Some Inequalities for the Coefficients in Generalized Fourier Expansions

Abstract

In this paper, we derive inequalities for the coefficients in generalized Fourier expansions of (m, n) convex functions in the sense of Popoviciu. Classical Fourier expansions as well as expansions relative to orthogonal polynomials are considered. The results presented here generalize the ones obtained by Niculescu and Rovenţa (Positivity 24(1):129–139, 2020). Some of the results obtained in deriving inequalities for these coefficients can be further used in obtaining Favard-type inequalities similar to the ones given in Wulbert (Math Comput Model 37(12–13):1383–1391, 2003). Favard type inequalities can be used in obtaining probabilistic inequalities which may be further used in fields such as statistical machine learning.