Comparison of inequalities for the minimax series via Tchebychev polynomials
By usig some properties of the Chebychev polynomials \(\{T_n\}\), the authors compare some expressions (called \((\alpha,\beta)\)-inequalities), of the form: \[ S^{[a,b]}(fg)\leq\alpha\bigl(\|f\|_{\infty,[a,b]}S^{[a,b]}(g)+\|g\|_{\infty,[a,b]}S^{[a,b]}(f)\bigr)+\beta S^{[a,b]}(f)S^{[a,b]}(g)\tag{*} \] where \(\alpha\) and \(\beta\) are real constants and \(S^{[a,b]}(fg)\) stands for the minimax series of a product of two functions \(f\) and \(g\) belonging to the Besov space \(B^1_{\infty,1}[a,b]\) [see \textit{N. Hayek} and \textit{F. Pérez Acosta}, Rev. Acad. Canar. Cienc. 6, No. 1, 119-127 (1994; Zbl 0835.41010); \textit{F. Pérez Acosta}, ibid. 8, No. 1, 113-133 (1996; Zbl 0888.41004)]. For the particular values (3,0) and (2,2), they show that these \((\alpha,\beta)\)-inequalities are valid on all \(B^1_{\infty,1}[a,b]\), and in the case (0,6), (*) holds for the functions in \(B^1_{\infty,1}[a,b]\) having zeros on \([a,b]\).
scientific article; zbMATH DE number 1326377