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5-1 is often used in applications. 5-2, 4) in order to prove that the operator T1 of the first kind preserves the Bergman space property. 1-1) with entire coefficients, the following holds. If h1 E AP(B1(0,1)), then w1 = T1f1 E AP(B1(0,1) X B2(0,1)), 1

4-2a. 5-10) l m 1 SJml(z) = M! (JC2 1 (Z1. 5-12) yields _ k1(Z, t)=k/e(Z, t) = 1 + 00 (-1)M C2m(zi. )m(zj - Sj)mt2m m-1 = j = 1,2. 2-6). 5-13) Lw=D1D2w+ Xn(n + 1) (1 + Aztz2) 2w=0, AEC, nEN is of importance, for instance in the case A = - 1, because of its relation to the wave equation, from which it can be obtained by separating variables in spherical coordinates and introducing suitable new variables in the resulting equation for the angular part. It has been investigated by many authors (references will be given in Chap.

On the other hand, that connection also suggests an extension of Bauer's work to solutions of more general equations. Indeed, our results [1969] have sparked a large number of publications by Bauer, Florian, Heersink, Jank, Ruscheweyh, Schroder, Watzlawek, the authors, and others, as can be seen from the list of references at the end of this book. The main topics in those papers are (i) existence, uniqueness and representation theorems for solutions, with application to initial value problems, and (ii) the study of function theoretic properties of solutions.

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