An estimate of the constant term of a nonnegative trigonometric polynomial with integer coefficients. (English. Russian original) Zbl 0871.42001
Math. Notes 59, No. 4, 451-453 (1996); translation from Mat. Zametki 59, No. 4, 627-629 (1996).
A sequence \(\Lambda= \{\lambda_n\}^\infty_{n=1}\) of positive numbers is said to be admissible if
\[
\sum^\infty_{n=1} {1\over \lambda_n} <\infty \quad \text{and} \quad \sum^\infty_{n=1} \sin\left({x \over\lambda_n} \right)\geq 0
\]
for all \(x\geq 0\). If, moreover,
\[
\sum^\infty_{n=1} \sin\left( {x\over\lambda_n} \right)>0
\]
for all \(x>0\), then the sequence \(\Lambda\) is said to be strictly admissible.
This definition was introduced by the first author. The aim of the present paper is to study admissible sequences and to use them to estimate the constant term of a nonnegative trigonometric polynomial with integer coefficients.
This definition was introduced by the first author. The aim of the present paper is to study admissible sequences and to use them to estimate the constant term of a nonnegative trigonometric polynomial with integer coefficients.
MSC:
42A05 | Trigonometric polynomials, inequalities, extremal problems |
42A32 | Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.) |
Keywords:
admissible sequences; constant term; nonnegative trigonometric polynomial with integer coefficientsReferences:
[1] | A. S. Belov, in:Constructive Function Theory and Applications. Conference Abstracts [in Russian], Makhachkala (1994), pp. 17–19. |
[2] | A. M. Odlyzko,J. London Math. Soc.,26, No. 3, 412–420 (1982). · doi:10.1112/jlms/s2-26.3.412 |
[3] | M. N. Kolountzakis,Proc. Amer. Math. Soc.,120, 157–163 (1984). · doi:10.1090/S0002-9939-1994-1169037-9 |
[4] | A. S. Belov, in:International Conference ”Function Spaces, Approximation Theory, and Nonlinear Analysis” dedicated to the 90th birthday of Academician S. M. Nikol’skii. Abstracts [in Russian], Moscow (1995), pp. 40–41. |
[5] | A. S. Belov,Mat. Zametki [Math. Notes],30, No. 4, 501–515 (1981). |
[6] | P. Erdös and G. Szekeres,Acad. Serbe Sci. Publ. Inst. Math.,13, 29–34 (1959). |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.