Multiplicative Characteristic Functions: Calculus and Applications

Abstract Very often, in Mathematics, we are dealing with objects which are decomposed into simpler objects. In this paper, we introduce the concept of “multiplicative characteristic function” on arbitrary set X and we give a general representation theorem of any function $$f:X\rightarrow [0,1]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mi>X</mml:mi> <mml:mo>→</mml:mo> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> as a uniform limit of finite products of such type of functions. This result may be fruitfully used in measure theory, distribution theory and so on. Some consequences in uniform approximation of continuous functions are presented here.