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The classical Gibbs phenomenon is a peculiarity that arises when approximating functions near a jump discontinuity with the Fourier series. Namely, the Fourier series "overshoots" (and "undershoots") the discontinuity by approximately 9% of the total jump. This same phenomenon, with the same value of the overshoot, has been shown to occur when approximating jump-discontinuous functions using specific families of orthogonal polynomials. In this paper, we extend these results and prove that the Gibbs phenomenon exists for approximations of functions with interior jump discontinuities with the two-parameter family of Jacobi polynomials Pn(a,b)(x). In particular, we show that for all a, b the approximation overshoots and undershoots the function by the same value as in the classical case – approximately 9% of the jump.
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Bahl, Riti, "Gibbs Phenomenon for Jacobi Approximations" (2021). Senior Projects Spring 2021. 165.
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