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On the minimum of norms of mixed derivatives of trigonometric polynomials with given number of harmonics. (English. Russian original) Zbl 0739.42013

Mosc. Univ. Math. Bull. 46, No. 2, 4-7 (1991); translation from Vestn. Mosk. Univ., Ser. I 1991, No. 2, 3-7 (1991).
Summary: The quantity \[ L_ N(r,q)=\inf_{K_ N}\left\|\left(\sum_{k\in K_ N}e^{i(k,t)}\right)^{(r)}\right\|_ q \] is considered, when \(K_ N\subset Z^ n\) is an arbitrary set of \(N\) harmonics, \(r=(r_ 1,\dots,r_ n)\in R_ +^ n\) is the order of the mixed derivative in the Weyl’s sense, \(1<q<\infty\). The order of the quantity \(L_ N(r,q)\) is defined for “small” smoothness, i.e., \(0<\min\{r_ 1,\dots,r_ n\}\leq 1/q, 2\leq q<\infty\). For “big” smoothness and the vector \(t=(t_ 1,\dots,t_ n)\) this value was found earlier by Galeev. For scalar \(t\), \(L_ N(r,q)\) was computed by Majorov, Konyagin, and Belinskij.

MSC:

42B99 Harmonic analysis in several variables
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