Periodic interpolation on uniform meshes

We present a unified approach to error estimates of periodic interpolation on equidistant, full, and sparse grids for functions from a scale of function spaces which includes L 2-Sobolev spaces, the Wiener algebra and the Korobov spaces.

A Class of Periodic Function Spaces and Interpolation on Sparse Grids

W. Gautschi's theory of attenuation factors for families of periodic functions in one variable is extended to families of functions in several variables. Again, the linearity and the translation invariance of the operator which maps the data space onto the family are crucial. Special results are obtained for tensor product families and for the interpolation by translates of one generating function. Interesting examples are provided by box splines, which include certain finite elements.

http://www.sam.math.ethz.ch/~mhg/pub/mhg-published/36-Gut87-NM51-attfacmvFa.pdf

Attenuation factors in multivariate fourier analysis

In previous papers, we proposed new filters based on polynomial interpolation to approximate the point values of a piecewise smooth function f on [0,1] from its Fourier coefficients and derived error estimates. We proved that if all the discontinuity points of f are nodes, we can reconstruct point values of f accurately, even close to the discontinuities. We use the new filters to develop iterative methods for detecting the discontinuity points and, therefore, accurately approximate the point values of the function from its Fourier coefficients.

Iterative methods based on polynomial interpolation filters to detect discontinuities and recover point values from Fourier data

We consider Boolean sums of univariate interpolation operators which define multivariate jth order blending interpolation operators on sparse grids. Sample spaces are defined as range of the blending operators. Sample and wavelet spaces have significantly lower dimension and good approximation order for certain function spaces. Fast decomposition and reconstruction algorithms for bivariate spline wavelets, based on algorithms for univariate functions, are described. Operation counts for the algorithms are given and it is shown that the complexity depends linearly on the dimension of sample spaces.

Fast Algorithms for Periodic Spline Wavelets on Sparse Grids

The exact Fourier coefficients c,(PM/) are proportional to the discrete Fourier coefficients dj-"\f) if Pn is a translation invariant operator which depends only on the values of/on an equidistant mesh of width 2it/n. The proportionality factors which depend only on P" but not on / are called attenuation factors and have been calculated for several operators P" of spline type. Here we analyze first the interpolation problem which is produced by the functions <j(- -lirj/n),j •» 0,..., n — 1, where a is a suitable 2»r-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 attenua- tion factors may be expressed in terms of the Fourier coefficients of a. We point especially to the case where o 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.

/pdf/interpolation-on-uniform-meshes-by-the-translates-of-one-23r3wkobig.pdf

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

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