Abstract
Full Text
MATHEMATICS
D. V. IONESCU
REPRESENTATION OF THE DIVIDED DIFFERENCE OF ORDER \((m,n)\) OF A FUNCTION OF TWO VARIABLES BY A DOUBLE INTEGRAL. I
(Presented by Academician A. N. Kolmogorov on 12 V 1961)
- Consider a function \(f(x)\) of class \(C^n\) on the interval \([x_0,x_n]\), and take in this interval nodes \(x_1,x_2,\ldots,x_{n-1}\) in such a way that \(x_0<x_1<\cdots<x_n\). We have proved \((^1)\) that the divided difference of order \(n\) at the nodes \(x_0,x_1,\ldots,x_n\) can be represented by the definite integral
\[ [x_0,x_1,\ldots,x_n; f]=\int_{x_0}^{x_n}\varphi(x) f^{(n)}(x)\,dx, \tag{1} \]
where the function \(\varphi(x)\) coincides on the intervals \([x_0,x_1]\), \([x_1,x_2]\), \(\ldots\), \([x_{n-1},x_n]\) with the polynomials \(\varphi_1(x),\varphi_2(x),\ldots,\varphi_n(x)\), defined by the equations
\[ \varphi_i^{(n-1)}(x)=(-1)^n(\alpha_0+\alpha_1+\cdots+\alpha_{i-1}); \]
\[ \alpha_h=(-1)^{n+h} \frac{V(x_0,x_1,\ldots,x_{h-1},x_{h+1},\ldots,x_n)} {V(x_0,x_1,\ldots,x_n)}, \tag{2} \]
\(i=1,2,\ldots,n\), and the boundary conditions
\[ \varphi_1^{(k)}(x_0)=0,\qquad \varphi_{i+1}^{(k)}(x_i)=\varphi_i^{(k)}(x_i),\qquad \varphi_n^{(k)}(x_n)=0, \tag{3} \]
\[ i=1,2,\ldots,n-1;\quad k=0,1,\ldots,n-1. \]
We have proved that the function \(\varphi(x)\) is positive on the interval \((x_0,x_n)\). Formula (1), as well as its generalization to the case of multiple nodes, has important applications in numerical analysis.
- In the present note we shall briefly set forth a generalization of formula (1) to the case of functions of two variables. We define the divided difference of order \((m,n)\) of the function \(f(x,y)\) at the nodes \((x_i,y_k)\), where \(x_0<x_1<\cdots<x_m,\ y_0<y_1<\cdots<y_n\) \((m>n)\), by means of the formula
\[ \begin{aligned} \left[ \begin{array}{c} x_0,x_1,\ldots,x_m\\ y_0,y_1,\ldots,y_n \end{array}; f \right] &= \sum_{i=0}^{m}\sum_{k=0}^{n}(-1)^{i+k} \frac{V(x_0,x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_n)} {V(x_0,x_1,\ldots,x_m)} \\ &\quad {}\times \frac{V(y_0,y_1,\ldots,y_{k-1},y_{k+1},\ldots,y_n)} {V(y_0,y_1,\ldots,y_n)} \, f(x_i,y_k). \end{aligned} \tag{4} \]
Define the function \(\Phi(x,y)\) in the rectangle \(D\{x_0\le x\le x_m,\ y_0\le y\le y_n\}\) in such a way that
\[ \left[ \begin{array}{c} x_0,x_1,\ldots,x_m\\ y_0,y_1,\ldots,y_n \end{array}; f \right] = \iint_D \Phi(x,y)\, \frac{\partial^{m+n}f}{\partial x^m\partial y^n}\,dx\,dy, \tag{5} \]
where the function \(f(x,y)\) is continuous in \(D\) together with its partial derivatives
\[ \partial^{2p-1}f/\partial x^p\partial y^{p-1},\quad \partial^{2p-1}f/\partial x^{p-1}\partial y^p,\quad \partial^{2p}f/\partial x^p\partial y^p,\quad \partial^{2n+r}f/\partial x^{n+r}\partial y^n, \tag{6} \]
where \(p=1,2,\ldots,n;\ r=1,2,\ldots,q;\ q=m-n\).
