zbMATH — the first resource for mathematics

An explicit interpolation formula for the Hardy space. (English. Russian original) Zbl 0954.30021
Sib. Math. J. 41, No. 2, 252-256 (2000); translation from Sib. Mat. Zh. 41, No. 2, 311-315 (2000).
Under consideration is the Hardy space \(H^2\) of functions defined in the circle \(\{z\: |z|<1\}\) which is endowed with the inner product \[ \langle f,g\rangle =\frac{1}{2\pi}\int\limits_{|z|=1}f(z)\overline{g(z)} |dz|. \] Given a function \(f\in H^2\), the problem is to obtain an explicit interpolation formula for the case of multiple nodes. Let \(\{a_i\}\) be a collection of interpolation points. Put \[ \begin{gathered} g_{a,\nu}(z)=\frac{z^{\nu}}{(1-z\overline{a})^{\nu+1}},\;\;G_{j\beta}^{i\alpha}= \frac{1}{\beta!}\frac{d^\beta}{dz^\beta} \left. \left(\frac{z^{\alpha}}{(1-z\overline{a_i})^{\alpha+1}}\right)\right|_{z=a_j}, \\ \beta_{i}^{(m_i-1-s)}=\frac{e^{-i\theta}}{s!}\frac{d^s}{dz^s}\left. \left((1-z\overline{a_i})^{m_i}\prod_{j\neq i} \biggl(\frac{1-z\overline{a_j}}{z-a_j}\biggr)^{m_j}\right)\right|_{z=a_i}, \end{gathered} \] where \(0\leq s\leq m_i-1\); moreover, \(\beta_i^{(\nu)}=0\) whenever \(\nu\geq m_i\). The corresponding interpolation formula is written as \[ \widehat{f}(z)=\sum\limits_{i,\alpha,\beta}\sum\limits_{j,\delta,\gamma} \frac{f^{(\alpha)}(a_i)}{\alpha!}g_{a_j,\gamma}(z)G_{i\beta}^{j\delta} \beta_i^{(\alpha+\beta)}\overline{\beta_j^{(\gamma+\delta)}}, \] with \(\widehat{f}\) being the corresponding approximation to \(f\).
30E05 Moment problems and interpolation problems in the complex plane
30E10 Approximation in the complex plane
30D55 \(H^p\)-classes (MSC2000)
Full Text: DOI EuDML
[1] Goffman C., Banach Spaces of Analytic Functions [Russian translation], Izdat. Inostr. Lit., Moscow (1963). · Zbl 0128.27601
[2] Meschkowski H., Hilbertsche Räume mit Kernfunktion Springer-Verlag, Berlin etc. (1962).
[3] Walsh J.L., Interpolation and Approximation by Rational Functions in the Complex Domain [Russian translation], Izdat. Inostr. Lit., Moscow (1961). · Zbl 0106.28103
[4] Wilf H.S. ”Advances in numerical quadrature”, in: Mathematical Methods for Digital Computers. Vol. 2, New York, 1967.
[5] Chawla M.M., ”On the inversion of a certain Gram matrix”, Calcolo,10, No. 3–4, 257–260 (1973). · Zbl 0285.15006 · doi:10.1007/BF02575846
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.