Some extremal properties of positive trigonometric polynomials
Tóm tắt
For n=8 an upper bound is given for the functional
$$V_n = \mathop {\inf }\limits_{t_n } \frac{{\alpha _1 + \alpha _2 + \cdots + \alpha _n }}{{\left( {\sqrt {\alpha _1 } - \sqrt {\alpha _0 } } \right)^2 }}$$
, which is defined on the class of even, nonnegative, trigonometric polynomials
$$t_n (\phi ) = \sum\nolimits_{k = 0}^n {\alpha _k } cos k\phi $$
, such that α
k
⩾ 0 (k=0, ...,n) α1>α0 :V
s ⩽ 34.54461566.
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