L2-approximation by the translates of a function and related attenuation factors

Abstract

Let[Figure not available: see fulltext.] be the k-dimensional subspace spanned by the translates φ{symbol}(·-2πj/k), j=0, 1, ..., k-1, of a continuous, piecewise smooth, complexvalued, 2π-periodic function φ{symbol}. For a given function f∈L2(-π, π), its least squares approximant Skf from[Figure not available: see fulltext.] can be expressed in terms of an orthonormal basis. If f is continuous, Skf 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[Figure not available: see fulltext.] include the space of trigonometric polynomials where φ{symbol} is the de la Valleé Poussin kernel, algebraic polynomial splines where φ{symbol} is the periodic B-spline, and trigonometric polynomial splines where φ{symbol} is the trigonometric B-spline. © 1992 Springer-Verlag.