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Landau and Kolmogoroff type polynomial inequalities

Journal article published in 1999 by Claudia Rr R. Alves, Dimitar K. Dimitrov ORCID
This paper is available in a repository.
This paper is available in a repository.

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Abstract

Let 0 < j < m n be integers. Denote by k k the norm kfk 2 = R 1 1 f 2 (x) exp( x 2 )dx: For various positive values of A and B we establish Kolmogoro type inequalities kf (j) k 2 Akf (m) k 2 +Bkfk 2 A k +B k ; with certain constants k e k ; which hold for every f 2 n (n denotes the space of real algebraic polynomials of degree not exceeding n). For the particular case j = 1 and m = 2, we provide a complete characterisation of the positive constants A and B, for which the corresponding Landau type polynomial inequalities kf 0 k 2 Akf 00 k 2 +Bkfk 2 A k +B k ; hold. In each case we determine the corresponding extremal polynomials for which equalities are attained. 1.