By Michael Ruzhansky, Sergey Tikhonov
Different features of interaction among harmonic research and approximation conception are lined during this quantity. the subjects integrated are Fourier research, functionality areas, optimization conception, partial differential equations, and their hyperlinks to fashionable advancements within the approximation thought. The articles of this assortment have been originated from occasions. the 1st occasion happened through the 9th ISAAC Congress in Krakow, Poland, 5th-9th August 2013, on the part “Approximation conception and Fourier Analysis”. the second one occasion was once the convention on Fourier research and Approximation thought within the Centre de Recerca Matem`tica (CRM), Barcelona, in the course of 4th-8th November 2013, equipped through the editors of this quantity. All articles chosen to be a part of this assortment have been rigorously reviewed.
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Additional info for Methods of Fourier Analysis and Approximation Theory
Bm is a regular mapping. By definition, a holomorphic mapping ' W Bn ! Bm /; a very close notion was earlier introduced in the Bloch-to-BMOA setting (see ). In fact, we show that the scheme applied in  is adaptable to the logarithmic Bloch spaces. The main result of the present paper is the following theorem. 2 Let ˛ < 12 , 0 < p Ä 1 and let ' W Bn ! Bm be a regular holomorphic mapping. Bm/ ! @Bn /: (5) Since every holomorphic mapping ' W Bn ! 3 Let ˛ < 12 , 0 < p Ä 1 and let ' W Bn ! B1 be a holomorphic mapping.
N C 1/ matrix BQ which is obtained from B with the addition of the row Ey, is unitary, and therefore also the columns of BQ are orthonormal. For every i; j Ä n C 1, hvi ; vj i ˙ q 1 q `2i 1 `2j D `i `j cos Âij ˙ q 1 q `2i 1 `2j D 0 (15) which implies (1). q Conversely, suppose that (1) holds. ˙ 1 `2j ; vj / are orthonormal for some choice of the sign ˙; therefore, the columns of the matrix BQ are orthonormal, and so also the rows are orthonormal, and B is a Parseval frame. 2 Let B D fv1 ; : : : vnC1 g be a nontrivial Parseval frame.
Bn ; Bm / consists of those holomorphic mappings ' W Bn ! r /; 0/ d n . / < 1; E. Doubtsov ( ) St. A. Steklov Mathematical Institute, Fontanka 27, St. ru © Springer International Publishing Switzerland 2016 M. Ruzhansky, S. 1007/978-3-319-27466-9_3 33 34 E. Doubtsov where ˇm denotes the Bergman metric on Bm . B1 / and ˇ1 is replaced by the Euclidean metric. r /j2 Ãp d n . w/ @wj is the radial derivative of f . Given a holomorphic mapping ' W Bn ! Bm / ! Bm /; z 2 Bn : Kwon  argued that various hyperbolic classes naturally arise in the studies of the composition operators C' defined on the Bloch space.