Estimation of errors of quadrature formula for singular integrals of Cauchy type with special forms

2011/03/05 by Israilov, M. I
#FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.1103.1034

Abstract

In this work, we consider the singular integrals of Cauchy type of the forms \ds J(f,x)= \frac√(1-x²)π∫_-1¹(f(t))/(√(1-t²)(t-x)) dt, -11. With the help of linear spline interpolation, we have proved the rate of convergence of the errors of QFs \reeq3 and \reeq4 for different classes (i.e. H^\a([-1,1],K), C^m,\a[-1,1], W^r[-1,1]) of density function f(t). It is shown that approximation by spline possesses more advantages than other kinds of approximation: it requires the minimum smoothness of density function f(x) to get good order of decreasing errors.

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