Summary
Let
be thek-dimensional subspace spanned by the translates ϕ(·−2πj/k),j=0, 1, ...,k−1, of a continuous, piecewise smooth, complexvalued, 2π-periodic function ϕ. For a given functionf∈L 2(−π, π), its least squares approximantS kf from
can be expressed in terms of an orthonormal basis. Iff is continuous,S kf can be computed via its discrete analogue by fast Fourier transform. The discrete least squares approximant is used to approximate Fourier coefficients, and this complements the works of Gautschi on attenuation factors. Examples of
include the space of trigonometric polynomials where ϕ is the de la Valleé Poussin kernel, algebraic polynomial splines where ϕ is the periodic B-spline, and trigonometric polynomial splines where ϕ is the trigonometric B-spline.
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Lee, S.L., Tan, R.C.E. & Tang, W.S. L 2-approximation by the translates of a function and related attenuation factors. Numer. Math. 60, 549–568 (1991). https://doi.org/10.1007/BF01385736
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DOI: https://doi.org/10.1007/BF01385736