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A spanning set for $ {\scr C}(I\sp n)$


Author: Thomas Bloom
Journal: Trans. Amer. Math. Soc. 321 (1990), 741-759
MSC: Primary 41A10; Secondary 32E30, 41A63
DOI: https://doi.org/10.1090/S0002-9947-1990-0984854-4
MathSciNet review: 984854
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Abstract: $ \mathcal{C}({I^n})$ denotes the Banach space of continuous functions on the unit $ n$-cube, $ {I^n}$, in $ {{\mathbf{R}}^n}$. Let $ \{ {a^i}\} $, $ i = 0,1,2, \ldots ,$, be a countable collection of $ n$-tuples of positive real numbers satisfying $ {\operatorname{lim}_i}a_j^i = + \infty $ for $ j = 1, \ldots ,n$. We canonically enlarge the family of monomials $ \{ {x^{{a^i}}}\} $ to a family of functions $ \mathcal{F}(A)$.

Conjecture. The linear span of $ \mathcal{F}(A)$ is dense in $ \mathcal{C}({I^n})$ if and only if $ \Sigma _{i = 0}^\infty 1/\left\vert {{a^i}} \right\vert = + \infty $. For $ n = 1$ this is equivalent to the Müntz-Szasz theorem. For $ n > 1$ we prove the necessity in general and the sufficiency under the additional hypothesis that there exist constants $ G$, $ N > 1$ such that $ \left\vert {{a^i}} \right\vert \leq G{\operatorname{exp}}({i^N})$ for all $ i$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1990-0984854-4
Keywords: Müntz-Szasz theorem, bounded holomorphic function on cones, uniqueness sets
Article copyright: © Copyright 1990 American Mathematical Society

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