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On homogeneous polynomials of several variables on the complex sphere. (English. Russian original) Zbl 0592.32003
Math. USSR, Sb. 54, 409-414 (1986); translation from Mat. Sb., Nov. Ser. 126(168), No. 3, 420-425 (1985).
Let \({\mathbb{C}}^ d\) be the complex space of the dimension d, \(S^ d\) the sphere in \({\mathbb{C}}^ d\), and E(d,N) a set of integers \(E(d,N)=\{(k_ 1,...,k_ d): k_ j\geq 0,\quad 1\leq j\leq d,\quad \sum^{d}_{j=1}k_ j=N\}.\) The main assertion of the author is the following
Theorem: For any \(\delta >0\) and \(d=2,3,..\). there is a constant K(\(\delta\),d) such that for any subset \(\Lambda\) \(\subset E(d,N)\) with \(| \Lambda | >\delta N^{d-1}\) there exists a polynomial \(P_{\Lambda}=P_{\Lambda}(z)=\sum_{(k_ 1,...,k_ d)\in \Lambda}a_{k_ 1,...,k_ d}z\quad_ 1^{k_ 1}...z_ d^{k_ d}\) which satisfies the inequality \[ \| P_{\Lambda}\|_{L^ 2(S^ d)}\leq \| P_{\Lambda}\|_{C(S^ d)}\leq K(\delta,d)\| P_{\Lambda}=_{L^ 2(S^ d)}. \] This is a generalization of the theorem given by J. Ryll and P. Wojtaszczyk in Trans. Am. Math. Soc. 276, 107-116 (1983; Zbl 0522.32004).
Reviewer: A.Dzuraev
32A40 Boundary behavior of holomorphic functions of several complex variables
32A15 Entire functions of several complex variables
12D99 Real and complex fields
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