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Lacunary interpolation by antiperiodic trigonometric polynomials. (English) Zbl 0931.42003

Summary: The problem of lacunary trigonometric interpolation is investigated. Does a trigonometric polynomial \(T\) exist which satisfies \(T(x_k)= a_k\), \(D^mT(x_k)= b_k\), \(0\leq k\leq n-1\), where \(x_k= k\pi/n\) is a nodal set, \(a_k\) and \(b_k\) are prescribed complex numbers, \(D= {d\over dx}\) and \(m\in\mathbb{N}\). Results obtained by several authors for the periodic case are extended to the antiperiodic case. In particular, solvability is established when \(n\) as well as \(m\) are even. In this case a periodic solution does not exist.

MSC:

42A15 Trigonometric interpolation
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