Cubature Remainder and Biorthogonal Systems

Estimating quadrature errors for analytic functions using kernel representations and biorthogonal systems

“…To refine the error bound (3.1) biorthogonal systems are used by employing higher order approximation degrees (see [10,11,13]). Let be given polynomials ϕ j ∈ P j and linear functionals L j on C[−1, 1], j = 0(1)s, then the system…”

“…Especially, the Chebyshev BOGS, which is applied with great benefit, is defined for each s ∈ N by (see [11])…”

“…Employing the polynomial of best approximation to the involved function, estimates containing approximation degrees are produced. An interesting refinement is obtained by applying polynomial biorthogonal systems to deduce estimates by evaluating higher approximation degrees (see [10,11,13]). Usually it might be a serious disadvantage to compute or to estimate these approximation degrees, and only in special cases the costs will be tolerable.…”