ArticleOriginal scientific text
Title
Polynomial selections and separation by polynomials
Authors 1
Affiliations
- Department of Mathematics, Technical University of Łódź, Branch in Bielsko-Biała, Willowa 2, 43-309 Bielsko-Biała, Poland
Abstract
K. Nikodem and the present author proved in [3] a theorem concerning separation by affine functions. Our purpose is to generalize that result for polynomials. As a consequence we obtain two theorems on separation of an n-convex function from an n-concave function by a polynomial of degree at most n and a stability result of Hyers-Ulam type for polynomials.
Keywords
separation theorem, set-valued function, selection, n-convex function, n-concave function, affine function, Helly's theorem, Lagrange interpolating polynomial
Bibliography
- E. Behrends and K. Nikodem, A selection theorem of Helly type and its applications, Studia Math. 116 (1995), 43-48.
- Z. Ciesielski, Some properties of convex functions of higher orders, Ann. Polon. Math. 7 (1959), 1-7.
- K. Nikodem and S. Wąsowicz, A sandwich theorem and Hyers-Ulam stability of affine functions, Aequationes Math. 49 (1995), 160-164.
- T. Popoviciu, Les Fonctions Convexes, Hermann, Paris, 1944.
- T. Popoviciu, Sur quelques propriétés des fonctions d'une variable réelle convexes d'ordre supérieur, Mathematica (Cluj) 8 (1934), 1-85.
- A. W. Roberts and D. E. Varberg, Convex Functions, Academic Press, New York, 1973.
- F. A. Valentine, Convex Sets, McGraw-Hill, New York, 1964.
- S. Wąsowicz, On affine selections of set-valued functions, J. Appl. Anal. 1 (1995), 173-179.
Pages:
75-82
Main language of publication
English
Received
1996-01-16
Accepted
1996-04-15
Published
1996
Exact and natural sciences