For a domain D, a pointpand a function f the integral M f (D; p)= Z D f(px)dx is called the moment of D with respect to p taken with the function f. Herepx denotes the distance of x to p. The Moment Theorem of László Fejes Tóth states the following: Let H be a convex polygon in E 2 with at most six sides and f a non-increasing function defined for non-negative reals. Let p1;pn be distinct points and let Di be the Dirichlet cell of pi relative to H. Then we have n P i=1 M f (Di; pi) 5 nM f (Hn; o); where Hn is a regular hexagon of area a(Hn)=a(H)=n centered at o. In the paper a stability criterion to the Moment Theorem is established.
We show that, in the classes of functions with values in a real or complex Banach space, the problem of Hyers-Ulam stability
of a linear functional equation of higher order (with constant coefficients) can be reduced to the problem of stability of
a first order linear functional equation. As a consequence we prove that (under some weak additional assumptions) the linear
equation of higher order, with constant coefficients, is stable in the case where its characteristic equation has no complex
roots of module one. We also derive some results concerning solutions of the equation.
Uniform stability and localization results for the higher order and singular Kobayashi metrics are established. As an application
we obtain the non-tangential weighted limits of these metrics in an h-extendible boundary point of a bounded domain in Cn.
We study the stability of Gabor frames with arbitrary sampling points in the time-frequency plane, in several aspects. We
prove that a Gabor frame generated by a window function in the Segal algebra S0(Rd) remains a frame even if (possibly) all the sampling points undergo an arbitrary perturbation, as long as this is uniformly
small. We give explicit stability bounds when the window function is nice enough, showing that the allowed perturbation depends
only on the lower frame bound of the original family and some qualitative parameters of the window under consideration. For
the perturbation of window functions we show that a Gabor frame generated by any window function with arbitrary sampling points
remains a frame when the window function has a small perturbation in S0(Rd) sense. We also study the stability of dual frames, which is useful in practice but has not found much attention in the literature.
We give some general results on this topic and explain consequences to Gabor frames.