Tauberian Theorems for Weighted Means of Double Sequences |
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Authors: | Chang-Pao Chen Jui-Ming Hsu |
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Institution: | (1) Department of Mathematics, National Tsing Hua University Hsinchu, 30043, Taiwan Republic of China;(2) Department of Mathematics, National Tsing Hua University Hsinchu, 30043, Taiwan Republic of China |
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Abstract: | Let p := {p
j
}
j=0
and q := {q
k
}
k–0
be complex (or real) sequences with the property that P
m
:=
j–0
m
p
j
0 for all m 0, Q
n
:=
k–0
n
q
k
0 for all n 0, and both of {P
m
}
m=0
and {Q
n
}
n=0
are varying away from 1. Assume that {s
mn
} is a double sequence in C(or one of R, a Banach space, and an ordered linear space), which is (N¯,p,q; ,) summable to a finite limit, where (,) =(1,1), (1,0), or (0,1). We give necessary and sufficient conditions under which {s
mn
} converges in Pringsheim's sense. These conditions are weaker than the two-dimensional analogues of Landau's condition and Schmidt's slow decrease condition. Our results generalize and extend 1 4, 12 15]. We also solve the problems posed in 3, 13, 14]. |
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Keywords: | |
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