Approximation theory in tensor product spaces (Q1064479)

The book is concerned with the approximation of multivariate functions by combinations of univariate ones, using tensor products of appropriate Banach function spaces. A good idea how tensor products occur naturally in these approximation problems is given by the following formulae: \(C(S\times T)=C(S)\otimes_{\lambda}C(T)\), (\(\lambda\) is the least of the reasonable crossnorms); \(L_ 1(S\times T)=L_ 1(S)\otimes_{\gamma}L_ 1(T),\) (\(\gamma\) is the greatest crossnorm); \(L_ 2(S\times T)=L_ 2(S)\otimes_{\beta}L_ 2(T),\) (\(\beta\) is the Hilbert crossnorm). The aim of the book is twofold: to provide an account of some recent results and to give an exposition for the readers not familiar with the subject. The book begins with a chapter giving a brief but thorough introduction to the theory of tensor products of Banach spaces. The approximation problems treated in the rest of the book are: proximinality (i.e. existence of best approximation); unicity of best approximation, alternating algorithms in Hilbert spaces (devised by J. von Neumann 1933), in C(S\(\times T)\) (S. P. Diliberto and E. G. Strauss 1951) and the \(L_ 1\)-version of this algorithm; projection constants and projections of least norm (minimal projections). The proofs are given in detail. Some results needed in the proofs are collected in two appendices: on Bochner integral and on miscellaneous results on Banach spaces. The book, dedicated to Robert Schatten (1911- 1977), the initiator of the theory of tensor products of Banach spaces, contains also a biographical sketch and a list of publications of R. Schatten. The book can be used for seminars and courses in approximation theory and as a reference book by specialists.