On a generalized moment problem. II
1984; American Mathematical Society; Volume: 91; Issue: 4 Linguagem: Inglês
10.1090/s0002-9939-1984-0746093-4
ISSN1088-6826
Autores Tópico(s)Advanced Topics in Algebra
ResumoRecently, we have extended the well-known Müntz-Szász theorem by showing that if f ( z ) f(z) is absolutely continuous and | f ′ ( x ) | ⩾ k > 0 |f’(x)| \geqslant k > 0 a.e. on ( a , b ) (a,b) , where a ⩾ 0 a \geqslant 0 and if { n p } \{ {n_p}\} is a sequence of positive numbers tending to infinity and satisfying ∑ p = 1 ∞ 1 / n p = ∞ \sum _{p = 1}^\infty 1/{n_p} = \infty , then the sequence { f ( x ) n p } \{ f{(x)^{{n_p}}}\} is complete on ( a , b ) (a,b) if and only if f ( x ) f(x) is strictly monotone on ( a , b ) (a,b) . We now apply Zarecki’s theorem to improve the condition " | f ′ ( x ) | ⩾ k > 0 |f’(x)| \geqslant k > 0 a.e. on ( a , b ) (a,b) " by the condition f ′ ( x ) ≠ 0 f’(x) \ne 0 a.e. on ( a , b ) (a,b) ". Furthermore, we extend some well-known theorems of Picone, Mikusiński, and Boas.
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