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.

The book disseminates recent progress in Fourier analysis and approximation theory The volume contains contributions in the areas of Fourier analysis, theory of functions, approximation theory, optimisation theory The refereed detailed articles cover a wide spectrum of topics ranging from theory to applications For the benefit of experts and students alike, the volume presents interactions of different aspects within the field, with an extensive introduction containing an overview of modern research in the topics of the book

Inhalt

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.