The arithmetic fourier transform and the visual pathway

The Arithmetic Fourier Transform is an algorithm for the computation of Fourier coefficients. In this paper it is extended to the computation of double Fourier coefficients by a repeated sum. We discuss the relevance of the repeated sum algorithm to signal processing by neurons in the visual pathway.     

Heat Transfer in a Region with Non-Uniform Boundary Conditions

Heat transfer in a rectangular region with non-uniform conditions on the walls is considered. The temperature is given on both vertical walls and a part of the upper wall. The remainder of the upper wall and the lower horizontal wall are perfectly insulated. This boundary value problem is reduced to dual Fourier series equations. Those equations are simplified under the assumption that the height of the region is greater than the length or comparable to it. An exact solution of the simplified equations is constructed by using the Schwinger transformation, which has been used successfully in analyzing the electro-dynamics of wave guides. Numerical solutions also are found using a commercial finite element solver and a finite difference solver written in FORTRAN. Results for the average temperature and the temperature distribution in the region for a variety of high temperature boundary locations are in very good agreement among the three solution techniques.     

On the norm of the Lp-Dunkl transform

In this work, we establish a Babenko–Beckner-type inequality for the Dunkl transform of a radial function and Dunkl transform associated to a finite reflection group generated by the sign changes. As applications, we establish a new version of Young’s-type inequality and uncertainty relation for Renyi entropy.     

Differential Equations of Fractional Order: Methods,Results and Problems. II

The article, being a continuation of the first one [A.A. Kilbas and J.J. Trujillo (2001). Differential equations of fractional order. Methods, results and problems, I. Applicable Analysis , 78 (1-2), 153-192.], deals with the so-called differential equations of fractional order in which an unknown function is contained under the operation of a derivative of fractional order. The methods and the results in the theory of such fractional differential equations are presented including the Dirichlet-type problem for ordinary fractional differential equations, studying such equations in spaces of generalized functions, partial fractional differential equations and more general abstract equations, and treatment of numerical methods for ordinary and partial fractional differential equations. Problems and new trends of research are discussed.     

Pseudo-differential operators involving Watson transform

A class of pseudo-differential operators (p.d.o.'s) associated with the general Fourier kernel studied by Hardy and Titchmarsh is defined. A symbol class T ^m is introduced. It is shown that the p.d.o.'s associated with the symbol are continuous linear mappings of the Braaksma and Schuitman space T(λ,μ) into itself. An integral representation of p.d.o. is obtained. Some special forms of the symbol are considered. It is shown that these p.d.o.'s and their products are bounded in certain Sobolev type space.     

On a general class of trigonometric functions and Fourier series†

We discuss a general class of trigonometric functions whose corresponding Fourier series can be used to calculate several interesting numerical series. Particular cases are presented.