Abstract Generated abstract
The paper extends Sobolev’s theory of cubature formulas for periodic functions to formulas that incorporate prescribed derivatives, viewed as multidimensional analogues of Hermite quadrature. It formulates the error functional on an n-dimensional torus with a general period matrix and derives an explicit expression for its norm in the dual Sobolev-type space of periodic functions using Fourier coefficients. The author then constructs choices of derivative coefficients that annihilate the error on selected sets of lattice points and obtains upper estimates for the error norm for rectangular and general periodic lattices. A numerical example for a two-dimensional periodic integral illustrates that, when derivatives are readily computable, the derivative-based cubature formula can substantially reduce the estimated error.
Full Text
UDC 518:517.392
MATHEMATICS
T. I. KHAITOV
CUBATURE FORMULAS WITH PRESCRIBED DERIVATIVES IN THE PERIODIC CASE
(Presented by Academician S. L. Sobolev, 19 V 1969)
In the present paper, the results of S. L. Sobolev on the integration of periodic functions from \(({}^{1-6})\) are generalized to the case of cubature formulas that are \(n\)-dimensional analogues of the Hermite quadrature formula.
Recall that a function \(f(x)\) is called periodic with period matrix \(H\), \(|H| = 1\), if for any integer vector \(B\) and all \(x\) one has \(f(x) = f(x + H\beta)\).
Identifying in Euclidean space \(E_n\) points that differ from one another by a period, we obtain the \(n\)-dimensional torus \(\Omega_0\). This torus is called the fundamental domain. The condition of fundamentality can be written in the form
\[ \sum_{\beta} \mathcal{E}_{\Omega_0}(x - H\beta) = 1 . \]
Here \(\mathcal{E}_{\Omega_0}(x)\) is the characteristic function of the domain \(\Omega_0\).
To each function \(\varphi(x) \in K\) from the basic space \(({}^{1,8})\) one can associate the function \(\mathring{\varphi}(x) \in \mathring{K}\), defined on the torus, by the formula
\[ \mathring{\varphi}(x) = \varphi(x) * \Phi^{(0)}(x) = \sum_{\beta} \varphi(x + \beta), \]
where
\[ \Phi^{(0)}(x) = \sum_{\beta} \delta(x + \beta). \]
If \(l(x)\) is a periodic functional in \(E_n\), then one can associate with it a functional \(\hat l(x)\), defined on the torus by the formula
\[ \int_{E_n} l(x)\varphi(x)\,dx = \int_{\Omega_0} \hat l(x)\mathring{\varphi}(x)\,dx . \]
These most important concepts are set forth fully, for example, in \(({}^{1})\).
In our case the error functional has the form
\[ l^{(t)}(x) = 1 - \sum_{|\alpha| \le 2t} (-1)^{|\alpha|} C^{(\alpha)} \Phi_H^{(\alpha)}(x); \]
\[ (l^{(t)}, \varphi) = \int_{\Omega_0} \varphi(x)\,dx - \sum_{|\alpha| \le 2t} C^\alpha D^\alpha \varphi(0); \tag{1} \]
\[ 2m > n;\qquad 2t \le m - [n/2] - 1;\qquad (l^{(t)}, 1) = 0 . \tag{2} \]
Here
\[ \Phi_H^{(\alpha)}(x) = \sum_{\beta} D^\alpha \delta(x - H\beta); \qquad \Phi_H^{(\dot{0})}(x) = \sum_{\beta} \delta(x - H\beta); \qquad |\alpha| = \alpha_1 + \]
\[ + \alpha_2 + \ldots + \alpha_n; \qquad D^\alpha = \partial^{|\alpha|}/\partial x_1^{\alpha_1}\partial x_2^{\alpha_2}\ldots \partial x_n^{\alpha_n}; \qquad C^{(\alpha)} \text{ are the coefficients of the cu-} \]
cubature formula. From \((l^{(t)}(x),\,1)=0\) it follows that \(C^{(0)}=1\). The following norms in \(\widetilde L_2^m(H)\) are used:
\[ \widetilde L_2^m(H)=\left\{f:\ f(x)=\sum_\beta f_\beta e^{2\pi i(\beta H^{-1},x)},\ \|f\|_{\widetilde L_2^m}^2=(2\pi)^{2m}\sum_\beta |f_\beta|^2|\beta H^{-1}|^{2m}\right\}, \tag{3} \]
and also
\[ \widetilde L_2^m(H)=\left\{f:\ f(x)=f(x+H\beta),\ \|f\|_{\widetilde L_2^m}^2= \int_{\Omega_0}\sum_{|\alpha|=m}\frac{|\alpha|!}{\alpha!}(D^\alpha f)^2\,dx\right\}, \]
\(\alpha!=\alpha_1!\alpha_2!\cdots\alpha_n!\); \(f_\beta\) are the Fourier coefficients of the function \(f(x)\). Weighted cubature formulas in \(\widetilde L_2^m\) with norm (3) were considered in (7).
