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 f is absolutely continuous on [ 0 , 1 ] [0,1] if and only if the sequence B . ( f ) B.(f) of Bernstein polynomials of f f converges to f f with respect to the total variation norm.
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