منابع مشابه
On Trigonometric Sums with Gaps
A well known theorem states as follows :' Let ni < n2 <. . ., nk+1 / nk > A > 1 be an infinite sequence of real numbers and S (ak + bk) a divergent series satisfying k=1 Then denotes the Lebesgue measure of the set in question. It seems likely that the Theorem remains true if it is not assumed that the n k are integers. On the other hand if nk ,f-n,.-1 is an arbitrary sequence of integers it is...
متن کاملRearrangements of Trigonometric Series and Trigonometric Polynomials
Abstract. The paper is related to the following question of P. L. Ul’yanov: is it true that for any 2π-periodic continuous function f there is a uniformly convergent rearrangement of its trigonometric Fourier series? In particular, we give an affirmative answer if the absolute values of Fourier coefficients of f decrease. Also, we study a problem how to choose m terms of a trigonometric polynom...
متن کاملLacunary Trigonometric Series. Ii
where E c [0, 1] is any given set o f positive measure and {ak} any given sequence of real numbers. This theorem was first proved by R. Salem and A. Zygmund in case of a -0, where {flk} satisfies the so-called Hadamard's gap condition (cf. [4], (5.5), pp. 264-268). In that case they also remarked that under the hypothesis (1.2) the condition (1.3) is necessary for the validity of (1.5) (cf. [4]...
متن کاملIntegral Concentration of Idempotent Trigonometric Polynomials with Gaps
We prove that for all p > 1/2 there exists a constant γp > 0 such that, for any symmetric measurable set of positive measure E ⊂ T and for any γ < γp, there is an idempotent trigonometrical polynomial f satisfying ∫ E |f |p > γ ∫ T |f |p. This disproves a conjecture of Anderson, Ash, Jones, Rider and Saffari, who proved the existence of γp > 0 for p > 1 and conjectured that it does not exists f...
متن کاملOn Lacunary Trigonometric Series.
1. Fundamental theorem. In a recent paper f I have proved the theorem that if a lacunary trigonometric series CO (1) X(a* cos nk6 + bk sin nk9) (nk+x/nk > q > 1, 0 ^ 0 ^ 2ir) 4-1 has its partial sums uniformly bounded on a set of 0 of positive measure, then the series (2) ¿(a*2 + bk2) k-l converges. The proof was based on the following lemma (which was not stated explicitly but is contained in ...
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ژورنال
عنوان ژورنال: Indiana University Mathematics Journal
سال: 1960
ISSN: 0022-2518
DOI: 10.1512/iumj.1960.9.59013