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Trigonometric series with positive partial sums. (English. Russian original) Zbl 0802.42008

Math. Notes 53, No. 3, 348-350 (1993); translation from Mat. Zametki 53, No. 3, 149-152 (1993).
In the present paper a trigonometric series with the spectrum \(A\) is considered, \(\sum_{\nu\in A} c_ \nu e^{i\nu x}\), \(A\subset\mathbb{Z}^ n\), \(c_ \nu= \overline{c_{-\nu}}\). By \(B^*= \{z: z= x-y, z\neq 0, x,y\in B\}\) the difference set of \(B\) is denoted. Let \(f(x)\) be a continuous function on \(T^ n= (-\pi,\pi]^ n\) with zero mean and \(M(f)= \min\{\max_{x\in T^ n} f(x); -\min_{x\in T^ n} f(x)\}\).
The main result is the following Theorem 1. Let \(A\subset \mathbb{Z}^ n\) be finite, \(B\subset A\), \(B^*\cap A= \emptyset\), \(t(x)= \sum_{\nu\in A} \widehat{t_ \nu} e^{i\nu x}\), \(\widehat t_ \nu=\overline{\widehat t}_{-\nu}\), \(\widehat t_ 0= 0\). Then \(M(f)\geq \Bigl\{\sum_{\nu\in B} |\widehat t_ \nu|^ 2\Bigr\}^{1/2}\).

MSC:

42B05 Fourier series and coefficients in several variables
42A20 Convergence and absolute convergence of Fourier and trigonometric series
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References:

[1] N. K. Bari, A Treatise on Trigonometric Series, Pergamon Press, Oxford, New York (1964). · Zbl 0154.06103
[2] Y. Katznelson, ?Trigonometric series with positive partial sums,? Bull. Am. Math. Soc.,71, 718-719 (1965). · Zbl 0133.02301
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