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Integral transforms, reproducing kernels and their applications. (English) Zbl 0891.44001

Pitman Research Notes in Mathematics Series. 369. Harlow: Longman. 280 p. (1997).
This is a systematic development of the contents of some thirty papers by the author published in the last 16 years, or so. The underlying idea is the observation that the inverse of an integral transformation can be represented by a simple abstract formula involving the reproducing kernel Hilbert space associated with the kernel of the transformation. To use (somewhat abbreviated) the author’s notation, let \(h_t(p)= h(t,p)\) be a complex valued function and let \(f(p)= \int F(t)\overline{h(t,p)}\) be the transformation with the kernel \(h\). The reproducing kernel defined by \(h\) is \(K(p,q)= \int h(t,q)\overline{h(t,p)}dt\) and the corresponding Hilbert space is \(H_K\) with the inner product \((.,.)_K\). Then, with some caveats, the inverse transformation is given by \(F(t)= (f,\overline h_t)_K\).
The monograph implements and develops this idea in numerous concrete cases arising in harmonic analysis – Fourier, Laplace and wavelet transforms, function theory (analytic continuation), spaces of holomorphic functions, theory of approximations, differential equations etc.
The first chapter is a nice introduction to the theory of proper functional Hilbert spaces (i.e. Hilbert spaces with reproducing kernels) and the bibliography consists of some 180 articles and 80 books.

MSC:

44A05 General integral transforms
30C40 Kernel functions in one complex variable and applications
30B40 Analytic continuation of functions of one complex variable
45A05 Linear integral equations
44-02 Research exposition (monographs, survey articles) pertaining to integral transforms
35A22 Transform methods (e.g., integral transforms) applied to PDEs
41A50 Best approximation, Chebyshev systems
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