Abstract
This paper considers the use of polynomial splines to approximate periodic functions with jump discontinuities of themselves and their derivatives when the information consists only of the first few Fourier coefficients and the location of the discontinuities. Spaces of splines are introduced which include, members with discontinuities at those locations. The main results deal with the orthogonal projection of such a spline space on spaces of trigonometric polynomials corresponding to the known coefficients. An approximation is defined based on inverting this projection. It is shown that when discontinuities are sufficiently far apart, the projection is invertible, its inverse has norm close to 1, and the approximation is nearly as good as directL 2 approximation by members of the spline space. An example is given which illustrates the results and which is extended to indicate how the approximation technique may be used to provide smoothing which which accurately represents discontinuities.
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References
W. Gautschi, Attenuation factors in practical Fourier analysis, Numer. Math. 18 (1972) 373–400.
B.W. Char, K.O. Geddes, G.H. Gonnet, B.L. Leong, M.B. Monagan, and S.W. Watt, Maple V Language Reference Manual (Springer-Verlag, New York, 1991).
L.L. Schumaker,Spline Functions: Basic Theory (Wiley, New York, 1981).
G. Wahba, Three topics in ill-posed problems, inInverse and Ill-Posed Problems, H.W. Engl and C.W. Groetsch (Eds.) (Academic Press, Boston, 1987).
R.K. Wright, “segmentd”, a Maple package for manipulating piecewise procedures, submitted to the Maple share library (1994).
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Wright, R.K. Spline fitting discontinuous functions given just a few Fourier coefficients. Numer Algor 9, 157–169 (1995). https://doi.org/10.1007/BF02143932
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DOI: https://doi.org/10.1007/BF02143932