Gauss-Tur’an quadratures of Kronrod type for generalized Chebyshev weight functions. (English) Zbl 1137.65014
Let \(w(t)\) be one of the following generalized Chebyshev weight functions
(a) \(w_1(t)=(1-t^2)^{-1/2}\), (c) \(w_3(t)=(1-t)^{-1/2}(1+t)^{1/2+s}\),
(b) \(w_2(t)=(1-t^2)^{1/2+s}\), (d) \(w_4(t)=(1-t)^{1/2+s}(1+t)^{-1/2}\).
The problem of estimating the errors when the integrals \(\int_{-1}^1 f(t)w(t)dt\) are approximated by Gauss-Turán quadrature formulae of Kronrod type is investigated in this paper. Some numerical results are presented to illustrate the efficiency of the estimates when \(f\) is analytic in a neighbourhood of the interval \([-1,1]\).
Explicit expressions are obtained for the generalized Stieltjes polynomials \(\hat{\pi}_{n+1}\) associated to \(w=w_i\), \(i=2,3,4\). For the case \(w=w_3\) the following equality holds (Theorem 2.2) \[ \hat{\pi}_{n+1}(t)=\frac{2^n (n!)^2}{(2n)!}(t-1)P_n^{(1/2,-1/2)}(t), \] where \(P_n^{(1/2,-1/2)}(t)\) is the orthogonal polynomial with respect to the weight function \(w(t)=\sqrt{(1-t)/(1+t)}\).
(a) \(w_1(t)=(1-t^2)^{-1/2}\), (c) \(w_3(t)=(1-t)^{-1/2}(1+t)^{1/2+s}\),
(b) \(w_2(t)=(1-t^2)^{1/2+s}\), (d) \(w_4(t)=(1-t)^{1/2+s}(1+t)^{-1/2}\).
The problem of estimating the errors when the integrals \(\int_{-1}^1 f(t)w(t)dt\) are approximated by Gauss-Turán quadrature formulae of Kronrod type is investigated in this paper. Some numerical results are presented to illustrate the efficiency of the estimates when \(f\) is analytic in a neighbourhood of the interval \([-1,1]\).
Explicit expressions are obtained for the generalized Stieltjes polynomials \(\hat{\pi}_{n+1}\) associated to \(w=w_i\), \(i=2,3,4\). For the case \(w=w_3\) the following equality holds (Theorem 2.2) \[ \hat{\pi}_{n+1}(t)=\frac{2^n (n!)^2}{(2n)!}(t-1)P_n^{(1/2,-1/2)}(t), \] where \(P_n^{(1/2,-1/2)}(t)\) is the orthogonal polynomial with respect to the weight function \(w(t)=\sqrt{(1-t)/(1+t)}\).
Reviewer: Jesus Illán González (Vigo)
MSC:
| 65D30 | Numerical integration |
| 65D32 | Numerical quadrature and cubature formulas |
| 41A55 | Approximate quadratures |
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