Lin, Chung and Han introduced in 2014 the LCH-basis in order to derive FFT-based multipoint evaluation and interpolation algorithms with respect to this polynomial basis. Considering an affine space of n = 2j points, their algorithms require O(n · log2 n) operations in F2r. The LCH-basis has then been extended over finite fields of characteristic p by Lin et al. in 2016 and an n-point evaluation algorithm has been derived for n = pj with complexity O(n · logp n · p). However, the problem of interpolating polynomials represented in such a basis over FPr has not been addressed.

Discrete Fourier theory has been applied successfully in digital communication theory. In this correspondence, we prove a new inequality linking the number of nonzero components of a complex valued function defined on a finite Abelian group to the number of nonzero components of its Fourier transform. We characterize the functions achieving equality. Finally, we compare this inequality applied to Boolean functions to the inequality arising from the minimal distance property of Reed-Muller codes.