An extremal nonnegative sine polynomial

• Autor(a) principal: Andreani, Roberto [UNESP]
• Data de Publicação: 2003
• Outros Autores: Dimitrov, Dimitar K. [UNESP]
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Institucional da UNESP
• Texto Completo: http://dx.doi.org/10.1216/rmjm/1181069926; http://hdl.handle.net/11449/67393
• topic: Convergence; Extremal polynomial ultraspherical polynomials; Nonnegative sine polynomial; Positive summability kernel

For any positive integer n, the sine polynomials that are nonnegative in [0, π] and which have the maximal derivative at the origin are determined in an explicit form. Associated cosine polynomials Kn (θ) are constructed in such a way that {Kn(θ)} is a summability kernel. Thus, for each Pi 1 ≤ P ≤ ∞ and for any 27π-periodic function f ∈ Lp [-π, π], the sequence of convolutions Kn * f is proved to converge to f in Lp[-ππ]. The pointwise and almost everywhere convergences are also consequences of our construction.