Quadrature rules for qualocation

Qualocation is a method for the numerical treatment of boundary integral equations on smooth curves which was developed by Chandler, Sloan and Wendland (1988‐2000) [1,2]. They showed that the method needs symmetric J–point–quadrature rules on [0, 1] that are exact for a maximum number of 1–periodic functions $$ G _{\alpha} (x) \ggleich \sum ^{\infty} _{k=1} k ^{-\alpha} \cos (2 \pi kx), \qquad \alpha > {1 \over {2}}. $$ The existence of 2–point–rules of that type was proven by Chandler and Sloan. For J ∈ {3, 4} such formulas have been calculated numerically in [2]. We show that the functions G α form a Chebyshev–system on [0, 1/2] for arbitrary indices á and thus prove the existence of such quadrature rules for any J.