AN APPROXIMATION TO THE PRODUCT OF DISCRETE FUNCTIONS

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  • 1 The University of Chicago
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Abstract

If one regards a discrete function as a vector, the “best” approximation to the product of two discrete functions (defined for the same set of values of the argument) is not necessarily the ordinary scalar product. The “best” approximation is shown to be an approximation-in-the-mean to the product of the trigonometric interpolation polynomials (cardinal functions) which correspond to the given discrete functions. This approximation arises naturally when the product is taken in the spectral domain. However, it can be approached by the ordinary scalar product provided the input functions are smoothed. The smoothing operator is linear and easily computed; it results in the suppression of all harmonics of wave length less than four times the mesh length.

Abstract

If one regards a discrete function as a vector, the “best” approximation to the product of two discrete functions (defined for the same set of values of the argument) is not necessarily the ordinary scalar product. The “best” approximation is shown to be an approximation-in-the-mean to the product of the trigonometric interpolation polynomials (cardinal functions) which correspond to the given discrete functions. This approximation arises naturally when the product is taken in the spectral domain. However, it can be approached by the ordinary scalar product provided the input functions are smoothed. The smoothing operator is linear and easily computed; it results in the suppression of all harmonics of wave length less than four times the mesh length.

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