The Gibbs Phenomenon for Best ~ 1 - Trigonometric Polynomial Approximation
نویسنده
چکیده
Abstract. The classical Gibbs phenomenon for the Fourier sections (best L2trigonometric polynomial approximants) of a jump function asserts that, near the jump, the~e sections "overshoot" the function by an asymptotically constant factor g (the L2-Gibbs constant). In this paper we show that, for a class of one-jump discontinuous functions, a similar phenomenon holds for the trigonometric polynomials of best L 1approximation. We determine the L I-Gibbs constant y, which is substantially smaller than g. Furthermore, we prove that uniform convergence of best L ,-approximants takes place on intervals that avoid the jump. In the analysis we obtain some strong uniqueness theorems for best L ,-approximants.
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