منابع مشابه
Integrals and Summable Trigonometric Series
is that of suitably defining a trigonometric integral with the property that, if the series (1.1) converges everywhere to a function ƒ(x), then f(x) is necessarily integrable and the coefficients, an and bn, given in the usual Fourier form. It is well known that a series may converge everywhere to a function which is not Lebesgue summable nor even Denjoy integrable (completely totalisable, [3])...
متن کاملSummable series and the Riemann rearrangement theorem
Let N be the set of positive integers. A function from N to a set is called a sequence. If X is a topological space and x ∈ X, a sequence a : N → X is said to converge to x if for every open neighborhood U of x there is some NU such that n ≥ NU implies that an ∈ U . If there is no x ∈ X for which a converges to x, we say that a diverges. Let a : N→ R. We define s(a) : N→ R by sn(a) = ∑n k=1 ak....
متن کاملConvergence Factors for Double Series Summable by Nörlund Means.
In a note' published in volume 21 of these PROCEEDINGS I gave necessary and sufficient conditions for convergence factors in simply infinite series summable by N6rlund means. The purpose of the present note is to furnish analogous results for double series. We are given a doubly infinite set of complex constants cij (i, j = 0, 1, 2, 3, .. .), where coo 0 0 and Zc,ix'y' is convergent for I x I <...
متن کاملHe Rearrangemens of C1-slummable Series
is absolutely convergent and has the sum s . Then, as is well known, every rearrangement, S' a ;, of (1) also converges and has the same sum s . If, however, (1) n=1 converges, but not absolutely, then, according to Riemann's classical rearrangement theorem [3, p . 235, or 2, p . 318j, for every real number s', there exists a rearrangement of (1) whose sum is s' . Assume, now, that (1) is C 1-s...
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...
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ژورنال
عنوان ژورنال: Acta Mathematica
سال: 1954
ISSN: 0001-5962
DOI: 10.1007/bf02392700