Atomic functions: Generalization to the multivariable case and promising applications

Abstract

517.95Main trends in the theory of atomic functions are outlined. These functions have been studied for more than 35 years and gave rise to new research areas in mathematical analysis, approximation theory, numerical methods, etc. Atomic functions are infinitely differentiable solutions with compact support to special functional-differential equations.Atomic functions whose investigation for the case of one independent variable has received significant attention until quite recently are infinitely differentiable compactly supported solutions to functional-differential equations of the formwhere L is a linear differential operator with constant coefficients. Atomic functions have been intensively investigated for more than 35 years (see [1] and reviews in [2-7]), and some new scientific directions arise in which typical properties inherent in atomic functions are necessarily required in theoretical and practical developments. Atomic functions and spaces generated by them are used in mathematical analysis [2,4,6], approximation theory [4, 5], in solving differential equations and boundary problems of mathematical physics [2, 3, 11, 15-21, 44, 48, 80], in numerical methods and other fields of applied mathematics [22, 46-47, 59-62], and also in technical realizations of bell-shaped functions [30, 31], digital signal processing [34-37, 41, 50-52], analysis and synthesis of antennas [49-52], etc. The history of atomic functions begins with [1], which was published by V. L. Rvachev and V. A. Rvachev in 1971 and in which, in particular, they used the method of contracting mappings to prove the existence and uniqueness of a compactly supported solution of the equation y x y x y x ¢ = + --( ) ( ) ( ) 2 2 1 2 2 1 under the conditions y ( ) 0 1 = and supp y x ( ) [ , ] = -1 1 . This problem was stated by V. L. Rvachev as long ago as in 1967 at a seminar on problems of applied mathematics, and its solution was connected with the advent of the most important atomic function up x ( ), up x itx t t dt k k k ( ) exp ( ) sin ( ) = -¥ ¥ = ¥ --ò Õ 1 2 2 2 1 p .The term "atomic function" was introduced in [2] in 1975. The paper considered the application of atomic functions in conjunction with the mathematical tools of the theory of R-functions in variational methods for solving problems of mathematical physics. In some works of foreign authors [9][10][11][12][13][14][15][16][17][18][19][20][21][22], the atomic function up x ( )also gained ground as "the Rvachev function." 893 1060-0396/07/4306-0893

Atomic Spacetime Model Based on Atomic AString Functions

“…Interestingly, Atomic AString Functions can be not only introduced into GR but deduced from GR to uphold A. Einstein's ideas of finite "regions of space" with "discrete energies" [1] [23]. The Atomization Theorems are not limited to spacetime but can also be applied to many physical theories including Quantum Mechanics, electromagnetism, elasticity, heat conductivity, soliton theories, and field theories [7]- [12] [22]- [27] [29]- [50]. Unified representation of fields composed of Atomic AString Functions may offer some novel variants of unified theory based on Atomic Solitons [2] [3] [4] [5] [23] [42] where, like in string theory, fields become interconnected having a common mathematical ancestor.…”

The Class of Atomic Exponential Basis Functions EFupn(x,ω)-Development and Application

“…Gotovac et al [15] showed the basic possibilities of using atomic functions in structural mechanics and numerical analysis. The work in [16] gives a generalization of atomic functions to the multivariable case. The use of Fup basis functions, which are atomic functions of the algebraic type, has been shown to solve the problem of signal processing [17], the initial value problem [18], the boundary value problems using the Fup Collocation Method [19], the boundary-initial value problems [20], elasto-plastic analysis of prismatic bars subjected to torsion [21], and modeling of groundwater flow and transport problems [22].…”

Maximum entropy algorithm with inexact upper entropy bound based on Fup basis functions with compact support

“…However, in practice, their application as basis functions is closer to splines or wavelets. In this study, we shall use Fup basis functions, which are one type of atomic basis functions; recent review in [12]. Gotovac and Kozulić [8] systemized the existing knowledge on atomic functions and presented the transformation of basis functions into a numerically applicable form.…”