We establish a relation between Gauss quadrature formulas on the interval [-1,1] that approximate integrals of the formI_s(F) = ∫_{x=-1..+1}F(x)s(x) dxand Szegö quadrature formulas on the unit circle of the complex plane that approximate integrals of the formI_w(f) = ∫_{t=-π..π} f(e^{it})w(t)dt.The weight s(x) is positive on [-1,1] while the weight w(t) is positive on [-π,π]. It is shown that if w(t)=s(cos t)|sin t|, then there is an intimate relation between the Gauss and Szegö quadrature formulas. Moreover, as a side result we also obtain an easy derivation for relations between orthogonal polynomials with respect to s(x) and orthogonal Szegö polynomials with respect to w(t). Inclusion of Gauss-Lobatto and Gauss-Radau formulas is natural.
展开▼
机译：我们在区间[-1,1]上建立高斯正交公式之间的关系，该公式近似形式为I_s（F）=∫_{x = -1 .. + 1} F（x）s（x）dx和Szegö积分的积分复平面的单位圆上的近似形式为I_w（f）=∫_{t =-π..π} f（e ^ {it}）w（t）dt的积分的公式。权重s（x）在[-1,1]上为正，而权重w（t）在[-π，π]上为正。结果表明，如果w（t）= s（cos t）| sin t |，那么高斯和Szegö正交公式之间存在密切关系。此外，作为附带结果，我们还可以轻松推导关于s（x）的正交多项式和关于w（t）的正交Szegö多项式之间的关系。包含高斯-洛巴托和高斯-拉多公式是很自然的。
展开▼
