MOMENT THEORY AND SOME INVERSE PROBLEMS IN POTENTIAL THEORY AND HEAT CONDUCTION

By EMMA MOORE

MATHEMATICAL PRELIMINARIES BANACH SPACES HILBERT SPACES SOME USEFUL FUNCTION SPACES SPACES OF CONTINUOUS FUNCTIONS SPACES OF INTEGRABLE FUNCTIONS SOBOLEV SPACES ANALYTIC FUNCTIONS AND HARMONIC FUNCTIONS FOURIER TRANSFORM AND LAPLACE TRANSFORM REGULARIZATION OF MOMENT PROBLEMS BY TRUNCATED EXPANSION AND BY THE TIKHONOV METHOD METHOD OF TRUNCATED EXPANSION A CONSTRUCTION OF REGULARIZED SOLUTIONS CONVERGENCE OF REGULARIZED SOLUTIONS AND ERROR ESTIMATES ERROR ESTIMATES USING EIGENVALUES OF THE LAPLACIAN METHOD OF TIKHONOV CASE 1: EXACT SOLUTIONS IN L2(?) CASE 2: EXACT SOLUTIONS IN L??(?), 1 < ??< ? CASE 3: EXACT SOLUTIONS IN H1(?) NOTES AND REMARKS BACKUS-GILBERT REGULARIZATION OF A MOMENT PROBLEM INTRODUCTION BACKUS-GILBERT SOLUTIONS AND THEIR STABILITY DE?NITION OF THE BACKUS-GILBERT SOLUTIONS STABILITY OF THE BACKUS-GILBERT SOLUTIONS REGULARIZATION VIA BACKUS-GILBERT SOLUTIONS DE?NITIONS AND NOTATIONS MAIN RESULTS THE HAUSDOR?MOMENT PROBLEM: REGULARIZATION AND ERROR ESTIMATES FINITE MOMENT APPROXIMATION OF (4 1) PROOF OF THEOREM 4 1 PROOF OF THEOREM 4 2 A MOMENT PROBLEM FROM LAPLACE TRANSFORM ANALYTIC FUNCTIONS: RECONSTRUCTION AND SINC APPROXIMATIONS RECONSTRUCTION OF FUNCTIONS IN H2(U): APPROXIMATION BY POLYNOMIALS RECONSTRUCTION OF AN ANALYTIC FUNCTION: A PROBLEM OF OPTIMAL RECOVERY CARDINAL SERIES REPRESENTATION AND APPROXIMATION: REFORMULATION OF MOMENT PROBLEMS TWO-DIMENSIONAL SINC THEORY APPROXIMATION THEOREMS REGULARIZATION OF SOME INVERSE PROBLEMS IN POTENTIAL THEORY ANALYTICITY OF HARMONIC FUNCTIONS CAUCHY’S PROBLEM FOR THE LAPLACE EQUATION SURFACE TEMPERATURE DETERMINATION FROM BOREHOLE MEASUREMENTS (STEADY CASE) THE BACKWARD HEAT EQUATION SURFACE TEMPERATURE DETERMINATION FROM BOREHOLE MEASUREMENTS: A TWO-DIMENSIONAL PROBLEM AN INVERSE TWO-DIMENSIONAL STEFAN PROBLEM: IDENTI?CATION OF BOUNDARY VALUES EPILOGUE REFERENCES