@article {IOPORT.03942133,
    author       = {Dahmen, W. and Micchelli, C.A. and Smith, P.W.},
    title        = {Asymptotically optimal sampling schemes for periodic functions.},
    year         = {1986},
    journal      = {Mathematical Proceedings of the Cambridge Philosophical Society},
    volume       = {99},
    issn         = {0305-0041},
    pages        = {171-177},
    publisher    = {Cambridge University Press, Cambridge},
    doi          = {10.1017/S0305004100064069},
    abstract     = {For $\phi \in L\sb*\sp 2[0,1)$ the space of $L\sp 2$-one periodic complex valued functions let ${\cal F}\sb{\phi}=\{\phi\sb*h:\Vert h\Vert =\int\sp{1}\sb{0}\vert h(t)\vert\sp 2dt\le 1\}$. This paper is concerned with the question when sampling a function $f\in {\cal F}\sb{\phi}$ at equidistant points gives optimal information for approximation. More precisely, denoting by A any mapping from ${\bbfR}\sp N$ into $L\sb*\sp 2[0,1)$ and by I any continuous linear mapping from $L\sb*\sp 2[0,1)$ into ${\bbfR}\sp N$ define $i\sb N({\cal F})=\inf\sb{A,I}\sup\sb{f\in {\cal F}}\Vert f-A(If)\Vert$ as well as $$ E({\cal F};\Delta\sb N)=\inf\sb{A}\sup\sb{f\in {\cal F}}\Vert f- A(f(0),f(\frac{1}{N}),...,f(\frac{N-1}{N}))\Vert. $$ The objective is to identify $\phi$ such that (1) $\overline{\lim}\sb{n\to \infty}E({\cal F}\sb{\phi},\delta\sb N)/i\sb N({\cal F}\sb{\phi})<\infty$. To this end, E(${\cal F}\sb{\phi},\Delta\sb N)$ is evaluated first precisely in terms of the Fourier coefficients of $\phi$. Furthermore, by using discrete Fourier transform with judiciously chosen attenuation factors an upper bound for E(${\cal F}\sb{\phi},\Delta\sb N)$ is obtained which is shown to be compatible with $i\sb N({\cal F}\sb{\phi})$ (in the sense of (1)) for a wide class of functions $\phi$. Finally, using the precise evaluation of E(${\cal F}\sb{\phi},\Delta\sb N)$ and the fact that $i\sb N({\cal F}\sb{\phi})$ is invariant under any rearrangement of the Fourier coefficients ${\hat \phi}$(k) while E(${\cal F}\sb{\phi},\Delta\sb N)$ is not an example is constructed for which (1) fails.},
    identifier   = {03942133},
}