منابع مشابه
On the Summation of Certain Iterated Series
The paper gives a unified treatment of the summation of certain iterated series of the form ∑∞ n=1 ∑∞ m=1 an+m, where (an)n∈N is a sequence of real numbers. We prove that, under certain conditions, the double iterated series equals the difference of two single series.
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A general summability method of different orthogonal series is given with the help of an integrable function θ. As special cases the trigonometric Fourier, Walsh-, Walsh-Kaczmarz-, Vilenkinand Ciesielski-Fourier series and the Fourier transforms are considered. For each orthonormal system a different Hardy space is introduced and the atomic decomposition of these Hardy spaces are presented. A s...
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Given nonnegative real sequences a = (αk)k∈Z and b = (βk)k∈Z we study the generated summation operator Sa,b(x) := ( αk [∑ l<k βlxl ]) k∈Z , x = (xk)k∈Z , regarded as a mapping from `p(Z) to `q(Z). We give necessary and sufficient conditions for the boundedness of Sa,b and prove optimal estimates for its entropy numbers relative to the summation properties of a and b. Our results are applied to ...
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Corresponding material in the book: Section 12.1, 12.2, 12.3. What students should definitely get: The summation notation and how it works, series, concepts of convergence. The use of telescoping and forward difference operator ideas to sum up series. The use of the integral test and other tests to determine whether a series converges and obtain numerical estimates. Convergence rules for ration...
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ژورنال
عنوان ژورنال: Proceedings of the London Mathematical Society
سال: 1902
ISSN: 0024-6115
DOI: 10.1112/plms/s1-35.1.284