The Sets Of Positivity Of Sine Series With Monotone Coefficients
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THE SETS OF POSITIVITY OF SINE SERIES WITH MONOTONE COEFFICIENTS K. OGANESYAN Universitat Aut` onoma de Barcelona, Centre de Recerca Matem` atica, Spain Lomonosov Moscow State University, Moscow Center for Fundamental and Applied Mathematics, Leninskie Gory 1, 119991 Moscow, Russia e-mail: [email protected] (Received January 13, 2020; accepted May 13, 2020)
Abstract. We study the sums of nondegenerate sine series with monotone coefficients and consider the sets of positivity of such functions. We obtain the sharp lower estimate of the measure of such a set on [π/2, π] and a new lower bound on its measure on [0, π]. It is shown that the latter measure is at least π/2 + 0.24 and in the case of fulfilling special conditions it is at least 2π/3, which is an unimprovable estimate.
1. Introduction Let f (x) be the sum of a nondegenerate sine series with monotone coefficients tending to zero, that is, (1.1)
f (x) =
∞
n=1
an sin nx,
a1 > 0, an ց 0.
Such series play an important role in various problems of harmonic analysis. We mention only the following three classic results. A [1]. Let p ∈ (1, ∞). Then f (x) ∈ Lp ([−π, π]) if and only if ∞Theorem p p−2 a n < ∞. n=1 n
Theorem B [1]. Let α ∈ (0, 1). Then f (x) ∈ Lip α if and only if an = O(n−1−α). Theorem C [9]. Let p ∈ (1, ∞), α ∈ (0, 1) and f (x) ∈ Lp ([−π, π]). Then f (x) ∈ Lip(α, p) if and only if an = O(n−1−α+1/p ). The present work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS” #19-8-2-28-1. Key words and phrases: sine series, monotone coefficients, measure of a set, set of positivity. Mathematics Subject Classification: SUBJ CLASS, . c 2020 0236-5294/$ 20.00 © � 0 Akad´ emiai Kiad´ o, ´ Budapest 0236-5294/$20.00 Akade ´miai Kiado , Budapest, Hungary
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K. K. OGANESYAN OGANESYAN
The results of Theorems A, B, C hold for cosine series with monotone coefficients as well. Similar results for a more general class of trigonometric series were obtained in [6], [7] and [10]. An important problem in studying series with monotone coefficients is to investigate properties of the sets of zeros of such functions. In particular, one of the important questions is how “large” such a set could be. In 1979 P. L. Ulyanov posed the following problems: 1) Could this set have the cardinality of the continuum? 2) Could it have positive measure? V. F. Gaposhkin [8] gave a positive answer to the first question and constructed an example of a series (1.1) with the set of zeros on [0, π] of the car−1,001 dinality of the continuum. More precisely, he considered ak = ∞ s=m s for 103(m−1) ≤ k < 103m . Later, M. I. Dyachenko [3] showed that series (1.1) can converge to zero on a set of positive measure. He constructed for a ∈ (a0 , π) the function 1 − x/a, x ∈ (0, a], fa (x) = 0, x ∈ (a, π), which has the Fourier expansion of the form (1.1), where the constant a0 ∈ (2π/3, 2.15) is found from the equation 2a0 − 4 sin a0 + sin 2a0 = 0. It was also shown [3] that the measure of the set of zeros of f (x) in [0, π] is les
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