To arrive at formula (5), we established two preliminary formulas and then studied a boundary-value problem for a system of partial differential equations.
- Suppose that the functions \(\varphi(x,y)\), \(f(x,y)\) are continuous in the rectangle \(\Delta\{x_1 \leq x \leq x_2,\ y_1 \leq y \leq y_2\}\), together with the partial derivatives of \(\varphi\)
\[ \frac{\partial^r\varphi}{\partial x^r},\qquad \frac{\partial^{2p+q-1}\varphi}{\partial x^{p+q}\partial y^{p-1}},\qquad \frac{\partial^{2p+q-1}\varphi}{\partial x^{p+q-1}\partial y^p},\qquad \frac{\partial^{2p+q}\varphi}{\partial x^{p+q}\partial y^p}, \tag{7} \]
where \(p=1,2,\ldots,n;\ r=1,2,\ldots,q\), and with the derivatives (6) of \(f\). Under these conditions, putting
\[ (-1)^q\partial^q\varphi/\partial x^q=\psi \tag{8} \]
and denoting
\[ \frac{\partial^{m+n}f}{\partial x^m\partial y^n}(x_i,y_k)=\overset{n}{f}_{m}(i,k),\quad \frac{\partial^{m+n}f}{\partial x^m\partial y^n}(x,y_k)=\overset{n}{f}_{m}(x,k),\quad \frac{\partial^{m+n}f}{\partial x^m\partial y^n}(x_i,y)=\overset{n}{f}_{m}(i,y), \tag{8'} \]
we proved the following formula:
\[ \begin{aligned} &\iint\limits_{\Delta}\varphi\, \frac{\partial^{m+n}f}{\partial x^m\partial y^n}\,dx\,dy -(-1)^q\iint\limits_{\Delta}f\, \frac{\partial^{m+n}\varphi}{\partial x^m\partial y^n}\,dx\,dy \\ &= \sum_{r=0}^{n-1} \Bigl[ \overset{\,n-r-1}{\psi}_{\,n-r-1}(2,2)\overset{r}{f}_{r}(2,2) - \overset{\,n-r-1}{\psi}_{\,n-r-1}(2,1)\overset{r}{f}_{r}(2,1) - \overset{\,n-r-1}{\psi}_{\,n-r-1}(1,2)\overset{r}{f}_{r}(1,2) \\ &\qquad\qquad + \overset{\,n-r-1}{\psi}_{\,n-r-1}(1,1)\overset{r}{f}_{r}(1,1) \Bigr] \\ &\quad +\sum_{r=0}^{n-1}\int_{x_1}^{x_2} \Bigl[ \overset{\,n-r-1}{\psi}_{\,n-r}(x,1)\overset{r}{f}_{r}(x,1) - \overset{\,n-r-1}{\psi}_{\,n-r}(x,2)\overset{r}{f}_{r}(x,2) \Bigr]\,dx \\ &\quad +\sum_{r=0}^{n-1}\int_{y_1}^{y_2} \Bigl[ \overset{\,n-r}{\psi}_{\,n-r-1}(1,y)\overset{r}{f}_{r}(1,y) - \overset{\,n-r}{\psi}_{\,n-r-1}(2,y)\overset{r}{f}_{r}(2,y) \Bigr]\,dy \\ &\quad +\sum_{j=0}^{q-1}(-1)^j\int_{y_1}^{y_2} \Bigl[ \overset{0}{\varphi}_{j}(2,y)\overset{n}{f}_{m-j-1}(2,y) - \overset{0}{\varphi}_{j}(1,y)\overset{n}{f}_{m-j-1}(1,y) \Bigr]\,dy . \tag{9} \end{aligned} \]
- Denote by \(D_i^k\) the rectangle determined by the inequalities \(x_i\leq x\leq x_{i+k}\), \(y_k\leq y\leq y_{k+1}\), and associate with each rectangle \(D_i^k\) a function \(\varphi_i^k(x,y)\), continuous in this rectangle together with its partial derivatives (7).
