Triangular function analysis

Abstract

This paper concerns triangular function analysis including triangular function series and triangular function transformation, which is very similar to Fourier analysis based on sine and cosine functions. Besides sine-cosine functions, triangular functions are frequently-used and easily-generated periodic functions in electronics as well, so it is an urgent practical problem 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 possess similar analysis properties. Any continuous periodic function may be approximated uniformly by linear combinations of triangular functions as well as trigonometric functions, and every function f( x) ϵ L 2[− π, π] has a triangular function series as well as a Fourier series. Since the triangular functions are nonorthogonal in L 2[− π, π], the orthonormalization is discussed so that a function f( x) ϵ L 2[− π, π] can be approximated best by a superposition of given finite triangular functions. Finally, we introduce the theory of the triangular function transformation in L 2(−∞, ∞), which has a close relation with Fourier transformation. These results form the theoretical foundation of the technique of triangular function analysis in modern electronics.