S. N. Bernstein’s Interpolation Process Based on Approximate Data

I. E. Naats

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

In the present paper we consider S. N. Bernstein’s interpolation process for the case when the initial data are approximate, i.e., the ordinates and the nodes at which they are taken are known with a certain error. The presence of the indicated errors substantially affects the quality of the approximation, as was already shown in [1].

S. N. Bernstein’s interpolation process is constructed on the basis of trigonometric interpolation polynomials according to the expression (the first process of S. N. Bernstein [2] is meant):

\[ U_q^{(n)}(x)=\frac{T_{n,0}(x)+T_{n,1}(x)+\cdots+T_{n,q}(x)}{q+1}, \tag{1} \]

where

\[ T_{n,p}(x)=\frac{a_0^{(n)}}{2}+\sum_{m=1}^{p}\left(a_m^{(n)}\cos mx+b_m^{(n)}\sin mx\right), \tag{2} \]

\[ \begin{aligned} a_0^{(n)}&=\frac{2}{2n+1}\sum_{k=0}^{2n} f(x_k),\\ &\text{and}\\ a_m^{(n)}&=\frac{2}{2n+1}\sum_{k=0}^{2n} f(x_k)\cos m x_k,\\ b_m^{(n)}&=\frac{2}{2n+1}\sum_{k=0}^{2n} f(x_k)\sin m x_k, \end{aligned} \tag{3} \]

for the system of equally spaced nodes

\[ x_k=\frac{2\pi k}{2n+1}. \tag{4} \]

The interpolating trigonometric polynomial (2) for the system (4) can also be written in the form

\[ T_{n,p}(x)=\frac{1}{2n+1}\sum_{k=0}^{2n} f(x_k)\, \frac{\sin^{2p+1}\frac{x-x_k}{2}}{\sin\frac{x-x_k}{2}} . \tag{5} \]

The polynomials (1), taking (2) and (3) into account, assume the following form:

\[ U_q^{(n)}(x)=\frac{1}{(2n+1)(q+1)} \sum_{k=0}^{2n} f(x_k) \left[ \frac{\sin\frac{q+1}{2}(x-x_k)} {\sin\frac{x-x_k}{2}} \right]^2 . \tag{6} \]

Let us consider the irreducible errors of the first and second kind for the interpolation process (6).

The irreducible error of the first kind is due to the presence of uncertainty in the sequence of readings \(y_k\) \((k=0,1,\ldots,2n)\). Considering the random set of sequences of readings (the set of realizations), one can write its properties as

\[ \begin{aligned} M\{y_0',y_1',\ldots,y_{2n}'\} &=M\{y_0'\},M\{y_1'\}\ldots M\{y_{2n}'\} =y_0,y_1,\ldots,y_{2n},\\ D\{y_0',y_1'\ldots y_{2n}'\} &=D\{y_0'\},D\{y_1'\}\ldots D\{y_{2n}'\} =\sigma_0^2,\sigma_1^2\ldots\sigma_{2n}^2 . \end{aligned} \tag{7} \]

In what follows we shall assume that

\[ \sigma_0=\sigma_1=\ldots=\sigma_{2n}. \tag{8} \]

For the random set of sequences of readings one can construct a random set of polynomials

\[ \{U_q^{(n)}(x)\}', \]

for which

\[ M\{U_q^{(n)}(x)\}'=U_q^{(n)}(x). \tag{9} \]

The irreducible error for this case will be defined by the expression

\[ \sigma_1^2=D[U_q^{(n)}(x)]'. \tag{10} \]

Using the general form of the interpolation polynomials

\[ U_q^{(n)}(x)=\sum_{k=0}^{2n} f(x_k)\cdot k_{n,q}(x-x_k), \tag{11} \]

under conditions (7) and (8), one can show that

\[ \sigma_1^2=\sigma_0^2\sum_{k=0}^{2n} k_{n,q}^2(x-x_k). \tag{12} \]

Analogously, one can formulate the problem also for the irreducible error of the second kind [1]. In this case the random set of poly-

of the polynomials \(\{U_q^{(n)}(x)\}''\) is constructed on a random set of sequences of interpolation nodes which, on the average, coincide with the sequence of equally spaced nodes (4). Then the value of the irreducible error of the second kind is determined by the expression

\[ \sigma_2^2 = D\{U_q^{(n)}(x)\}'' = d^2 \sum_{k=0}^{2n} y_k^2 \bigl[k'_{n,q}(x-x_k)\bigr]^2 . \tag{13} \]

Considering the sequence of readings to be bounded in the aggregate by some constant \(M\), we obtain the inequality

\[ \sigma_2^2 < d^2 M^2 \sum_{k=0}^{2n} \bigl[k'_{n,q}(x-x_k)\bigr]^2 . \tag{14} \]

In order to obtain concrete values of the estimates according to (12) and (14), it is necessary to compute the values of the corresponding sums for the kernel \(k_{n,q}(x-x_k)\).

