On generalized shift bases for the wiener disc algebra |
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| Authors: | James R Holub |
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| Affiliation: | (1) Department of Mathematics, Virginia Polytechnic Institute and State University, 24061 Blacksburg, VA, USA |
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| Abstract: | LetW(D) denote the set of functionsf(z)=Σ
n=0
∞
A
n
Z
n
a
nzn for which Σn=0
∞|a
n
|<+∞. Given any finite set lcub;f
i
(z)rcub;
i=1
n
inW(D) the following are equivalent: (i) The generalized shift sequence lcub;f
1(z)z
kn
,f
2(z)z
kn+1, …,f
n
(z)z
(k+1)n−1rcub;
k=0
∞
is a basis forW(D) which is equivalent to the basis lcub;z
m
rcub;
m=0
∞
. (ii) The generalized shift sequence is complete inW(D), (iii) The function
![$$F(z) = \det \left {\begin{array}{*{20}c} {f_1 (z)f_1 (wz)...f_1 (w^{n - 1} z)} \\ {f_2 (z)wf_2 (wz)...w^{n - 1} f_2 (w^{n - 1} z)} \\ {f_n (z)w^{n - 1} f_n (wz)...(w^{n - 1} )^{n - 1} f_n (w^{n - 1} z)} \\ \end{array} } \right]$$](/content/b726r0616w026n9l/11856_2007_Article_BF02786518_TeX2GIFIE1.gif)
has no zero in |z|≦1, wherew=e
2πiti
/n. |
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| Keywords: | |
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