Theorem. If \(l^{(t)}(x)\) is a functional of the form (1), (2) from \(\widetilde L_2^{m*}\), then
\[ \|l^{(t)}\|_{\widetilde L_2^{m*}}^2 =(2\pi)^{-2m}\sum_{\beta\ne0}|\beta H^{-1}|^{-2m}\Psi_{H^{-1}}^2(C_{H^{-1}},\beta), \tag{4} \]
\[ \Psi_{H^{-1}}(C_{H^{-1}},\beta) =\sum_{s=0}^t\sum_{|\alpha|=2s}(-1)^s(2\pi)^{2s}(\beta H^{-1})^\alpha C_{H^{-1}}^{(\alpha)} . \tag{5} \]
Proof. We estimate \((l^{(t)},\varphi)\) for \(\varphi\in\widetilde L_2^m\), applying the Cauchy inequality:
\[ \begin{aligned} |(l^{(t)},\varphi)| &=\left|(l^{(t)},\sum_\beta \varphi_\beta e^{2\pi i(\beta H^{-1},x)})\right| \\ &=\left|\sum_{\beta\ne0}\varphi_\beta\left(-\sum_{|\alpha|\le 2t}(2\pi i)^{|\alpha|}(\beta H^{-1})^\alpha C_{H^{-1}}^{(\alpha)}\right)\right| \\ &=\left|\sum_{\beta\ne0}(2\pi)^m|\beta H^{-1}|^m\varphi_\beta \left(-(2\pi)^{-m}\sum_{|\alpha|\le 2t}(2\pi i)^{|\alpha|}(\beta H^{-1})^\alpha C_{H^{-1}}^{(\alpha)}|\beta H^{-1}|^{-m}\right)\right| \\ &\le \left[(2\pi)^{-2m}\sum_{\beta\ne0}|\beta H^{-1}|^{-2m} \left|\sum_{|\alpha|\le 2t}(2\pi i)^{|\alpha|}(\beta H^{-1})^\alpha C_{H^{-1}}^{(\alpha)}\right|^2\right]^{1/2} \|\varphi\|_{\widetilde L_2^m}. \end{aligned} \tag{6} \]
Put
\[ u(x)=-(2\pi)^{-2m}\sum_{\beta\ne0}\left\{|\beta H^{-1}|^{-2m} \sum_{|\alpha|\le 2t}(2\pi i)^{|\alpha|}(\beta H^{-1})^\alpha C_{H^{-1}}^{(\alpha)} \times e^{2\pi i(\beta H^{-1},x)}\right\}; \]
then for \(u(x)\) equality is attained in (6):
\[ |(l^{(t)},u)|= \left[(2\pi)^{-2m}\sum_{\beta\ne0}|\beta H^{-1}|^{-2m} \left|\sum_{|\alpha|\le 2t}(2\pi i)^{|\alpha|}(\beta H^{-1})^\alpha C_{H^{-1}}^{(\alpha)}\right|^2\right]^{1/2} \|u\|_{\widetilde L_2^m}. \tag{7} \]
If in (7) \(\beta\) is replaced by \(-\beta\), then the terms with odd \(|\alpha|\) cancel each other. The theorem is proved.
