IRREMOVABLE ERROR OF INTERPOLATION BY TRIGONOMETRIC POLYNOMIALS ON A GRID OF NODES IN THE PLANE

I. E. NAATS

(Submitted by the scientific seminar of the computing laboratory)

In the present work the results of paper [1] are extended to a system of points in the plane. Here, as before, it is assumed that the interpolation nodes in the plane are determined experimentally, i.e., cannot be known with arbitrary precision. Therefore the formulation of the problem in the case under consideration will be exactly the same as in [1], namely: to estimate the variance of the deviation of the set of interpolating trigonometric polynomials constructed on a system of random grids in the plane, which on the average deviate little from a certain uniform grid of nodes in the square \((2\pi \times 2\pi)\), from the polynomial constructed on the nodes of this grid.

Let there be an interpolating polynomial

\[ T_{mn}(x,y)=\sum_{i=0}^{2m}\sum_{j=0}^{2n} f(x_i,y_j)D_m(x-x_i)D_n(y-y_j) \tag{1} \]

in the system of points in the plane \(\{x_i,y_j\}\), where

\[ x_i=\frac{2\pi i}{M}\quad (i=0,1,2\ldots 2m), \qquad M=2m+1, \]

\[ y_j=\frac{2\pi j}{N}\quad (j=0,1,2\ldots 2n), \qquad N=2n+1 \tag{2} \]

and

\[ D_m(t)=\frac{\sin(2m+1)\dfrac{t}{2}}{(2m+1)\sin\dfrac{t}{2}}. \tag{3} \]

The magnitude of the deviation of the interpolating polynomial constructed on the grid of nodes

\[ \left. \begin{aligned} x_{iq}&=x_i+\varepsilon_{xiq}\\ y_{jq}&=y_j+\varepsilon_{yjq} \end{aligned} \right\}, \tag{4} \]

fixed in the \(q\)-th experiment (realization), and the polynomial constructed from the system of equally spaced nodes (2), is written in the following form

\[ \Delta_{mnq}(x,y)=\sum_{i=0}^{2m}\sum_{j=0}^{2n} f(x_i,y_j)[D_m(x-x_i+\varepsilon_{xiq})\times \tag{5} \]

\[ \times D_n(y-y_j+\varepsilon_{yjq})-D_m(x-x_i)D_n(y-y_j)]. \]

In what follows we shall assume that the systems of random variables

\[ \varepsilon_{xi}(i=0,1\ldots 2m)\quad \text{and}\quad \varepsilon_{yj}(j=0,1\ldots 2n) \]

possess the following properties:

\[ \left. \begin{aligned} M[\varepsilon_{xi}\cdot \varepsilon_{xp}]&= \begin{cases} 0, & \text{for } i\ne p,\\ d_x^2, & \text{for } i=p, \end{cases} \qquad (i,p=0,1\ldots 2m),\\ M[\varepsilon_{yj}\cdot \varepsilon_{yk}]&= \begin{cases} 0, & \text{for } j\ne k,\\ d_y^2, & \text{for } j=k, \end{cases} \qquad (j,k=0,1\ldots 2n),\\ M[\varepsilon_{xi}\cdot \varepsilon_{yj}]&=0\quad (i=0,1\ldots 2m),(j=0,1\ldots 2n). \end{aligned} \right\} \tag{6} \]

The last relation states the absence of correlation in the errors of determining the coordinates of the nodes in the plane in the directions of the \(x\)- and \(y\)-axes.

