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On the trigonometric interpolation and the entire interpolation. (English) Zbl 0724.42005

Summary: We study a kind of interpolation problems on a given nodal set by trigonometric polynomials of order n and entire functions of exponential type according as the nodal set is \(\{2k\pi /n\}^{n-1}_{k=0}\) or \(\{2k\pi /\sigma \}^{+\infty}_{k=-\infty}\), respectively. We establish some equivalent conditions and find the explicit forms of some interpolation functions for the interpolation problems. As a special case, the explicit forms of fundamental functions of (0,m)-interpolation by trigonometric case or entire functions case (in \(B^ 2_{\sigma})\), respectively, if they exist, may follow from our results. Besides, we also consider the convergence of the interpolation functions as stated above.

MSC:

42A15 Trigonometric interpolation
30E05 Moment problems and interpolation problems in the complex plane
30D99 Entire and meromorphic functions of one complex variable, and related topics
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