Abstract. A notion of Paley-Wiener spaces on combinatorial graphs is introduced. It is shown that functions from some of these spaces are uniquely determined by their values on some sets of vertices which are called the uniqueness sets. Such uniqueness sets are described in terms of Poincare-Wirtingertype inequalities. A reconstruction algorithm of Paley-Wiener functions from uniqueness sets which uses the idea of frames in Hilbert spaces is developed. Special consideration is given to the n-dimensional lattice, homogeneous trees, and eigenvalue and eigenfunction problems on finite graphs.

Abstract. In the last decade, methods based on various kinds of spherical wavelet bases have found applications in virtually all areas where analysis of spherical data is required, including cosmology, weather prediction, and geodesy. In particular, the so-called needlets (=band-limited Parseval frames) have become an important tool for the analysis of Cosmic Microwave Background (CMB) temperature data. The goal of the present paper is to construct band-limited and highly localized Parseval frames on general compact homogeneous manifolds. Our construction can be considered as an analogue of the well-known ϕ-transform on Euclidean spaces.

Abstract. We consider a generalization of entire functions of spherical exponential type and Lagrangian splines on manifolds. An analog of the PaleyWiener theorem is given. We also show that every spectral entire function on a manifold is uniquely determined by its values on some discrete sets of points.The main result of the paper is a formula for reconstruction of spectral entire functions from their values on discrete sets using Lagrangian splines.

Notions of interpolating variational splines and Paley-Wiener spaces are introduced on a combinatorial graph G. Both of these definitions explore existence of a combinatorial Laplace operator on G. The existence and uniqueness of interpolating variational spline on a graph is shown. As an application of variational splines the paper presents a reconstruction algorithm of Paley-Wiener functions on graphs from their uniqueness sets.1991 Mathematics Subject Classification. 42C99, 05C99, 94A20, 41A15; Secondary 94A12 .

No abstract