Put \(y=Hx\) or \(x=H^{-1}y\); then
\[ \|l^{(t)}\|_{\widetilde L_2^{m*}}^2 =(2\pi)^{-2m}\sum_{\beta\ne0}(\beta,\beta^*)^{-m}\Psi^2(c,\beta), \tag{8} \]
\[ \Psi(c,\beta)=1-\sum_{s=1}^t\sum_{|\alpha|=2s}(-1)^{s-1}(2\pi)^{2s}\beta^\alpha C^{(\alpha)}. \tag{9} \]
Here \(\beta^*\) is conjugate to \(\beta\), and \(\beta^\alpha=\beta_1^{\alpha_1}\beta_2^{\alpha_2}\cdots\beta_n^{\alpha_n}\). Put \(t\ge n\), \(r_0=[t/n]\), \(p=t-nr_0\), \(B=\{\beta:\ |\beta_i|\le r_0\ (i=1,2,\ldots,n)\}\), and solve the following problem.
Problem. Find the coefficients \(C^{(\alpha)}\) for (9) so that \(\Psi(c,\beta)=0\) for all integer \(\beta\in B\).
The solution of this problem can be represented in the form
\[ \Psi(c,\beta)=r_0!^{-2n}(r_0+1)^{-2p} \prod_{s=1}^n\prod_{k=1}^{r_0}(\beta_s^2-k^2) \prod_{s=1}^p[\beta^2-(r_0+1)^2], \]
which grows as \(\beta\to\infty\). Applying the inequality ((9), p. 29), we obtain—
we have the following estimate:
\[ |\Psi(c,\beta)| \leqslant r_0!^{-2n}(r_0+1)^{-2p} n^{-nr_0}p^{-p} \left(\sum_{s=1}^n \beta_s^2\right)^t,\qquad \beta\notin B. \tag{10} \]
Substituting (10) into (8), we obtain
\[ \|l^{(t)}\|_{\widetilde L_2^{m*}}^2 \leqslant (2\pi)^{-2m} r_0!^{-4m}(r_0+1)^{-4p} n^{-2nr_0}p^{-2p} \sum_{\beta\notin B}(\beta,\beta^*)^{-m+2t}. \tag{11} \]
Let now a periodic lattice be given with period matrix \(H^{-1}\), \(|H^{-1}|=1\). On the basis of (2) (pp. 64–78), we may regard \(H^{-1}\) as a triangular matrix.
Put
\[ B_{H^{-1}}=\left\{\beta H^{-1}:\left|\sum_{i=j}^n h_{ij}\beta_i\right|\leqslant r_1\ (j=1,2,\ldots,n)\right\}. \]
Renumber all \(\beta H^{-1}\in B_{H^{-1}}\) and, for convenience, introduce the notation
\[ \beta^{(k)}H^{-1}=\gamma^{(k)}\in B_{H^{-1}};\qquad a_s=\prod_{k=1}^{\sigma_s}\gamma_s^{(k)};\qquad \gamma_s^{(k)}=\sum_{i=s}^n h_{is}\beta_i^{(k)};\qquad |\gamma_s^{(k)}|\leqslant r_1; \]
\[ s=1,2,\ldots,n;\quad k=1,2,\ldots,\sigma_s;\quad h_{ij}\text{ is an element of the matrix }H^{-1},\ r_1\text{ is chosen} \]
so that the condition \(\sum_{k=1}^n \sigma_k=t\) is satisfied.
