Asymptotically sharp error bounds for a quadrature rule for Cauchy principal value integrals based on piecewise linear interpolation

For the numerical evaluation of Cauchy principal value integrals of the form $int_{ - 1}^1 {fleft( x right)left( {x - lambda } right)^{ - 1} dx} $ with λ∈ (?1,1) and f∈C’[?1’,1], we investigate the quadrature formula Q _n+1 ^Spl1 [·;λ] obtained by replacing the integrand function f by its piecewise linear interpolant at an equidistant set of nodes as proposed by Rabinowitz (Math. Comp., 51:741–747,1988). We give upper bounds for the Peano—type error constants $$rho _s left( {R_{n + 1}^{Spl1} left[ { bullet ;lambda } right]} right): = supleft{ {left| {int_{ - 1}^1 {frac{{fleft( x right)}}{{x - lambda }}dx - Q_{n + 1}^{Spl1} left[ {f;lambda } right]} } right|:f in C'left[ { - 1,1} right],left| {f^{left( s right)} } right|_infty leqslant 1} right}$$ for s∈{1,2}. These are the best possible constants in inequalities of the type $$left| {int_{ - 1}^1 {frac{{fleft( x right)}}{{x - lambda }}dx - Q_{n + 1}^{Spl1} left[ {f;lambda } right]} } right|: leqslant c_{1,n + 1} left( lambda right)left| {f^{left( 1 right)} } right|_infty $$ . Furthermore, we prove that our upper bounds are asymptotically sharp.