R-, Atomic and WA-system Functions Theory in Physical Problems

Abstract

Review is concerned on results based on the works of outstanding Soviet mathematician academician of the National Academy of Sciences of Ukraine V. L. Rvachev. In 1960-70 in Kharkiv he worked on numerical solution of problems in mechanics. During this work Rvachev laid foundation of R-function theory. It presents mathematical apparatus for solution of inverse problem of analytical geometry by means of Boolean algebra. In particular it allows construction of equations which describe complex geometrical figures. It is widely applied in boundary value problems solution, numerical modeling of electromagnetic and thermal fields and in image processing. Another great invention of V. L. Rvachev is atomic function theory. Note 50-anniversary from publication of the first paper describing atomic function in 1971. Atomic functions are solutions of functional-differential equations of special form and may be represented as infinite convolutions of rectangular (in simplest case) pulses of decreasing length. They are both infinitely smooth and compactly supported. This unique property makes them widely applicable as basis and weight functions. We note some specific applications of atomic functions. Generalization of sampling theorem based on atomic functions was presented by V. F. Kravchenko. Generalized Kravchenko-Kotelnikov series have fast convergence and small truncation error. Family of wavelets based on scaled sums of shifts of atomic functions was presented by D. V. Churikov and A. V. Yurin. These wavelets are also called WA-systems. Recent work performed by K. A. Budunova is concerned on construction of FIR and IIR digital and analogue filters with magnitude response approximating atomic functions. In the review fundamental and applied works which develop generalize and apply V. L. Rvachev’s ideas of R- and atomic function theory are discussed.