Triangular function analysis

This paper concerns triangular function analysis including triangular funct ion series and triangular function transformation, which is very similar to Fourier analysis based on sine and cosine functions. Besides sine-cosine f unctions, triangular functions are frequently-used and easily-generated per iodic functions in electronics as well, so it is an urgent practical proble m to study the basic properties of triangular functions and the fundamental theory of triangular function analysis. We show that triangular functions and sine-cosine functions not only have the similar graphs, but also posses s similar analysis properties. Any continuous periodic function may be appr oximated uniformly by linear combinations of triangular functions as well a s trigonometric functions, and every function f(x) is an element of L-2[-pi ,pi] has a triangular function series as well as a Fourier series. Since th e triangular functions are nonorthogonal in L-2[-pi,pi], the orthonormaliza tion is discussed so that a function f(x) is an element of L-2[-pi,pi] can be approximated best by a superposition of given finite triangular function s. Finally, we introduce the theory of the triangular function transformati on in L-2(-infinity, infinity), which has a close relation with Fourier tra nsformation. These results form the theoretical foundation of the technique of triangular function analysis in modern electronics. (C) 1999 Elsevier S cience Ltd. All rights reserved.