# 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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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 $$\sum^{\infty}_{i=0}{\frac{\lambda_i+1/p}{(\lambda_i+1/p)^2+1}} = \infty.$$