Best monotone approximation in 𝐿₁[0,1]

Huotari, Robert; Legg, David

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

Best monotone approximation in $L_ 1[0,1]$
HTML articles powered by AMS MathViewer

by Robert Huotari and David Legg

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

DOI: https://doi.org/10.1090/S0002-9939-1985-0784179-X

PDF | 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.

References

Richard B. Darst and Robert Huotari, Best $L_1$-approximation of bounded, approximately continuous functions on $[0,1]$ by nondecreasing functions, J. Approx. Theory 43 (1985), no. 2, 178–189. MR 775785, DOI 10.1016/0021-9045(85)90125-X
R. B. Darst, D. A. Legg, and D. W. Townsend, The Pólya algorithm in $L_{\infty }$ approximation, J. Approx. Theory 38 (1983), no. 3, 209–220. MR 705541, DOI 10.1016/0021-9045(83)90129-6
I. P. Natanson, Theory of functions of a real variable, Frederick Ungar Publishing Co., New York, 1955. Translated by Leo F. Boron with the collaboration of Edwin Hewitt. MR 0067952
D. Landers and L. Rogge, Natural choice of $L_{1}$-approximants, J. Approx. Theory 33 (1981), no. 3, 268–280. MR 647853, DOI 10.1016/0021-9045(81)90076-9
Takuro Shintani and T. Ando, Best approximants in $L^{1}$ space, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 33 (1975/76), no. 1, 33–39. MR 380963, DOI 10.1007/BF00539858