APPROXIMATION IN THE MEAN BY TRIGONOMETRIC POLYNOMIALS AT APPROXIMATE NODES

I. E. NAATS

(Presented by the scientific seminar of the Department of Higher Mathematics, TPI)

In processing experimental data for the purpose of constructing empirical dependencies, a discrete analogue of the Fourier series in a trigonometric system of functions is used rather often. In this case the initial data—namely, the values of the ordinates of the function being approximated, as well as the nodes at which these ordinates are taken—are known only approximately. This occurs, for example, in the problem of automatically constructing the relief of a certain physical field, monitored by periodically taking discrete readings from sensors located in this field [1].

In the present work an estimate is given for the root-mean-square error of approximation by trigonometric polynomials that arises from uncertainty in fixing the approximation nodes. This part of the total approximation error has been called the irreducible error of the second kind, in contrast to the irreducible error of the first kind, which is caused by the error in measuring the ordinates [2]. We shall give the nature of the uncertainty of the values of the nodes on the interval of approximation a statistical meaning; that is, we shall assume that there is a random set of sequences of nodes \(\{x_0', x_1', \ldots, x_n'\}\) which, on the average, coincides with some selected system of nodes \(\{x_0, x_1, \ldots, x_n\}\). Mathematically this fact may be expressed as follows:

\[ M\{x_0', x_1', \ldots, x_n'\}=M\{x_0'\}, M\{x_1'\}\ldots M\{x_n'\}=x_0, x_1\ldots x_n, \tag{1} \]

\[ D\{x_0', x_1'\ldots x_n'\}=D\{x_0'\}, D\{x_1'\}\ldots D\{x_n'\}=d_0^2, d_1^2\ldots d_n^2. \tag{2} \]

Here \(M\) and \(D\) are, respectively, the operators of mathematical expectation and dispersion.

Such a formulation of the problem fully corresponds to reality, since in essence it describes the measurement procedure for determining the location of the nodes on the interval of approximation, if in the final analysis one is dealing with an experiment in a physical field or with the control of some process on a finite time interval.

In what follows we shall assume that

\[ d_0=d_1=d_2=\ldots d_n=d \tag{3} \]

and that the nodes \(x_0, x_1, \ldots, x_n\) are distributed uniformly on the interval of approximation. Then the trigonometric polynomial for some function \(f(x)\), constructed—

determined from the observations \(y_0, y_1, \ldots, y_{2n}\) and the nodes \(x_0, x_1, \ldots, x_{2n}\), and best in the sense of the mean-square criterion, has the form

\[ T_n(x)=\frac{a_0}{2}+\sum_{k=1}^{n}(a_k\cos kx+b_k\sin kx), \tag{4} \]

where

\[ \left. \begin{aligned} a_0&=\frac{2}{2n+1}\cdot\sum_{i=0}^{2n} y_i,\\ a_k&=\frac{2}{2n+1}\sum_{i=0}^{2n} y_i\cos kx_i,\\ b_k&=\frac{2}{2n+1}\sum_{i=0}^{2n} y_i\sin kx_i. \end{aligned} \right\} \tag{5} \]

On the set of random sequences \(\{x'_0, x'_1,\ldots,x'_{2n}\}\), with fixed observations \(y_0, y_1,\ldots,y_{2n}\), one can construct a set of random polynomials \(\{T'_n(x)\}\) according to (4) and (5) such that

\[ M[T'_n(x)]=T_n(x) \tag{6} \]

for all \(x\), and also a set of deviations \(\{\Delta'_n(x)\}\), defined as

\[ \Delta'_n(x)=T'_n(x)-T_n(x)=\sum_{k=1}^{n}(\alpha_k\cos kx+\beta_k\sin kx), \tag{7} \]

where

\[ \left. \begin{aligned} \alpha_k&=\frac{2}{2n+1}\sum_{i=0}^{2n} y_i[\cos kx'_i-\cos kx_i],\\ \beta_k&=\frac{2}{2n+1}\sum_{i=0}^{2n} y_i[\sin kx'_i-\sin kx_i]. \end{aligned} \right\} \tag{8} \]

It follows from (6) that

\[ M[\Delta'_n(x)]=0. \tag{9} \]

The irreducible error of the second kind \(\sigma_2\) over the whole interval of approximation (say, of length \(2\pi\)), caused by the inaccuracy in fixing the nodes, will be expressed by the following expression:

\[ \sigma_2^2=\frac{1}{2\pi}\int_{0}^{2\pi}D[\Delta'_n(x)]\,dx. \tag{10} \]

