On a new type of mixed interpolation

Abstract We approximate every function f by a function f n ( x ) of the form a cos kx + b sin kx + Σ n −2 i =0 c i x i so that f ( jh ) = f n ( jh ) for the n + 1 equidistant points jh , j = 0,…, n . That interpolation function f n ( x ) is proved to be unique and can be written as the sum of the n th-degree interpolation polynomial based on the same points and two correction terms. The error term is also discussed. The results for this mixed type of interpolation reduce to the known results of the polynomial case as the parameter k is tending to 0. This new interpolation theory will be used in the future for the construction of quadrature rules and multistep methods for ordinary differential equations.

[1]  A. C. Allison,et al.  Exponential-fitting methods for the numerical solution of the schrodinger equation , 1978 .

[2]  Ben P. Sommeijer,et al.  Linear Multistep Methods with Reduced Truncation Error for Periodic Initial-value Problems , 1984 .

[3]  M. Beiner,et al.  Direct numerical integration of coupled one-dimensional Schrödinger equations , 1972 .

[4]  David Levin,et al.  Procedures for computing one- and two-dimensional integrals of functions with rapid irregular oscillations , 1982 .

[5]  Ulf Torsten Ehrenmark,et al.  A three-point formula for numerical quadrature of oscillatory integrals with variable frequency , 1988 .

[6]  Families of backward differentiation methods based on trigonometric polynomials , 1986 .

[7]  D. G. Bettis,et al.  Stabilization of Cowell's method , 1969 .

[8]  Beny Neta,et al.  Families of methods for ordinary differential equations based on trigonometric polynomials , 1984 .

[9]  E. A. Flinn.  A Modification of Filon's Method of Numerical Integration , 1960, JACM.

[10]  Tom Lyche,et al.  Chebyshevian multistep methods for ordinary differential equations , 1972 .

[11]  Philip Rabinowitz,et al.  Methods of Numerical Integration , 1985 .

[12]  L. Filon.  III.—On a Quadrature Formula for Trigonometric Integrals. , 1930 .

[13]  Yudell L. Luke,et al.  On the computation of oscillatory integrals , 1954, Mathematical Proceedings of the Cambridge Philosophical Society.

[14]  W. Gautschi.  Numerical integration of ordinary differential equations based on trigonometric polynomials , 1961 .

[15]  E. O. Tuck,et al.  A Simple “Filon-trapezoidal” rule , 1967 .

[16]  D. G. Bettis.  Numerical integration of products of fourier and ordinary polynomials , 1970 .