Attenuation factors in practical Fourier analysis

Summary

Given a 2 π-periodic functionf, it is desired to approximate itsn-th Fourier coefficientc _n (f) in terms of function valuesf _μ atN equidistant abscisses

$$x_\mu = \mu 2\pi /N, \mu = 0, 1, ..., N - 1.$$

A time-honored procedure consists in interpolatingf at these points by some 2 π-periodic function ϕ and aproximatingc _n (f) byc _n (ϕ). In a number of cases, where ϕ is piecewise polynomial, it has been known that$c_n (\varphi ) = \tau _n \hat c_n (f)$ where$\hat c_n (f)$ is the trapezoidal rule approximation ofc _n (f) andτ _n is independent off. Our interest is in the factorsτ _n , called attenuation factors. We first clarify the conditions on the approximation processP:f→ϕ under which such attenuation factors arise. It turns out that a necessary and sufficient condition is linearity and translation invariance ofP. The latter means that shifting the periodic dataf={f _μ } one place to the right has the effect of shifting ϕ=P f by the same amount. An explicit formula forτ _n is obtained for any processP which is linear and translation invariant. For interpolation processes it suffices to obtain a factorizationc _n (ϕ)=ω(n)ψ _f (n), where ω does not depend onf andψ _f (n) has periodN. This also implies existence of attenuation factorsτ _n , which are expressible in terms of ω. The results can be extended in two directions: First, the processP may also approximate successive derivative valuesf ^(x) _μ , ϰ=0,1,...,k−1, of the functionf, in which case formulas of the type$c_n (\varphi ) = \sum\limits_{x = 0}^{k - 1} {\tau _{_{n_{,x} } } \hat c} (f^{(x)} )$ emerge. Secondly,P may be translation invariant overr subintervals,r>1, in which case$c_n (\varphi ) = \sum\limits_{\varrho = 0}^{r - 1} {\tau _{n,\varrho } \hat c_n + \varrho N/} r(f)$. All results are illustrated by a number of examples, in which ϕ are polynomial and nonpolynomial spline interpolants, including deficient splines, as well as other piecewise polynomial interpolants. These include approximants of low, and medium continuity classes permitting arbitrarily high degree of approximation.