Applications of the general theory of reproducing kernels (Q2712255)

In this paper, the author presents a survey of his results published in [``Integral transforms, reproducing kernels and their applications'', Pitman Research Notes in Mathematics Series. 369. Harlow: Longman. (1997; Zbl 0891.44001)] related to reproducing kernels and linear transforms theory. The basic model of a linear system is an integral transformation \(F\mapsto f(p) = \int_T \overline{h(t,p)} F(t) d\mu(t)\) with \(F,h(\cdot,p) \in L^2(T;d\mu)\) and the ``output'' \(f\) in a naturally constructed reproducing kernel Hilbert space \(H_K\). Next, the author considers the inversion problem and presents a mechanism that obtains the minimum \(L^2(T;d\mu)\)-norm inverse. The inverse problem is further studied for Fredholm integral equations of the first kind. Throughout the paper, the author presents generalized forms of isoperimetric inequalities. These are obtained as consequences of \(\|f \|_{H_K}^2 \leq \|F \|_{L^2(T;d\mu)}^2\). The exact statements and proofs are referred to loc. cit.NEWLINENEWLINEFor the entire collection see [Zbl 0957.00034].