Geodesic
Approximation of functions by algebraic polynomials in the $L_p$ metric
Izvestiya. Mathematics , Tome 5 (1971) no. 4, pp. 889-914.

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We introduce a new method of approximation of nonperiodic functions by algebraic polynomials. In particular, by this method we establish necessary and sufficient conditions for a function on the interval $[-1,1]$ to satisfy Hölder's condition in the $L_p$ metric. 
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V. P. Motornyi. Approximation of functions by algebraic polynomials in the $L_p$ metric. Izvestiya. Mathematics , Tome 5 (1971) no. 4, pp. 889-914. http://geodesic.mathdoc.fr/item/IM2_1971_5_4_a8/

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