wiley & Sons, Ltd.     

Trace class criteria for bilinear Hankel forms of higher weights

In this paper we give a complete characterization of higher weight Hankel forms, on the unit ball of , of Schatten-von Neumann class , . For this purpose we give an atomic decomposition for certain Besov-type spaces. The main result is then obtained by combining the decomposition and our earlier results.

     

Hausdorff moment problem and Maximum Entropy: On the existence conditions

Different existence conditions of the Maximum Entropy solution to finite Hausdorff moment problem have been formulated in literature. Through a counterexample we prove that the most cited one is uncorrect. We do not bound ourselves to a crude counterexample, as we think that a detailed explanation is of interest by itself. It clarifies the difference existing between the finite and infinite Hausdorff moment problem existence conditions.