Complex Analysis - Toeplitz's Theorem and Its Application скачать в хорошем качестве

Complex Analysis - Toeplitz's Theorem and Its Application

(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.