Analysis of errors in some recent numerical quadrature formulas for periodic singular and hypersingular integrals via regularization

Recently, we derived some new numerical quadrature formulas of trapezoidal rule type for the singular integrals I ( 1 ) [ u ] = ∫ a b ( cot π ( x − t ) T ) u ( x ) d x and I ( 2 ) [ u ] = ∫ a b ( csc 2 π ( x − t ) T ) u ( x ) d x , with b − a = T and u ( x ) a T-periodic continuous function on R . These integrals are not defined in the regular sense, but are defined in the sense of Cauchy Principal Value and Hadamard Finite Part, respectively. With h = ( b − a ) / n , n = 1 , 2 , … , the numerical quadrature formulas Q n ( 1 ) [ u ] for I ( 1 ) [ u ] and Q n ( 2 ) [ u ] for I ( 2 ) [ u ] are Q n ( 1 ) [ u ] = h ∑ j = 1 n f ( t + j h − h / 2 ) , f ( x ) = ( cot π ( x − t ) T ) u ( x ) , and Q n ( 2 ) [ u ] = h ∑ j = 1 n f ( t + j h − h / 2 ) − T 2 u ( t ) h − 1 , f ( x ) = ( csc 2 π ( x − t ) T ) u ( x ) . We provided a complete analysis of the errors in these formulas under the assumption that u ∈ C ∞ ( R ) and is T-periodic. We actually showed that, I ( 1 ) [ u ] − Q n ( 1 ) [ u ] = O ( n − μ ) and I ( 2 ) [ u ] − Q n ( 2 ) [ u ] = O ( n − μ ) as n → ∞ , ∀ μ > 0 . In this note, we analyze the errors in these formulas under the weaker assumption that u ∈ C s ( R ) for some finite integer s. By first regularizing these integrals, we prove that, if u ( s + 1 ) is piecewise continuous, then I ( 1 ) [ u ] − Q n ( 1 ) [ u ] = o ( n − s − 1 / 2 ) as n → ∞ , if s ⩾ 1 , and I ( 2 ) [ u ] − Q n ( 2 ) [ u ] = o ( n − s + 1 / 2 ) as n → ∞ , if s ⩾ 2 . We also extend these results by imposing different smoothness conditions on u ( s + 1 ) . Finally, we append suitable numerical examples.