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

In this work, we consider the singular integrals of Cauchy type of the forms $$\text{ds}J(f,x)=\frac{\sqrt{1-x^2}}{\pi }{\int }_{-1}^1\frac{f(t)}{\sqrt{1-t^2}(t-x)}dt,-1<x<1and$$\ds \Phi(f,z)= -\frac{\sqrt{z^2-1}}{\pi}\int_{-1}^1\frac{f(t)}{\sqrt{1-t^2}(t-z)},dt, \ \ \qquad z\notin [-1,1].$$whichareunderstoodasCauchyprincipalvalueintegrals\mathrm{.}Quadratureformulas(QFs)forsingularintegrals(SIs)\text{re}eq1and\text{re}eq2areoftheforms$$\ds J(f,x)= \sum_{k=0}^{N}A_k(x)f(t_k)+ R_N(f,x), \ \ \qquad-1<x<1. and \ds \Phi(f,z)= \sum_{k=0}^{N}B_k(z)f(t_k)+ R_N^*(f,z), z\notin [-1,1]$$where$z$iscomplexvariablewith$|Re(z)|>1$\mathrm{.}Withthehelpoflinearsplineinterpolation,wehaveprovedtherateofconvergenceoftheerrorsofQFs\text{re}eq3and\text{re}eq4fordifferentclasses(i\mathrm{.}e\mathrm{.}$H^\a([-1,1],K), C^{m,\a}[-1,1], W^r[-1,1]$)ofdensityfunction$f(t)$\mathrm{.}Itisshownthatapproximationbysplinepossessesmoreadvantagesthanotherkindsofapproximation:itrequirestheminimumsmoothnessofdensityfunction$f(x)$ to get good order of decreasing errors.