Reconstruction of Piecewise Smooth Functions from Non-uniform Grid Point Data |
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Authors: | Anne Gelb |
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Affiliation: | (1) Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287, USA |
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Abstract: | Spectral series expansions of piecewise smooth functions are known to yield poor results, with spurious oscillations forming
near the jump discontinuities and reduced convergence throughout the interval of approximation. The spectral reprojection
method, most notably the Gegenbauer reconstruction method, can restore exponential convergence to piecewise smooth function
approximations from their (pseudo-)spectral coefficients. Difficulties may arise due to numerical robustness and ill-conditioning
of the reprojection basis polynomials, however. This paper considers non-classical orthogonal polynomials as reprojection
bases for a general order (finite or spectral) reconstruction of piecewise smooth functions. Furthermore, when the given data
are discrete grid point values, the reprojection polynomials are constructed to be orthogonal in the discrete sense, rather
than by the usual continuous inner product. No calculation of optimal quadrature points is therefore needed. This adaptation
suggests a method to approximate piecewise smooth functions from discrete non-uniform data, and results in a one-dimensional
approximation that is accurate and numerically robust.
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Keywords: | Orthogonal polynomials piecewise smooth functions Gibbs phenomenon Gegenbauer reconstruction reprojection non-uniform grid point approximation |
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