To the functions \(\varphi_i^k(x,y)\), by means of formula (8), there correspond functions \(\psi_i^k(x,y)\). Applying formula (9) to each rectangle \(D_i^k\) and to the functions \(\varphi_i^k(x,y)\), \(f(x,y)\), and adding the resulting expressions, we shall find the basic formula of the present work:
\[ \begin{aligned} &\iint\limits_D \Phi(x,y) \frac{\partial^{m+n}f}{\partial x^m\partial y^n}\,dx\,dy - \sum_{i=0}^{m-1}\sum_{k=0}^{n-1} \iint\limits_{D_i^k} \frac{\partial^{2n}\psi_i^k}{\partial x^n\partial y^n}\,f\,dx\,dy \\ &= \sum_{r=0}^{n-1}\sum_{s=0}^{m}\sum_{t=0}^{n} A_{s,t,r}\, \frac{\partial^{2r}f}{\partial x^r\partial y^r}(x_s,y_t) + \sum_{r=0}^{n-1}\sum_{i=0}^{m-1}\sum_{t=0}^{n} \int_{x_i}^{x_{i+1}} M_{i,t,r}(x,y_t)\, \frac{\partial^{2r}f}{\partial x^r\partial y^r}(x,y_t)\,dx \\ &\quad + \sum_{r=0}^{n-1}\sum_{k=0}^{n-1}\sum_{s=0}^{m} \int_{y_k}^{y_{k+1}} N_{s,k,r}(x_s,y)\, \frac{\partial^{2r}f}{\partial x^r\partial y^r}(x_s,y)\,dy \\ &\quad + \sum_{j=0}^{q-1}\sum_{k=0}^{n-1}\sum_{s=0}^{m} (-1)^j \int_{y_k}^{y_{k+1}} P_{s,j,k}(x_s,y)\, \frac{\partial^{2n+q-j-1}f}{\partial x^{\,n+q-j-1}\partial y^n}(x_s,y)\,dy, \tag{10} \end{aligned} \]
where we have put
\[ A_{0,0,r}=\underset{n-r-1}{\overset{n-r-1}{\psi}}{}^{\,0}_{0}(0,0); \qquad A_{m,0,r}=-\underset{n-r-1}{\overset{n-r-1}{\psi}}{}^{\,0}_{m-1}(m,0); \]
\[ A_{s,0,r}=\underset{n-r-1}{\overset{n-r-1}{\psi}}{}^{\,0}_{s}(s,0) - \underset{n-r-1}{\overset{n-r-1}{\psi}}{}^{\,0}_{s-1}(s,0); \]
\[ A_{0,t,r}=\underset{n-r-1}{\overset{n-r-1}{\psi}}{}^{\,t}_{0}(0,t) - \underset{n-r-1}{\overset{n-r-1}{\psi}}{}^{\,t-1}_{0}(0,t); \qquad A_{m,t,r}= -\underset{n-r-1}{\overset{n-r-1}{\psi}}{}^{\,t}_{m-1}(m,t) + \underset{n-r-1}{\overset{n-r-1}{\psi}}{}^{\,t-1}_{m-1}(m,t); \]
\[ A_{s,t,r}= \underset{n-r-1}{\overset{n-r-1}{\psi}}{}^{\,t}_{s}(s,t) - \underset{n-r-1}{\overset{n-r-1}{\psi}}{}^{\,t}_{s-1}(s,t) - \underset{n-r-1}{\overset{n-r-1}{\psi}}{}^{\,t-1}_{s}(s,t) + \underset{n-r-1}{\overset{n-r-1}{\psi}}{}^{\,t-1}_{s-1}(s,t); \tag{11} \]
\[ A_{0,n,r}=-\underset{n-r-1}{\overset{n-r-1}{\psi}}{}^{\,n-1}_{0}(0,n); \qquad A_{m,n,r}=\underset{n-r-1}{\overset{n-r-1}{\psi}}{}^{\,n-1}_{m-1}(m,n); \]
\[ A_{s,n,r}= \underset{n-r-1}{\overset{n-r-1}{\psi}}{}^{\,n-1}_{s-1}(s,n) - \underset{n-r-1}{\overset{n-r-1}{\psi}}{}^{\,n-1}_{s}(s,n); \]
