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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
##### MSC:
 32A40 Boundary behavior of holomorphic functions of several complex variables 32A15 Entire functions of several complex variables 12D99 Real and complex fields
##### Keywords:
homogeneous polynomials; complex sphere
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