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MATHEMATICS
I. I. Ibragimov, R. M. Aliev
BEST QUADRATURE FORMULAS FOR CERTAIN CLASSES OF FUNCTIONS
(Presented by Academician I. N. Vekua on 16 XI 1964)
Let \(W_{L_p}^{(r)}(M_r; 0, 1)\) \((p \geqslant 1)\) be the class of functions \(f(x)\) having absolutely continuous derivatives of order \(r-1\) and a derivative of order \(r\) satisfying the condition
\[ \left\| f^{(r)}(x) \right\|_{L_p} = \left(\int_0^1 \left| f^{(r)}(x) \right|^p dx\right)^{1/p} \leqslant M_r \quad (p \geqslant 1). \tag{1} \]
In the case \(p=\infty\), instead of condition (1) we assume that for the piecewise continuous derivative \(f^{(r)}(x)\) the inequality
\[ \left| f^{(r)}(x) \right| \leqslant M_r \quad (0 \leqslant x \leqslant 1) \tag{2} \]
holds.
The classes \(W_{L_p}^{(r)}(M_r; 0, 1)\) in the cases \(p=1, \infty\) are denoted, respectively, by \(W_L^{(r)}(M_r; 0, 1)\) and \(W^{(r)}(M_r; 0, 1)\).
S. M. Nikol’skii \((^1)\), for functions \(f(x)\) belonging to the class \(W^r(M_r; 0, 1)\) and satisfying additionally the condition \(f(0)=f'(0)=\cdots=f^{(r-1)}(0)=0\), constructed the best quadrature formula of the form
\[ \int_0^1 f(x)\,dx \approx \frac{1}{r!} \sum_{k=0}^{N-1} \sum_{l=0}^{r-2} A_k^{(l)}(r-l-1)!\, f^{(l)}(x_k), \tag{3} \]
where \(0 \leqslant x_0 \leqslant x_1 < \cdots < x_{N-1} \leqslant 1\).
In the present article we construct best quadrature formulas of the form (3), exact for polynomials of degree \(r-1\), in the classes of functions \(W^{(r)}(M_r; 0, 1)\), \(W_L^{(r)}(M_r; 0, 1)\), and \(W_{L_2}^{(r)}(M_r; 0, 1)\), where \(r\) is an even positive integer.
From the fact that the quadrature formula (3) is exact for polynomials of degree \(r-1\), it follows that for a function \(f(x)\in W^{(r)}(M_r; 0, 1)\) the inequality
\[ |R(f)| = \left| \int_0^1 f(x)\,dx - \frac{1}{r!} \sum_{k=0}^{N-1} \sum_{l=0}^{r-2} A_k^{(l)}(r-l-1)!\, f^{(l)}(x_k) \right| \leqslant \frac{M_r}{r!}\int_0^1 |K(t)|\,dt, \]
where
\[ K(t) = (1-t)^r - \sum_{k=0}^{N-1} \sum_{l=0}^{r-2} A_k^{(l)} E_{r-l}(x_k-t); \]
\[ E_r(u) = \begin{cases} u^{r-1}, & \text{for } u>0,\\ 0, & \text{for } u\leqslant 0. \end{cases} \]
Hence it follows that
\[ \mathcal{E}_N\!\left(W^{(r)}\right) = \min_{(A_k^{(l)},\,x_k)} \left\{ \sup_{f\in W^{(r)}} |R(f)| \right\} = \frac{M_r}{r!} \min_{(A_k^{(l)},\,x_k)} \int_0^1 |K(t)|\,dt, \tag{4} \]
where the function \(K(t)\) on the intervals \([0,x_0]\) and \([x_{N-1},1]\) is defined by the equality
\[ K(t)= \begin{cases} t^r, & \text{for } 0\leqslant t\leqslant x_0,\\ (1-t)^r, & \text{for } x_{N-1}\leqslant t\leqslant 1. \end{cases} \]
Let \(x_k-x_{k-1}=2h_k\) and \(c_k=(x_{k-1}+x_k)/2\) \((k=1,2,\ldots,N-1)\). Note that the polynomial \(g_k(x)\) with coefficient of \(x^r\) equal to one, and least deviating from zero in the metric of the space \(L\) on the interval \((x_{k-1},x_k)\), has the form
\[ g_k(x)=h_k^r Q_r\left(\frac{x-c_k}{h_k}\right), \]
where
\[ Q_r(x)=\frac{\sin[(r+1)\arccos x]}{2^r\sqrt{1-x^2}}\quad (-1\leqslant x\leqslant 1). \]
In order that the polynomials \(g_k(x)\) and \(g_{k+1}(x)\) coincide at the point \(x_k=c_k+h_k=c_{k+1}-h_{k+1}\), it is necessary and sufficient that the condition \(h_k=h_{k+1}\) be satisfied \((k=1,2,\ldots,N-2)\). Further, it is not difficult to show that
\[ h=2x_0/\sqrt[r]{\,r+1\,};\qquad x_0=1-x_{N-1}; \tag{5} \]
whence, and from the equality \(x_{N-1}=x_0+(N-1)2h\), we find
\[ x_k=(\sqrt[r]{\,r+1\,}+4k)\omega_N\qquad (k=0,1,\ldots,N-1), \tag{6} \]
