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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 \(f\).
MSC:
30E05 Moment problems and interpolation problems in the complex plane
30E10 Approximation in the complex plane
30D55 \(H^p\)-classes (MSC2000)
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References:
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