Splines and other Approximations
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In this chapter we move away from the Fourier series and the general orthogonal series expansions. We will consider the appearance of the Gibbs phenomenon in the piecewise-linear approximation of functions with jump discontinuities that was first considered by Foster and Richards . In this case, they demonstrated that the size of the overshoot is larger, i.e. about 13.4% compared to the classical 8.95% value of the Fourier trigonometric series and integrals approximations. For both cases here the usual measure of L 2 (or the least square)-approximation is employed. They pointed out that the (lower) overshoot of 8.95% is not exclusive for the trigonometric polynomial, but rather due to the latter being the best least square approximation. This piecewise-linear approximation is seen as a special case of using splines of degree one approximation. Richards  extended this analysis to high order splines where he showed that the size of the overshoot decreases with increasing the order m of the spline ∅ m+i (x) considered. Indeed it turns out that the size of the overshoot approaches the classical one of 8.95% as the order m + 1 (degree m) of the employed spline approaches infinity. This result was sought and proved almost completely by Richards , then was finalized later by Shim and Volkmer  and Foster and Richards .
KeywordsFourier Series Discrete Fourier Transform Trigonometric Polynomial Jump Discontinuity Spline Approximation
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Additional References (From Appendix A.)
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