On the Determination of Quadrature Formulae of Highest Degree of Precision for Approximating Fourier Coefficients

Let Q₀(x), Q₁(x),..., Q_n(x),... be a sequence of polynomialswhich are orthogonal with respect to the inner product . In estimating the Fourier coefficients a₁ = $$\langlef,{Q}_{i}\rangle $$, it is natural to use a quadrature formulaof highest possible degree of precision to approximate $${\int}_{a}^{b}W\left(x\right)f\left(x\right)dx$$ where W(x) = w(x)Q_i(x).Since the weight function W(x) changes sign i times in (a, b),the usual results of quadrature theory do not apply. This paperdevelops a procedure which is an initial attempt to determinewhat degree of precision is attainable.