Regarding the absolute value of the deviations \(\varepsilon_{xi}\) and \(\varepsilon_{yj}\) over the set of experiments as sufficiently small in comparison with the interpolation step along the axes, we write for the deviation \(\Delta_{mnq}(x,y)\) the following approximate equality:

\[ \Delta_{mnq}(x,y)=\sum_{i=0}^{2m}\sum_{j=0}^{2n} f(x_i,y_j)[\varepsilon_{xiq}D_m'(x-x_i)D_n(y-y_j)+ \]

\[ +\varepsilon_{yjq}D_m(x-x_i)D_n'(y-y_j)]. \tag{7} \]

Taking (6) into account, it is not difficult to show that

\[ M[\Delta_{mn}(x,y)]=0, \tag{8} \]

and the expression for the variance is reduced to the following form

\[ D[\Delta_{mn}(x,y)]=\sum_{i=0}^{2m}\sum_{j=0}^{2n} f^2(x_i,y_j)[d_x^2 D_m'^2(x-x_i)\times \]

\[ \times D_n^2(y-y_j)+d_y^2D_m^2(x-x_i)D_n'^2(y-y_j)]. \tag{9} \]

Further assuming \(f(x_i,y_j)\) to be bounded in their totality by a constant \(C\), and taking into account that for the grid (2)

\[ \sum_{i=0}^{2m}D_m'^2(x-x_i)=\frac{m(m+1)}{3}, \]

\[ \sum_{i=0}^{2m}D_m^2(x-x_i)=1, \]

for the variance we obtain the estimate

\[ D[\Delta_{mn}(x,y)] \leq C^2\left[d_x^2\frac{m(m+1)}{3}+d_y^2\frac{n(n+1)}{3}\right]. \tag{10} \]

We shall take as the irremovable error the upper bound of the values \(D[\Delta_{mn}(x,y)]\); then

\[ \sigma_2=C\cdot\sqrt{d_x^2\frac{m(m+1)}{3}+d_y^2\frac{n(n+1)}{3}} . \tag{11} \]

In the particular case when \(M=N\) (respectively \(m=n\)) and \(d_x=d_y=d\), we have

\[ \sigma_2=\sqrt{\frac{2}{3}}\,Cdn \tag{12} \]

and, correspondingly, for the relative error

\[ \xi_2=\frac{\sigma_2}{C}=\sqrt{\frac{2}{3}}\,dn . \tag{13} \]

Thus, as in the case of one variable [1], the irremovable error of the second kind for the problem of approximation in the plane increases if, as the number of readings is increased, the error in determining the initial data—and especially the abscissas of interpolation—is not decreased. Since otherwise the total approximation error also increases, the costs of increasing the volume of the initial data become, from some point on, inexpedient.

Now consider the estimate of the variance of the quantity \(\Delta_{mn}(x,y)\) in the presence of correlation between the quantities \(\varepsilon_{xi}\) and \(\varepsilon_{yj}\), i.e., when

\[ M[\varepsilon_{xi}\cdot\varepsilon_{yj}]=d_{xy}^{2}. \tag{14} \]

In this case we have

\[ \begin{aligned} D[\Delta_{mn}(x,y)]\leq C^2\Bigg[& d_x^2\frac{m(m+1)}{3}+d_y^2\frac{n(n+1)}{3}+{}\\ &+d_{xy}^{2}\sum_{i=0}^{2m}D'_m(x-x_i)D_m(x-x_i) \sum_{j=0}^{2n}D'_n(y-y_j)D_n(y-y_j) \Bigg]. \end{aligned} \tag{15} \]

Using the known identity

\[ \frac{1}{2}+\cos\alpha+\cos2\alpha+\ldots\cos m\alpha = \frac{\sin(2m+1)\frac{\alpha}{2}}{2\sin\frac{\alpha}{2}}, \]

it is not difficult to prove that

\[ \sum_{i=0}^{2m}D'_m(x-x_i)D_m(x-x_i)\equiv0, \tag{16} \]

for equidistant nodes \(x_i(i=0,1,\ldots,2m)\). Therefore, in (15) the last

the term is identically equal to zero, and the resulting estimate for the variance of the deviation is analogous to (10). In other words, the value of the variance does not depend on the correlation between the errors in determining the coordinates of the grid of nodes. It is not difficult to see that this is a consequence of the orthogonality of trigonometric polynomials on a discrete set of equidistant points.

References

1. I. E. Naatz. Irremovable error of interpolation by trigonometric polynomials. (This collection).