\[ M_{i,0,r}(x,y_0)=\underset{n-r}{\overset{n-r-1}{\psi}}{}^{\,0}_{i}(x,0); \qquad M_{j,n,r}=-\underset{n-r}{\overset{n-r-1}{\psi}}{}^{\,n-1}_{i}(x,n); \]
\[ M_{i,t,r}(x,y_t)= \underset{n-r}{\overset{n-r-1}{\psi}}{}^{\,t}_{i}(x,t) - \underset{n-r}{\overset{n-r-1}{\psi}}{}^{\,t-1}_{i}(x,t); \tag{12} \]
\[ N_{0,k,r}(x_0,y)=\underset{n-r-1}{\overset{n-r}{\psi}}{}^{\,k}_{0}(0,y); \qquad N_{m,k,r}=-\underset{n-r-1}{\overset{n-r}{\psi}}{}^{\,k}_{m-1}(m,y); \]
\[ N_{s,k,r}(x_s,y)= \underset{n-r-1}{\overset{n-r}{\psi}}{}^{\,k}_{s}(s,y) - \underset{n-r-1}{\overset{n-r}{\psi}}{}^{\,k}_{s-1}(s,y); \tag{13} \]
\[ P_{0,j,k}(x_0,y)=-\underset{j}{\overset{0}{\varphi}}{}^{\,k}_{0}(0,y); \qquad P_{m,j,k}(x_m,y)=\underset{j}{\overset{0}{\varphi}}{}^{\,k}_{m-1}(m,y); \]
\[ P_{s,j,k}(x_s,y)= \underset{j}{\overset{0}{\varphi}}{}^{\,k}_{s-1}(s,y) - \underset{j}{\overset{0}{\varphi}}{}^{\,k}_{s}(s,y), \tag{14} \]
\[ (s=1,2,\ldots,m-1;\quad t=1,2,\ldots,n-1), \]
where the function \(\Phi(x,y)\) coincides in each rectangle \(D_i^k\) with \(\varphi_i^k(x,y)\).
- Formula (10) leads us to the consideration of the boundary-value problem: determine the functions \(\psi_i^k(x,y)\) by the partial differential equations
\[ \partial^{2n}\psi_i^k/\partial x^n\partial y^n=0 \tag{15} \]
in \(D_i^k\), and by boundary conditions such that on the right-hand side of formula (10) only the values of the function \(f(x,y)\) at the nodes \((x_i,y_k)\) remain.
We have imposed the following boundary conditions:
\(N_{s,k,0}(x_s,y)=0\), \(M_{i,t,0}(x,y_t)=0\), \(A_{s,t,0}(x_s,y_t)=C_s^t\) for \(s=0,1,\ldots,m-1;\ t=0,1,\ldots,n-1;\ i=0,1,\ldots,m-1;\ k=0,1,\ldots,n-1;\ C_s^t\) are constants, which we shall determine below. This will lead to the partial differential equations
\[ \frac{\partial^{2(n-1)}\psi_i^k}{\partial x^{\,n-1}\partial y^{\,n-1}} = \sum_{\alpha=0}^{i}\sum_{\beta=0}^{k} C_{\alpha}^{\beta}. \tag{16} \]
Next we add the boundary conditions \(N_{s,k,r}(x_s,y)=0\), \(M_{i,t,r}(x,y_t)=0\); \(A_{s,t,r}(x_s,y_t)=0\) for \(s=0,1,\ldots,m-1\); \(t=0,1,\ldots,n-1\); \(i=0,1,\ldots,m-1\); \(k=0,1,\ldots,n-1\); \(r=1,2,\ldots,n-1\), which leads to
\[ \psi_i^k(x,y)=\frac{1}{[(n-1)!]^2}\sum_{\alpha=0}^{i}\sum_{\beta=0}^{k} C_\alpha^\beta (x-x_\alpha)^{n-1}(y-y_\beta)^{n-1}. \tag{17} \]
In order to determine the functions \(\varphi_i^k(x,y)\), one integrates the equations
\[ \partial^q \varphi_i^k/\partial x^q=(-1)^q\psi_i^k, \tag{18} \]
