ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: 39, 26 Gilányi, Attila, Lajos Kossuth University, Debrecen, Hungary On polynomial functions, monomial functions and the Dinghas interval derivative A real function f is called a polynomial function of degree n if Δ_{y}^{n+1}f(x) = 0 for all x, y ∈ IR, f is said to be a monomial function of degree n if Δ_{y}^{n}f(x) − n!f(y) = 0 (x, y ∈ IR), where n is a positive integer and Δ denotes the well-known difference operator. In this talk characterization and stability theorems for functions satisfying these equations are presented using, for a nonnegative real number α, the generalization of the Dinghas interval derivative

D^{n,α}f(ξ) = lim_{(x,y)→(ξ,0)}_{x≤ξ≤x+ny} \frac{Δ_{y}^{n}f(x)}{y^{α}}

and the operator

D̃^{n,α}f(ξ) = lim_{(x,y)→(ξ,0)}_{x≤ξ≤x+ny} \frac{Δ_{y}^{n}f(x) − n!f(y)}{y^{α}}.

The given results are connected to some works of A. Dinghas and P. Volkmann on the interval derivative above.