Polynomial selections and separation by polynomials

• Autorzy: Szymon Wąsowicz
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

separation theorem; set-valued function; selection; n-convex function; n-concave function; affine function; Helly's theorem; Lagrange interpolating polynomial

Kategorie tematyczne

• 52A35: Helly-type theorems and geometric transversal theory
• 26D07: Inequalities involving other types of functions
• 26A51: Convexity, generalizations
• 26E25: Set-valued functions
• 54C65: Selections
• 39B72: Systems of functional equations and inequalities

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1996
• Tom: 120
• Numer: 1
• Strony: 75-82

Daty

wydano

1996

otrzymano

1996-01-16

poprawiono

1996-04-15

Twórcy

autor

Szymon Wąsowicz

• Department of Mathematics, Technical University of Łódź, Branch in Bielsko-Biała, Willowa 2, 43-309 Bielsko-Biała, Poland

Bibliografia

• [1] E. Behrends and K. Nikodem, A selection theorem of Helly type and its applications, Studia Math. 116 (1995), 43-48.
• [2] Z. Ciesielski, Some properties of convex functions of higher orders, Ann. Polon. Math. 7 (1959), 1-7.
• [3] K. Nikodem and S. Wąsowicz, A sandwich theorem and Hyers-Ulam stability of affine functions, Aequationes Math. 49 (1995), 160-164.
• [4] T. Popoviciu, Les Fonctions Convexes, Hermann, Paris, 1944.
• [5] T. Popoviciu, Sur quelques propriétés des fonctions d'une variable réelle convexes d'ordre supérieur, Mathematica (Cluj) 8 (1934), 1-85.
• [6] A. W. Roberts and D. E. Varberg, Convex Functions, Academic Press, New York, 1973.
• [7] F. A. Valentine, Convex Sets, McGraw-Hill, New York, 1964.
• [8] S. Wąsowicz, On affine selections of set-valued functions, J. Appl. Anal. 1 (1995), 173-179.