Showing papers by "Walter Gautschi published in 1971"

Attenuation factors in practical Fourier analysis

Abstract: Given a 2 ?-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 ?-periodic function ? and aproximatingc n (f) byc n (?). In a number of cases, where ? 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) and? n is independent off. Our interest is in the factors? n , called attenuation factors. We first clarify the conditions on the approximation processP:f?? 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 ?=P f by the same amount. An explicit formula for? n is obtained for any processP which is linear and translation invariant. For interpolation processes it suffices to obtain a factorizationc n (?)=?(n)? f (n), where ? does not depend onf and? f (n) has periodN. This also implies existence of attenuation factors? n , which are expressible in terms of ?. The results can be extended in two directions: First, the processP may also approximate successive derivative valuesf μ (x) , ?=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 ? 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.