FUNDAMENTALS OF HARMONIC ANALYSIS

By J. FAVIR

CONTENTS I FOURIER ANALYSIS FOURIER SERIES PERIODIC FUNCTIONS EXPONENTIALS THE BESSEL INEQUALITY CONVERGENCE IN THE L2-NORM UNIFORM CONVERGENCE OF FOURIER SERIES PERIODIC FUNCTIONS REVISITED EXERCISES HILBERT SPACES PRE-HILBERT AND HILBERT SPACES ?2-SPACES ORTHONORMAL BASES AND COMPLETION FOURIER SERIES REVISITED THE FOURIER TRANSFORM CONVERGENCE THEOREMS CONVOLUTION THE TRANSFORM THE INVERSION FORMULA PLANCHEREL’S THEOREM THE POISSON SUMMATION FORMULA THETA SERIES DISTRIBUTIONS DE?NITION THE DERIVATIVE OF A DISTRIBUTION TEMPERED DISTRIBUTIONS FOURIER TRANSFORM II LCA GROUPS FINITE ABELIAN GROUPS THE DUAL GROUP LCA GROUPS METRIC SPACES AND TOPOLOGY COMPLETION THE DUAL AS LCA GROUP PONTRYAGIN DUALITY PLANCHEREL THEOREM HAAR INTEGRATION FUBINI’S THEOREM III NONCOMMUTATIVE GROUPS MATRIX GROUPS GLN(C) AND U(N) REPRESENTATIONS THE EXPONENTIAL 10 THE REPRESENTATIONS OF SU(2) 10 1 THE LIE ALGEBRA 10 2 THE REPRESENTATIONS 10 3 EXERCISES 11 THE PETER -WEYL THEOREM 11 1 DECOMPOSITION OF REPRESENTATIONS 11 2 THE REPRESENTATION ON HOM(V?, V?) 11 3 THE PETER -WEYL THEOREM 11 4 A REFORMULATION 11 5 EXERCISES 12 THE HEISENBERG GROUP 12 1 DE?NITION 12 2 THE UNITARY DUAL 12 3 HILBERT-SCHMIDT OPERATORS 12 4 THE PLANCHEREL THEOREM FOR H 12 5 A REFORMULATION 12 6 EXERCISES