Variation norm convergence of function sequences
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- by Randolph Constantine PDF
- Proc. Amer. Math. Soc. 45 (1974), 339-345 Request permission
Abstract:
We prove that a pointwise convergent sequence of convex functions with a continuous limit converges with respect to the total variation norm. This yields a theorem on convexity-preserving operators which has as a corollary the result that a complex function $f$ is absolutely continuous on $[0,1]$ if and only if the sequence $B.(f)$ of Bernstein polynomials of $f$ converges to $f$ with respect to the total variation norm.References
- J. R. Edwards and S. G. Wayment, Representations for transformations continuous in the $\textrm {BV}$ norm, Trans. Amer. Math. Soc. 154 (1971), 251–265. MR 274704, DOI 10.1090/S0002-9947-1971-0274704-4
- Wassily Hoeffding, The $L_{1}$ norm of the approximation error for Bernstein-type polynomials, J. Approximation Theory 4 (1971), 347–356. MR 288466, DOI 10.1016/0021-9045(71)90001-3
- G. G. Lorentz, Bernstein polynomials, Mathematical Expositions, No. 8, University of Toronto Press, Toronto, 1953. MR 0057370
Additional Information
- © Copyright 1974 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 45 (1974), 339-345
- MSC: Primary 41A30; Secondary 26A51
- DOI: https://doi.org/10.1090/S0002-9939-1974-0352808-3
- MathSciNet review: 0352808