Complex Analysis - Toeplitz's Theorem and Its Application
Автор: MathWU悟数
Загружено: 2026-01-08
Просмотров: 141
(1) Prove Toeplitz's theorem.
Suppose (a_nk) is an infinite matrix of complex numbers (n, k = 1, 2, ...) which satisfies
(i) \sum_{k=1}^{infinity} |a_nk| not greater than A for n = 1, 2, ...;
(ii) lim_{n to infinity} a_nk = 0 for k = 1, 2, ...;
(iii) lim_{n to infinity} (\sum_{k=1}^{infinity} a_nk) = 1.
Then, for any positive integer n and any convergent {zeta_n} the series \sum_{k=1}^{infinity} a_nk zeta_k is convergent. Moreover, if z_n = \sum_{k=1}^{infinity} a_nk zeta_k, then lim_{n to infinity} z_n exists and equal lim_{n to infinity} zeta_n.
(2) Prob 3.3 in "Theory of Functions of a Complex Variable (Vol 1)" - A.I. Markushevich, p.55
Prov that if lim_{n to infinity} z_n = zeta, then
lim_{n to infinity} (z_1 + ... + z_n)/n = zeta.
More generally, prove that if lim_{n to infinity} z_n = zeta, then
lim_{n to infinity} (c_1z_1 + ... + c_nz_n)/(c_1 + ... + c_n) = zeta,
where c_1, c_2, ..., c_n, ... is any sequence of positive numbers such that
lim_{n to infinity} (c_1 + ... + c_n) = + infinity.
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