, Volume 39, Issue 3, pp 439-450

Lacunary Interpolation by Antiperiodic Trigonometric Polynomials

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Abstract

The problem of lacunary trigonometric interpolation is investigated. Does a trigonometric polynomial T exist which satisfies T(x k) = a k, D m T(x k) = b k, 0 ≤ kn − 1, where x k = kπ/n is a nodal set, a k and b k are prescribed complex numbers, \(D = \frac{d}{{dx}}\) and mN. 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.