THEORY OF APPROXIMATION OF FUNCTIONS OF A REAL VARIABLE
CONTENTS ABSTRACT EXACT AND APPROXIMATE SAMPLING THEOREMS SAMPLING IN REPRODUCING KERNEL HILBERT SPACE BOAS-TYPE FORMULAS AND SAMPLING IN BANACH SPACES WITH APPLICATIONS TO ANALYSIS ON MANIFOLDS ON WINDOW METHODS IN GENERALIZED SHANNON SAMPLING OPERATORS GENERALIZED SAMPLING APPROXIMATION FOR MULTIVARIATE DISCONTINUOUS SIGNALS AND APPLICATIONS TO IMAGE PROCESSING SIGNAL AND SYSTEM APPROXIMATION FROM GENERAL MEASUREMENTS SAMPLING IN IMAGE REPRESENTATION AND COMPRESSION SPARSE SIGNAL PROCESSING SIGNAL SAMPLING AND TESTING UNDER NOISE SUPEROSCILLATIONS GENERAL MODULI OF SMOOTHNESS AND APPROXIMATION BY FAMILIES OF LINEAR POLYNOMIAL OPERATORS VARIATION AND APPROXIMATION IN MULTIDIMENSIONAL SETTING FOR MELLIN INTEGRAL OPERATORS THE LEBESGUE CONSTANT FOR SINC APPROXIMATIONS SIX (SEVEN) PROBLEMS IN FRAME THEORY FIVE GOOD REASONS FOR COMPLEX-VALUED TRANSFORMS IN IMAGE PROCESSING FREQUENCY DETERMINATION USING THE DISCRETE HERMITE TRANSFORM FRACTIONAL OPERATORS, DIRICHLET AVERAGES, AND SPLINES A DISTRIBUTIONAL APPROACH TO GENERALIZED STOCHASTIC PROCESSES ON LOCALLY COMPACT ABELIAN GROUPS ON A DISCRETE TURÁN PROBLEM FOR `-1 RADIAL FUNCTIONS CHAPTER INTRODUCTION SAMPLING THEORY CAN BE DESCRIBED BRIE?Y AS THE STUDY OF RECONSTRUCTION OF CONTINU- OUS FUNCTIONS FROM VALUES ON A DISCRETE SAMPLING SET BY MEANS OF A SAMPLING SERIES INVOLVING THE SAMPLE VALUES THE SAMPLING SET CAN BE EITHER REGULAR (EQUALLY SPACED), FOR EXAMPLE, WHEN IT IS A SUBGROUP, OR IRREGULAR, WHEN IT IS A UNION OF SUBGROUPS AND THEIR TRANSLATES THE MODERN THEORY BEGAN WITH THE WHITTAKER–KOTELNIKOV–SHANNON (WKS) THEOREM [34], A CORNERSTONE OF COMMUNICATION THEORY AND ANALOGUE/DIGITAL SIGNAL PROCESSING THE THEOREM ESTABLISHES A THEORETICALLY EXACT RECONSTRUCTION OF CONTINUOUS, ?NITE ENERGY, BANDLIMITED SIGNALS (I E , USING MORE MATHEMATICAL LANGUAGE, OF CONTINUOUS SQUARE-INTEGRABLE FUNCTIONS F W R ! C WITH BOUNDED SPECTRUM) FROM VALUES ON A SAMPLING SET THAT CONSISTS OF A DISCRETE SUBGROUP OF R DISCOVERED MANY TIMES [24], IT HAS BEEN STUDIED AND EXTENDED BY NUMEROUS