MathDB
analysis

Source: miklos schweitzer 1995 q3

October 5, 2021
real analysis

Problem Statement

Denote ⟨x⟩\langle x\rangle the distance of the real number x from the nearest integer. Let f be a linear, 1 periodic, continuous real function. Prove that there exist natural n and real numbers a1,...,an,b1,...,bn,c1,...,cna_1 , ..., a_n , b_1 , ..., b_n , c_1 , ..., c_n such that f(x)=∑i=1nci⟨aix+bi⟩f(x) = \sum_{i = 1}^n c_i \langle a_ix + b_i \rangle for every x iff there is a k such that ∑j=12kf(x+j2k)\sum_{j = 1}^{2^k} f \left(x+{j\over2^k}\right) is constant.
Back to ProblemsView on AoPS