Best monotone approximation in 𝐿₁[0,1]

Huotari, Robert; Legg, David

doi:10.1090/S0002-9939-1985-0784179-X

Best monotone approximation in $L_ 1[0,1]$
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by Robert Huotari and David Legg PDF

Proc. Amer. Math. Soc. 94 (1985), 279-282 Request permission

Abstract:

If $f$ is a bounded Lebesgue measurable function on [0,1] and $1 < p < \infty$, let ${f_p}$ denote the best ${L_p}$-approximation to $f$ by nondecreasing functions. It is shown that ${f_p}$ converges almost everywhere as $p$ decreases to one to a best ${L_1}$-approximation to $f$ by nondecreasing functions. The set of best ${L_1}$-approximations to $f$ by nondecreasing functions is shown to include its supremum and infimum.

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