Dot product rearrangements
Let a=(an), x=(xn) denote nonnegative sequences; x=(xπ(n)) denotes the rearranged sequence determined by the permutation π, a⋅x denotes the dot product ∑anxn; and S(a,x) denotes {a⋅xπ:π is a permuation of the positive integers}. We examine S(a,x) as a subset of the nonnegative real line in certain s...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Hindawi Limited
1983-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1155/S0161171283000368 |
| Summary: | Let a=(an), x=(xn) denote nonnegative sequences; x=(xπ(n)) denotes the rearranged sequence determined by the permutation π, a⋅x denotes the dot product ∑anxn; and S(a,x) denotes {a⋅xπ:π is a permuation of the positive integers}. We examine S(a,x) as a subset of the nonnegative real line in certain special circumstances. The main result is that if an↑∞, then S(a,x)=[a⋅x,∞] for every xn↓≠0 if and only if an+1/an is uniformly bounded. |
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| ISSN: | 0161-1712 1687-0425 |