Periodic and quasi-periodic functions

Abstract

Let f(ø) =f₁(ø),…,f_n(ø)) be an ℝⁿ-valued function of a variable ø = (ø₁,…, ø _m ) that is continuous and periodic with period 2π with respect to each variable ø_α(α = 1,…, m). The set of all such functions forms a linear space which we denote by C(T _m ).This space can be turned into a complete normed space by introducing the norm

$$|f\left( \phi \right){{|}_{r}} = \mathop{{\max }}\limits_{{0 \leqslant \rho \leqslant r}} |{{D}^{\rho }}f\left( \phi \right){{|}_{0}}$$

where

$$ \parallel f{\parallel ^2} = \sum\limits_{i = 1}^n | {f_i}{|^2} $$

is the Euclidean norm in the space ℝⁿ of the function f(ø).