Interpolation on uniform meshes by the translates of one function and related attenuation factors

Abstract:

The exact Fourier coefficients ${c_j}({P_n}f)$ are proportional to the discrete Fourier coefficients $d_j^{(n)}(f)$ if ${P_n}$ is a translation invariant operator which depends only on the values of f on an equidistant mesh of width $2\pi /n$. The proportionality factors which depend only on ${P_n}$ but not on f are called attenuation factors and have been calculated for several operators ${P_n}$ of spline type. Here we analyze first the interpolation problem which is produced by the functions $\sigma ( \bullet - 2\pi j/n),j = 0, \ldots ,n - 1$, where $\sigma$ is a suitable $2\pi$-periodic generating function. It is essential that the associated interpolation matrix is of discrete convolution type. Thus, we can derive conditions guaranteeing the unique solvability of the interpolation problem and representations of the interpolating function. Then the attenuation factors may be expressed in terms of the Fourier coefficients of $\sigma$. We point especially to the case where $\sigma$ is a reproducing kernel in a suitable Hilbert space. Here we get attenuation factors of a new type which are generated by interpolation with analytic functions.