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A Family of Approximation Formulas and Some Applications

Anry Nersessian
Organization: National Academy of Sciences of Armenia
Department: Institute of Mathematics
Journal / Anthology

Numerical Functional Analysis and Optimization
Year: 2000
Volume: 21
Issue: 1-2
Page range: 227-240

The general scheme, suggested in [1] using a basis of an infinite-dimensional space and allowing to construct finite-dimensional orthogonal systems and interpolation formulas, is improved in the paper. This results particularly in a generalization of the well-known scheme by which periodic interpolatory wavelets are constructed. A number of systems which do not satisfy all the conditions for multiresolution analysis but have some useful properties are introduced and investigated. Starting with general constructions in Hilbert spaces, we give a more careful consideration to the case connected with the classic Fourier basis. Convergence of expansions which are similar to partial sums of the summation method of Fourier series, as well as convergence of interpolation formulas are considered. Some applications to fast calculation of Fourier coefficients and to solution of integro-differential equations are given. The corresponding numerical results have been obtained by means of Mathematica 3.0 system.

*Applied Mathematics > Numerical Methods
*Applied Mathematics > Numerical Methods > Approximation Theory