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 Calderón-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.

Different facets of interplay between harmonic analysis and
approximation theory are covered in this volume. The topics included are
Fourier analysis, function spaces, optimization theory, partial differential
equations, and their links to modern developments in the approximation theory.
The articles of this collection were originated from two events. The first event
took place during the 9^th ISAAC Congress in Krakow, Poland, 5th-9th August
2013, at the section "Approximation
Theory and Fourier Analysis". The second
event was the conference on Fourier Analysis and Approximation Theory in the
Centre de Recerca Matemàtica (CRM), Barcelona, during 4th-8th November 2013,
organized by the editors of this volume. All articles selected to be part of
this collection were carefully reviewed.