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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)
##### Keywords:
Hardy space; interpolation; Gram matrix; multiple node
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##### References:
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