where the \(\psi_i^k\) are given by formulas (17), with the boundary conditions \(P_{s,j,k}(x_s,y)=0\) for \(k=0,1,\ldots,n-1\); \(s=0,1,\ldots,m-1\); \(j=0,1,\ldots,q-1\), which gives
\[ \varphi_i^k(x,y)=\frac{(-1)^{m-n}}{(m-1)!(n-1)!}\sum_{\alpha=0}^{i}\sum_{\beta=0}^{k} C_\alpha^\beta (x-x_\alpha)^{m-1}(y-y_\beta)^{n-1}. \tag{19} \]
- Introducing the new constants \(A_{m,t,0}=C_m^t\), \(A_{s,n,0}=C_s^n\), \(A_{m,n,0}=C_m^n\) for \(t=0,1,\ldots,n-1\); \(s=0,1,\ldots,m-1\), we see that all the constants \(C_\alpha^\beta\) are determined by the new boundary conditions \(M_{i,n,r}(x,y_n)=0\), \(N_{m,k,r}(x_m,y)=0\), \(P_{m,j,k}(x_m,y)=0\) for \(i=0,1,\ldots,m-1\); \(k=0,1,\ldots,n-1\); \(j=0,1,\ldots,q-1\); \(r=1,2,\ldots,n-1\). In this case formula (10) reduces to
\[ \iint_D \Phi(x,y)\frac{\partial^{m+n}f}{\partial x^m\partial y^n}\,dx\,dy = \sum_{i=0}^{m}\sum_{k=0}^{n} C_i^k f(x_i,y_k), \tag{20} \]
and the constants \(C_i^k\) are determined by the equations:
\[ \sum_{\alpha=0}^{m} C_\alpha^k x_\alpha^s=0,\qquad \sum_{\beta=0}^{n} C_i^\beta y_\beta^t=0 \quad (s=0,1,\ldots,m-1;\ t=0,1,\ldots,n-1). \tag{21} \]
From the first equations (21) we obtain
\[ C_i^k=(-1)^i V(x_0,x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_m)\lambda_k \tag{22} \]
for \(i=0,1,\ldots,m\); \(k=0,1,\ldots,n\), where the \(\lambda_k\) are constants. Writing that all the remaining equations (21) are satisfied, we obtain equations for determining \(\lambda_k\):
\[ \sum_{\beta=0}^{n}\lambda_\beta y_\beta^t=0 \quad (t=0,1,\ldots,n-1). \tag{23} \]
We find
\[ \lambda_k=(-1)^k\lambda V(y_0,y_1,\ldots,y_{k-1},y_{k+1},\ldots,y_n), \tag{24} \]
and, choosing
\[ \lambda=\frac{1}{V(x_0,x_1,\ldots,x_m)\,V(y_0,y_1,\ldots,y_n)}, \]
finally
\[ C_i^k=(-1)^{i+k} \frac{V(x_0,\ldots,x_{i-1},x_{i+1},\ldots,x_m)}{V(x_0,x_1,\ldots,x_m)} \frac{V(y_0,y_1,\ldots,y_{k-1},y_{k+1},\ldots,y_n)}{V(y_0,y_1,\ldots,y_n)}, \]
and this proves that the right-hand side of formula (20) coincides with the divided difference of order \((m,n)\) at the nodes \((x_i,y_k)\) of the function \(f(x,y)\). Thus formula (5) is proved.
In the following note we shall prove that the function \(\Phi(x,y)\) of formula (5), which vanishes on the sides of the rectangle \(D\), has constant sign inside this rectangle.
Cluj
Romanian People’s Republic
Received
25 IV 1961
REFERENCES
- D. V. Ionescu, Cuadraturi numerice, Bucuresti, 1957, chap. III.