Abstract.
An extremal problem for the coefficients of sine polynomials, which are nonnegative in [0,π] , posed and discussed by Rogosinski and Szegő is under consideration. An analog of the Fejér—Riesz representation of nonnegative general trigonometric and cosine polynomials is proved for nonnegative sine polynomials. Various extremal sine polynomials for the problem of Rogosinski and Szegő are obtained explicitly. Associated cosine polynomials kn (θ) are constructed in such a way that { kn (θ) } are summability kernels. Thus, the Lp, pointwise and almost everywhere convergence of the corresponding convolutions, is established.
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Dimitrov, D., Merlo, C. Nonnegative Trigonometric Polynomials. Constr. Approx. 18, 117–143 (2001). https://doi.org/10.1007/s00365-001-0004-x
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DOI: https://doi.org/10.1007/s00365-001-0004-x
Key words.
- Nonnegative trigonometric polynomials
- Extremal polynomials
- Summability kernel
- Fejér—Riesz-type theorem
- Lp Convergence
- Pointwise and almost everywhere convergence