On the evaluation of certain two-dimensional singular integrals with Cauchy kernels. (English) Zbl 0541.41033

Dagnino, C.; Palamara Orsi, A.

Numer. Math. 46, 121-130 (1985).

The authors consider product formulas of interpolatory type for the two- dimensional Cauchy principal value integral: \[ I(f;\eta_ 1,\eta_ 2)=\int^{1}_{-1}\int^{1}_{-1}\omega(x,y)\frac{f(x,y)}{(x-\eta_ 1)(y-\eta_ 2)}dx dy \] where: \(\eta_ 1,\eta_ 2\in(-1,1)\), \(\omega(x,y)=\omega_ 1(x).\omega_ 2(y)\) and \(\omega_ 1(x)\) and \(\omega_ 2(y)\) are two absolutely integrable weight functions. The integral is approximated by \[ F_{n,m}(f;\eta_ 1,\eta_ 2)=\sum^{n}_{i=0}\sum^{m}_{j=0}A_ i^{(1)}(\eta_ 1)A_ j^{(2)}(\eta_ 2)f(x_ i,y_ j) \] where the nodes \(\{x_ i\}\) and \(\{y_ j\}\) are the zeros of the Chebyshev polynomials of the first kind, commonly named ”classical” abscissas, or the Clenshaw points, often called ”practical” abscissas. Convergence results for these rules are presented.

MSC:

41A55	Approximate quadratures
41A17	Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
42B20	Singular and oscillatory integrals (Calderón-Zygmund, etc.)

Keywords:

two-dimensional singular integrals; Cauchy kernels; product formulas; weight functions; Chebyshev polynomials

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