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| The Lebesgue Constants and Gibbs Phenomenon for Non-negative (f , dn)Summability Method |
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Citation： |
Shi Xianliang.The Lebesgue Constants and Gibbs Phenomenon for Non-negative (f , dn)Summability Method[J].Chinese Annals of Mathematics B,1982,3(3):365~374 |
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Net amount： 1020 |
Authors： |
Shi Xianliang; |
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| Abstract： |
The ( f,d_n) -summability method is defined as follows^[1,4]: Let f be a nonconstant
function, analytic in |z | < R for R > l, and let {d_n} be a sequence of complex numbers,such that for all n,$d_n \ne -f(1)$.Suppose that the elements of the metrix A = (a_nk) are given by the relations
$a_00=1,a_0k=0(k \geq 1)$
$[\prod\limits_{j = 1}^n {\frac{{f(z) + {d_j}}}{{f(1) + {d_j}}} = \sum\limits_{k = 0}^\infty {{a_{nk}}{z^k}} } \]$
A sequence {S_n} is said to be ( f, d_n), —summable to s, if \sigma_n = \sum\limits_{k=0}^\infty \arrow s as n \arrow \infty. The
( f, d_n) —summability method is said to be non-negative if for all n, d_n> 0 and the
Maclaurin coefficients of f are real and non-negative. The Lebesgue constants for the
( f,d_n)-method are defined by
$L_n(A)=2/\pi \int_0^\pi /2 {\frac{|\sum\limits_{k=0}^\infty {a_nk sin(2k+1)t|}{sint}dt}$
In this parer we prove the following two theorems. |
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