A system of exponential functions with shift and the Kostyuchenko problem

We introduce a system of exponential functions with shift, study its basis properties in L ₂, and examine its connections with the Kostyuchenko problem.     

Generalized arcwise-connected functions and characterizations of local-global minimum properties

In this paper, new classes of generalized convex functions are introduced, extending the concepts of quasi-convexity, pseudoconvexity, and their associate subclasses. Functions belonging to these classes satisfy certain local-global minimum properties. Conversely, it is shown that, under some mild regularity conditions, functions for which the local-global minimum properties hold must belong to one of the classes of functions introduced.Dedicated to R. BellmanThe authors are indebted to I. Kozma, N. Megiddo, and A. Tamir for valuable discussions and to S. Schaible for valuable remarks. This research was partially supported by the Fund for the Encouragement of Research at the Technion.     

Applications of autoreproducing kernel grammian moduli to S (U,H)-valued stationary random functions

It is shown that the analytical characterizations of q-variate interpolable and minimal stationary processes obtained by H. Salehi (Ark. Mat., 7 (1967), 305–311; Ark. Mat., 8 (1968), 1–6; J. Math. Anal. Appl., 25 (1969), 653–662), and later by A. Weron (Studia Math., 49 (1974), 165–183), can be easily extended to Hilbert space valued stationary processes when using the two grammian moduli that respectively autoreproduce their correlation kernel and their spectral measure. Furthermore, for these processes, a Wold-Cramér concordance theorem is obtained that generalizes an earlier result established by H. Salehi and J. K. Scheidt (J. Multivar. Anal., 2 (1972), 307–331) and by A. Makagon and A. Weron (J. Multivar. Anal., 6 (1976), 123–137).     

A note on functions whose local minima are global

In this note, a simple proof of a theorem concerning functions whose local minima are global is presented and some closedness properties of this class of functions are discussed.The authors would like to thank Dr. Tatsuro Ichiishi of CORE for outlining the new proof of Theorem 2.1.This research was done while the author was a research fellow at the Center for Operations Research and Econometrics, University of Louvain, Heverlee, Belgium.     

Some remarks on discrete aperiodic Schrödinger operators

We consider Schrödinger operators onl ²( ) with deterministic aperiodic potential and Schrödinger operators on the l²-space of the set of vertices of Penrose tilings and other aperiodic self-similar tilings. The operators onl ²( ) fit into the formalism of ergodic random Schrödinger operators. Hence, their Lyapunov exponent, integrated density of states, and spectrum are almost-surely constant. We show that they are actually constant: the Lyapunov exponent for one-dimensional Schrödinger operators with potential defined by a primitive substitution, the integrated density of states, and the spectrum in arbitrary dimension if the system is strictly ergodic. We give examples of strictly ergodic Schrödinger operators that include several kinds of almost-periodic operators that have been studied in the literature. For Schrödinger operators on Penrose tilings we prove that the integrated density of states exists and is independent of boundary conditions and the particular Penrose tiling under consideration.