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	NCTS PDE & Analysis Seminar

11:00 - 12:00, January 26, 2016 (Tuesday)
R440, Astronomy-Mathematics Building, NTU
(台灣大學天文數學館 440室)
Recent results on nonnegative trigonometric polynomials
Man-Kam Kwong (The Hong Kong Polytechnic University)

T.H. Fejér conjectured in 1910 that, for any positive integer ${\[ n \]}$ and ${\[ x \in 0 \]}$

${\[ \sum_{k=1}^{n}\frac{\sin (kx)}{k}> 0 \]}$

D. Jackson offered the first proof in 1911, followed shortly by T.H. Gronwall. W.H. Young in 1913 established the cosine counterpart

${\[ 1+\sum_{k=1}^{n}\frac{\cos (kx)}{k}> 0 \]}$

These results have stimulated the research of many mathematicians who discovered remarkable new proofs, generalizations, refinements and numerous variants. The results have important applications in the study of Fourier series, especially the Gibb's phenomenon, univalent functions, signal processing, etc.

In 1958, L. Vietoris published a deep result: If ${\[ a_{k}\searrow 0 \]}$ and ${\[ a_{2k}\leq \frac{2k-1}{2k}a_{2k-1} \]}$ , then

${\[ \sum_{k=1}^{n}a_{k}\sin (kx)> 0 \]}$ and ${\[ \sum_{k=0}^{n}a_{k}\cos (kx)> 0 \]}$

It plays a role in the proof of the Bieberbach conjecture.

We survey recent developments in this field. Contributors include Alzer, Askey, Belov, Brown, Dimitrov, Hewitt, Koumandos, …