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Polynomial approximations of functions with endpoint singularities and product integration formulas

Authors: Giuseppe Mastroianni and Giovanni Monegato
Journal: Math. Comp. 62 (1994), 725-738
MSC: Primary 65D32; Secondary 41A10
MathSciNet review: 1201069
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Abstract: Several problems of mathematical physics lead to Fredholm integral equations of the second kind where the kernels are either weakly or strongly singular and the known terms are smooth. These equations have solutions which are smooth in the whole interval of integration except at the endpoints where they have mild singularities. In this paper we derive new pointwise and uniform polynomial approximation error estimates for that type of function. These estimates are then used to obtain bounds for the remainder terms of interpolatory product rules, based on the zeros of classical Jacobi orthogonal polynomials, that have been proposed for the discretization of integrals of the form

$\displaystyle \int_{ - 1}^1 {k(x,y)f(x)dx,} $

appearing in the integral equations mentioned above.

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