Substituting (8) into (10) and carrying out the corresponding transformations, we obtain

\[ \sigma_2^2=\frac{1}{2}\sum_{k=1}^{n}\left(M[\alpha_k^2]+M[\beta_k^2]\right). \tag{11} \]

We shall assume that

\[ |x'_i-x_i|=\varepsilon_i\ll x_{i+1}-x_i \tag{12} \]

for all \(i\); then, for \(\alpha_k\) and \(\beta_k\), one may use the approximations

\[ \left. \begin{aligned} \alpha_k&=-\frac{2}{2n+1}\cdot k\sum_{i=0}^{2n} y_i\varepsilon_i'\sin kx_i,\\ \beta_k&=\frac{2}{2n+1}\cdot k\sum_{i=0}^{2n} y_i\varepsilon_i'\cos kx_i. \end{aligned} \right\} \tag{13} \]

Using (13), it is not difficult to show that

\[ \left. \begin{aligned} M[\alpha_k^2]&=\frac{4}{(2n+1)^2}k^2d^2\sum_{i=0}^{2n} y_i^2\sin^2 kx_i,\\ M[\beta_k^2]&=\frac{4}{(2n+1)^2}k^2d^2\sum_{i=0}^{2n} y_i^2\cos^2 kx_i. \end{aligned} \right\} \tag{14} \]

Substituting (14) into (12), we find

\[ \sigma_2^2=\frac{n(n+1)d^2}{3(2n+1)}\sum_{i=0}^{2n} y_i^2. \tag{15} \]

If we assume that the function \(f(x)\) is bounded on the interval of approximation by some constant \(M\), i.e., \(|f(x)|\le M\) for all \(x\), which is quite natural in practical problems, then for the quantity \(\sigma_2\) we obtain the estimate

\[ \sigma_2^2\le \frac{n(n+1)}{3}M^2d^2 \tag{16} \]

or, for sufficiently large \(n\),

\[ \sigma_2\le \frac{ndM}{\sqrt{3}}. \tag{17} \]

It is convenient to introduce the relative error \(\xi_2\) according to the expression

\[ \xi_2=\frac{\sigma_2}{M}\le \frac{nd}{\sqrt{3}}. \tag{18} \]

Regarding the value of the irreducible error of the second kind according to (17) (likewise (18)), the following conclusions may be drawn: the irreducible error of the second kind, caused by the error in determining the nodes on the interval of approximation, increases with increasing degree of the approximating polynomial. As a result, and this is very important to note, beginning with certain \(n\) (likewise with \(N=2n+1\) sample values), the total approximation error, which consists of the removable error (i.e., the error of the method, determined by the convergence of the polynomial to the function \(f(x)\)) and the irreducible errors of the first and second kinds, will increase despite a further increase in the volume of the initial data, i.e., the number of sample values \(N\). Consequently, when increasing the number of sample values it is necessary to increase the accuracy of measurement both of the function values and of the positions of the nodes, in order to obtain acceptable values of the approximation error.

In conclusion, we note that the increase in the irreducible error of the second kind indicates the instability of the process of approximation by trigonometric polynomials with respect to variation of the mutual arrangement of the nodes \(x_i\) on the interval of approximation. This same process remains stable, as was shown in [3], with respect to variation of the sequence-

of the samples in the error band, i.e., the irreducible error of the first kind does not depend on the degree of the approximating polynomial. By a special method the approximation process can be made more stable. The simplest of these is smoothing of series computed from approximate data. In particular, if the degree of the polynomial \(n\) is chosen from the relation \(N=r(2n+1)\), where \(r\) is an integer greater than one, as occurs in trigonometric regression, then the corresponding estimate for \(\xi_2\) takes the form

\[ \xi_2 < \frac{\delta n}{r\sqrt{3}}, \tag{19} \]

i.e., \(\xi_2\) proves to be \(r\) times smaller.

The results obtained may serve as a basis for planning the volume of experimental data in the case when approximation of experimental dependences is carried out by the method of least squares using trigonometric polynomials.

REFERENCES

1. I. E. Naats. Some questions in the design of devices for recording spatially distributed parameters. Izv. TPI, vol. 138, Tomsk State University Press. Tomsk, 1965.

2. N. S. Berezin, N. P. Zhidkov. Methods of Computation, vol. 1, Fizmatgiz. Moscow, 1962.

3. I. E. Naats. Application of trigonometric interpolation in problems of discrete measurement. Izv. TPI, vol. 168, Tomsk State University Press. Tomsk, 1968.