Polynomial series versus sinc expansions for functions with corner or endpoint singularities. (English) Zbl 0608.65010

In a review of methods that use ”Whittaker cardinal” or ”sinc” functions, F. Stenger [SIAM Rev. 23, 165-224 (1981; Zbl 0461.65007)] shows that these basis functions - in combination with a change-of-variable - are a powerful tool for approximating a function with weak singularities at the ends of the interval. Although Stenger himself is careful to note that the same optimal convergence rate can be obtained with other basis functions, he does not elaborate or give examples.
In this note, we show that the change-of-variable - not the use of sinc functions - is the key to success in coping with endpoint singularities. We explicitly construct approximations using mapped orthogonal polynomials which have the property of ”exponential” or ”infinite order” convergence for f(x) which has weak singularities at the endpoints but is regular on the interior of the interval.


65D15 Algorithms for approximation of functions
41A30 Approximation by other special function classes
41A25 Rate of convergence, degree of approximation
41A10 Approximation by polynomials


Zbl 0461.65007
Full Text: DOI Link


[1] Stenger, F., SIAM Rev., 23, 165 (1981) · Zbl 0461.65007
[2] Lund, J. R.; Riley, B. V., IMA J. Numer. Anal., 4, 83 (1984) · Zbl 0544.65057
[3] Boyd, J. P., J. Comput. Phys., 54, 382 (1984) · Zbl 0551.65006
[4] Elliott, D. E., Math. Comput., 18, 274 (1964) · Zbl 0119.32904
[5] Boyd, J. P., J. Comput. Phys., 45, 43 (1982) · Zbl 0488.65035
[6] Strang, G.; Fix, G. J., An Analysis of the Finite Element Method (1971), Prentice-Hall: Prentice-Hall New York, Chapter 8
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