Omitting the rather laborious and cumbersome computations, we write the final result

\[ \sum_{k=0}^{2n} k_{n,q}^2(x-x_k) = \frac{1}{(2n+1)^2(q+1)^2} \sum_{k=0}^{2n} \left[ \frac{\sin \frac{q+1}{2}(x-x_k)} {\sin \frac{x-x_k}{2}} \right]^4 = \frac{4}{3}\, \frac{q\left(q-\frac{1}{2}\right)} {(2n+1)(q+1)} \tag{15} \]

and, correspondingly,

\[ \sum_{k=0}^{2n} \bigl[k'_{n,q}(x-x_k)\bigr]^2 = \frac{q(q^2+3q+2)(2q^3-2q+3)} {18(q+1)(2n+1)} . \tag{16} \]

As a result, for the irreducible errors we obtain the following expressions:

\[ \sigma_1^2 = \sigma_0^2\, \frac{4q\left(q-\frac{1}{2}\right)} {3(2n+1)(q+1)} \tag{17} \]

and

\[ \sigma_2^2 < \frac{d^2 M^2}{18}\cdot \frac{q(q^2+3q+2)(2q^3-2q+3)} {(2n+1)(q+1)} . \tag{18} \]

Let us discuss the results obtained.

For sufficiently large \(n\) and \(q\), for the irreducible error \(\sigma_1\) one can give a simpler expression, namely:

\[ \sigma_1 = \sigma_0 \sqrt{\frac{2q}{3n}} . \tag{19} \]

From this it is clear that, as \(q\) and \(n\) increase (\(q<n\)), the value \(\sigma_1\) remains within certain limits determined by the value of the measurement error \(\sigma_0\).

Consequently, S. N. Bernstein’s first process for approximate observations remains stable as the order of the interpolation polynomial increases. Analogous results are also valid for the trigonometric interpolation polynomial [3]. Here it is necessary to note the following. Since the general structure of the approximation error is determined by the expression

\[ \sigma^{2}=\sigma_{n}^{2}+\sigma_{1}^{2}+\sigma_{2}^{2}, \tag{20} \]

where \(\sigma_n\) is the root-mean-square approximation error for exact data, determined only by the convergence of the interpolation process, it becomes possible to estimate the required amount \(N\) of initial data for given values of \(\sigma\), \(\sigma_0\), and \(d\). In doing so, it is usually assumed that all three components \(\sigma_n\), \(\sigma_1\), and \(\sigma_2\) must be of one and the same order.

For the irremovable error of the second kind \(\sigma_2\), for large \(n\) and \(q\) \((q<n)\), one can also give the simpler inequality

\[ \sigma_{2}\leq dM\frac{q^{2}\sqrt{q}}{3\sqrt{2n+1}}. \tag{21} \]

It follows from this that the irremovable error of the second kind in S. N. Bernstein’s interpolation process increases very sharply with increasing \(q\). This means that S. N. Bernstein’s process is unstable with respect to displacements of the nodes on the interval of interpolation. This fact is explained as follows. As \(q\) increases in S. N. Bernstein’s interpolation process, there is an intensification of the localization of the kernel \(k_{n,q}(x-x_k)\) in the region of the central maximum located at the point \(x_k\). Although this has a favorable effect on the convergence of the polynomial to the function being approximated, the influence of a random variation of the sequence of nodes relative to the system of equidistant nodes \(\{x_k\}\) on the value of the polynomial at any point of the interval of approximation becomes significant. Analytically, this is reflected in an increase in the values of the derivative of the kernel in the \(\delta\)-neighborhood of the point \(x_k\) as \(q\) increases.

Thus, an interpolation process constructed by the summation method over a system of trigonometric interpolation polynomials becomes more sensitive to uncertainty in the fixing of the interpolation nodes. Therefore, the use of such complex approximation apparatuses as S. N. Bernstein’s interpolation process is justified only in the case of very accurate measurements of the initial data.

REFERENCES

1. I. E. Naats. Approximation in the mean by trigonometric polynomials with approximate nodes. (This collection).

2. I. P. Natanson. Constructive Theory of Functions. Moscow, 1949.

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