Solving the preceding problem for this case, we shall have the following estimates for (5) and (4):
\[ \Psi_{H^{-1}}(C_{H^{-1}},\beta) = \prod_{s=1}^n\prod_{k=1}^{\sigma_s} \frac{\gamma_s^2-\gamma_s^{(k)2}}{a_1^2a_2^2\ldots a_n^2}; \qquad |\Psi_{H^{-1}}| \leqslant \frac{\sigma_0^{2t}t^{-2t}}{a_1^2a_2^2\ldots a_n^2}\, |\beta H^{-1}|^{2t}; \]
\[ \|l^{(t)}\|_{\widetilde L_2^{m*}}^2 \leqslant \frac{(2\pi)^{-2m}\sigma_0^{4t}t^{-4t}}{a_1^4a_2^4\ldots a_n^4} \sum_{\beta H^{-1}\notin B_{H^{-1}}} |\beta H^{-1}|^{-2m+4t}; \qquad \sigma_0=\max(\sigma_1,\sigma_2,\ldots,\sigma_n). \tag{12} \]
We inscribe a ball in \(B\) and \(B_{H^{-1}}\). The radius of the ball is, obviously, respectively equal to \(r_0\) and \(r_1\). The estimate
\[ \sum_{\beta\notin B} r^{-2m+4t} \leqslant \frac{\varkappa_n(1+o(1))}{(r_0+1)^{2m-4t-n}}, \qquad \varkappa_n=\frac{2\pi^{n/2}}{(2m-4t-n)\Gamma(n/2)}, \qquad r^2=\sum_{s=1}^n \beta_s^2, \]
belongs to L. V. Voĭtishek. Substituting this into (11) and (12), we obtain an estimate of the norm of the functional \(l^{(t)}(x)\), respectively for the case of rectangular lattices and lattices with matrix \(H^{-1}\):
\[ \|l^{(t)}\|_{\widetilde L_2^{m*}}^2 \leqslant \frac{\varkappa_n n^{-2nr_0}p^{-2p}r_0!^{-4n}} {(2\pi)^{2m}(r_0+1)^{2m-4t+4p-n}} + O\!\left(\frac{1}{(r_0+1)^{2m-4t-n}}\right), \tag{13} \]
\[ \|l^{(t)}\|_{\widetilde L_2^{m*}}^2 \leqslant \frac{\varkappa_n\sigma_0^{4t}t^{-4t}} {a_1^4a_2^4\ldots a_n^4(2\pi)^{2m}(r_1+1)^{2m-4t-n}} + O\!\left(\frac{1}{(r_1+1)^{2m-4t-n}}\right). \]
Estimate (13) can be transformed into the form
\[ \|l^{(t)}(x)\|_{\widetilde L_2^{m*}}^2 \leqslant \frac{r_0^{-4n}(r_0+1)^{-2m-4p+4t+n}} {(2m-4t-n)n^{2nr_0}} \|l^{(0)}(x)\|_{\widetilde L_2^{m*}}^2, \tag{14} \]
where \(l^{(0)}(x)\) is obtained from \(l^{(t)}(x)\) for \(t=0\).
Example. Put in (14) \(n=2,\ t=2,\ m=6,\ r_0=1,\ p=0\); then
\[ \|l^{(2)}(x)\|_{\widetilde L_2^{6*}}^2 \ll \frac{1}{128}\|l^{(0)}(x)\|_{\widetilde L_2^{m*}}^2 . \]
Now let
\[ I=\frac{1}{4\pi^2}\int_0^{2\pi}\int_0^{2\pi} e^{\sin(x+y)}\,dx\,dy . \]
Computations with step \(h=\pi/3\) gave the values
\[ \|l^{(0)}\|_{\widetilde L_2^{6*}}=0.0000450\ldots,\qquad \|l^{(2)}\|_{\widetilde L_2^{6*}}=0.000000041\ldots . \]
Remark. In cases where computing the derivatives of \(\varphi(x)\) does not present difficulties, it is advantageous to apply the cubature formula considered in this note.
The author expresses his deep gratitude to the supervisor of this work, S. L. Sobolev.
Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR
Novosibirsk
Received
13 V 1969
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