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The Full Muntz Theorem in C[0,1] and L_1[0,1]

Borwein, Peter and Erdelyi, Tamas (1994) The Full Muntz Theorem in C[0,1] and L_1[0,1]. [Preprint]

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      Abstract

      The main result of this paper is the establishment of the "full Muntz Theorem" in C[0,1]. This characterizes the sequences $\{\lambda_i\}^\infty_{i=1}$ of distinct, positive real numbers for which $$\text{\rm span}\{1, x^{\lambda_1},x^{\lambda_2}, \ldots \}$$ is dense in C[0,1]. The novelty of this result is the treatment of the most difficult case when $\inf_i{\lambda_i} = 0$ while $\sup_i{\lambda_i}=\infty$. The paper settles the $L_\infty$ and $L_1$ cases of the following. Conjecture (Full Muntz Theorem in L_p[0,1]) Let p \in [1,\infty]. Suppose $\{\lambda_i\}^\infty_{i=0}$ is a sequence of distinct real numbers greater than $-1/p$. Then $$\text{\rm span}\{x^{\lambda_0}, x^{\lambda_1}, \ldots \}$$ is dense in L_p[0,1] if and only if $$\sum^{\infty}_{i=0}{\frac{\lambda_i+1/p}{(\lambda_i+1/p)^2+1}} = \infty.$$

      Item Type: Preprint
      Additional Information: pubdom FALSE
      Uncontrolled Keywords: Muntz's theorem, denseness
      Subjects: 26-xx Real functions > 26Dxx Inequalities
      41-xx Approximations and expansions > 41Axx Approximations and expansions
      30-xx Functions of a complex variable > 30Bxx Series expansion
      Faculty: UNSPECIFIED
      Depositing User: Users 1 not found.
      Date Deposited: 16 Nov 2003
      Last Modified: 21 Apr 2010 11:13
      URI: https://docserver.carma.newcastle.edu.au/id/eprint/84

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