Modular parts of a function

摘要

An application of finite Fourier series, modular parts generalize the even-odd partition of a function. The sum of all the modular parts of a function f is again f, and each modular part displays argumentation: fn⧹p(xi4m⧸p)=i4mn⧸pfn⧹p(x). Moreover, if a function argumentates, it is a modular part. If f is analytic and f(x)=∑ajxj, then fn⧹p(x)=∑ajp+nxjp+n. Other elementary properties and a procedure for calculating (fm⧹p)n⧹q are derived. A countermodular part fn∨p involving the Musèan hypernumber ε is defined which has the property fn∨p=[(1−ε)/2]fn⧹p+[(1+ε)/2]f.