Different facets of interplay between harmonic analysis andapproximation theory are covered in this volume. The topics included areFourier analysis, function spaces, optimization theory, partial differentialequations, and their links to modern developments in the approximation theory.The articles of this collection were originated from two events. The first eventtook place during the 9th ISAAC Congress in Krakow, Poland, 5th-9th August2013, at the section ApproximationTheory and Fourier Analysis . The secondevent was the conference on Fourier Analysis and Approximation Theory in theCentre de Recerca Matem?tica (CRM), Barcelona, during 4th-8th November 2013,organized by the editors of this volume. All articles selected to be part ofthis collection were carefully reviewed.
1. Introduction.- 2. Fourier analysis.- 2.1. Parseval frames.- 2.2. Hyperbolic Hardy classes and logarithmic Bloch spaces.- 2.3. Logan's and Bohman's extremal problems.- 2.4. Weighted estimates for the Hilbert transform.- 2.5. Q-Measures and uniqueness sets for Haar series.- 2.6. O-diagonal estimates for Calder?n-Zygmund operators.- 3. Function spaces of radial functions.- 3.1. Potential spaces of radial functions.-
3.2. On Leray's formula.- 4. Approximation theory.- 4.1. Approximation order of Besov classes.- 4.2. Ulyanov inequalities for moduli of smoothness.- 4.3. Approximation order of Besov classes.- 5. Optimization theory and related topics.- 5.1. The Laplace-Borel transform.- 5.2. Optimization control problems.- 2 Michael Ruzhansky and Sergey Tikhonov.-5.3. Optimization control problems for parabolic equation.- 5.4. Numerical modeling of the linear filtration.- References.
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