Generalized Padé-type approximation and integral representations
The author replace the multidimensional Cauchy kernel by the Bergman kernel function \(K_\Omega(t,x)\) into an open bounded subset \(\Omega\) of \(\mathbb{C}^n\) and by using interpolating generalized polynomials for \(K_\Omega(z,x)\) define generalized Padé-type approximants to any \(f\) in the space \(OL^2 (\Omega)\) of all analytic functions on \(\Omega\) which are of class \(L^2\). The characteristic property of such an approximant is that its Fourier series representation with respect to an orthonormal basis for \(OL^2(\Omega)\) matches the Fourier series expansion of \(f\) as far as possible. The errors formula and the convergence problem is studied. It is showed that the generalized Padé-type approximants have integral representations which give rise to the consideration of an integral operator which maps every \(f\in OL^2 (\Omega)\) to a generalized Padé-type approximant to \(f\). The study conclude with the extension of these ideas into every functional Hilbert space \(H\) and also with the definition and properties of the generalized Padé-type approximants to a linear operator of \(H\) into itself. There are contained two examples making use of generalized Padé-type approximants and a large list of references.
self-summable countable orthonormal set in a Hilbert space
scientific article; zbMATH DE number 1896711