On a problem concerning integrability of trigonometric series
نویسندگان
چکیده
منابع مشابه
On Integrability of Functions Defined by Trigonometric Series
The goal of the present paper is to generalize two theorems of R.P. Boas Jr. pertaining to L (p > 1) integrability of Fourier series with nonnegative coefficients and weight x . In our improvement the weight x is replaced by a more general one, and the case p = 1 is also yielded. We also generalize an equivalence statement of Boas utilizing power-monotone sequences instead of {n}.
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The theorem proved in this paper is a generalization of some results, concerning integrability of trigonometric series, due to R.P. Boas, L. Leindler, etc. This result can be considered as an example showing the utility of the notion of power-monotone sequences.
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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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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...
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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]...
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ژورنال
عنوان ژورنال: Journal of Mathematical Analysis and Applications
سال: 1970
ISSN: 0022-247X
DOI: 10.1016/0022-247x(70)90280-5