where
\[ \omega_N=[2\sqrt[r]{\,r+1\,}+4(N-1)]^{-1},\qquad h=2\omega_N. \tag{7} \]
The coefficients \(A_k^{(l)}\) are determined analogously to how this was done by S. M. Nikol’skii \((^1)\):
\[ A_k^{(2i+1)}=0\qquad (i=0,1,\ldots,(r-4)/2), \]
\[ A_k^{(2i)}=\frac{2h^{2i+1}}{(r-2i-1)!}\,Q_r^{(r-2i-1)}(1)\qquad (i=0,1,\ldots,(r-2)/2) \tag{8} \]
\[ (k=1,2,\ldots,N-2), \]
\[ A_0^{(l)}=A_{N-1}^{(l)} = \frac{h^{l+1}}{(r-l-1)!} \left( \frac{r!}{(l+1)!}[Q_r(1)]^{(l+1)/r} + (-1)^l Q^{(r-l-1)}(1) \right). \]
It can be proved that only for the coefficients \(A_k^{(l)}\) and the nodes \(x_k\) determined respectively by equalities (8) and (6) does the integral \(\int_0^1 |K(t)|\,dt\) attain its minimum. Thus, the following assertion holds:
1. The quadrature formula of the form (3), exact for polynomials of degree \((r-1)\), whose coefficients \(A_k^{(l)}\) and nodes \(x_k\) are determined respectively by equalities (6) and (8), is the unique best quadrature formula for functions from the class \(W^{(r)}(M_r;0,1)\). Moreover,
\[ \mathcal{E}_N(W^{(r)})=\frac{M_r}{r!}\,\omega_N^r. \]
By similar reasoning the following assertions are proved:
2. The quadrature formula of the form (3), exact for polynomials of degree \(r-1\), whose coefficients \(A_k^{(l)}\) are determined by equalities (8), and the nodes \(x_k\) \((0\leqslant x_0<x_1<\cdots<x_{N-1}\leqslant 1)\) by the equalities
\[ x_k=(1+2k\sqrt[r]{\,2^{r-1}\,})\omega_N\qquad (k=0,1,\ldots,N-1), \]
where
\[ \omega_N=\frac{1}{2}\left[1+\sqrt[r]{2^{r-1}(N-1)}\right]^{-1},\qquad h=\sqrt[r]{2^{r-1}}\omega_N, \]
is the unique best quadrature formula for functions from the class \(W_L^{(r)}(M_r;0,1)\) and, moreover,
\[ \mathcal E_N\left(W_L^{(r)}\right)=\frac{M_r}{r!}\omega_N^r. \]
Let us note that in the proof of this assertion the polynomial \(Q_r(x)\), used in the proof of the first assertion, is replaced by the Chebyshev polynomial of the first kind \(T_r(x)\).
- The quadrature formula of the form (3), exact for polynomials of degree \(r-1\), and whose coefficients \(A_k^{(l)}\) are determined by the equalities (8), while the nodes \(x_k\) are given by the equalities
\[ x_k=\left(1+k\sqrt[r]{\frac{(2r)!}{(r!)^2}}\right)\omega_N, \]
where
\[ \omega_N=\left[2+\sqrt[r]{\frac{(2r)!}{(r!)^2}}(N-1)\right]^{-1},\qquad h=\frac{1}{2}\sqrt[r]{\frac{(2r)!}{(r!)^2}}\omega_N, \]
is the unique best quadrature formula for functions from the class \(W_{L_2}^{(r)}(M_r;0,1)\), with
\[ \mathcal E_N\left(W_{L_2}^{(r)}\right)=\frac{M_r}{r!}(2r+1)^{-1/2}\omega_N^r \]
and \(Q_r(x)\) is replaced by a polynomial of the form
\[ X_r(x)=\frac{(r!)^2}{(2r)!} \left[ \sum_{k=0}^{r/2} (-1)^k \frac{(2r-2k)!}{(r-2k)!(r-k)!k!} x^{r-2k} \right]. \]
Let us note that from assertions 1 and 2, for \(r=2\), follow the best quadrature formulas constructed by T. A. Shaidaeva \((^3)\). Moreover, assertion 3 for \(r=2\) coincides with the corresponding assertion of T. A. Shaidaeva in the case \(p=2\). Finally, assertion 3 in the case \(r=2\) is a direct generalization of the corresponding assertion of G. Ya. Doronin \((^2)\).
Institute of Mathematics and Mechanics
Academy of Sciences of the Azerbaijan SSR
Received
11.XI.1964
CITED LITERATURE
- S. M. Nikol’skii, Quadrature Formulas, 1958.
- G. Ya. Doronin, Collection of Scientific Works of the Dnepropetrovsk Institute of Civil Engineering, 1–2, 210 (1955).
- T. A. Shaidaeva, Proceedings of the V. A. Steklov Mathematical Institute, Academy of Sciences of the USSR, 53, 313 (1959).