Irremovable errors of a local smoothed trigonometric series
Abstract Generated abstract
This note examines unavoidable errors of the second kind in mean approximation by trigonometric polynomials when the approximating series is locally smoothed. Using the averaging operation corresponding to sigma multipliers, the author derives expressions and bounds for the smoothed error and compares them with the error of the original unsmoothed series. The analysis indicates that standard local smoothing reduces the error, especially for small polynomial degrees, but gives only a modest gain as the degree grows because the averaging interval tends to zero. The paper further argues that using a larger averaging interval can improve stability, provided it remains small relative to the number of nodes to avoid excessive distortion of the approximating series.
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UNAVOIDABLE ERRORS OF A LOCALLY SMOOTHED TRIGONOMETRIC SERIES
I. E. NAATS
(Presented by the scientific seminar of the Department of Higher Mathematics, TPI)
In [1] it is shown that the value of the unavoidable error of the second kind in approximation in the mean by a trigonometric polynomial increases as the degree of the polynomial increases, i.e., the convergence of the approximation as a whole is violated. In the present note we consider one of the methods for reducing the influence of uncertainty in the values of the nodes on the accuracy and stability of the approximation; its essence consists in local smoothing of the approximating trigonometric series constructed from experimental data.
The operation of local smoothing consists in replacing the original series by a new series according to the relation [2]
\[ \overline{T}_n(x)=\frac{1}{h}\int_{-\frac{h}{2}}^{\frac{h}{2}}T_n(x-\xi)\,d\xi, \tag{1} \]
where \(h\) is the averaging interval. In other words, the original series \(T_n(x)\) is replaced by the new \(\overline{T}_n(x)\), obtained from the former by averaging over an interval of length \(h\) with center at the point \(x\).
If the value of \(h\) is chosen from the condition
\[ h=\frac{2\pi}{N}=\frac{2\pi}{2n+1}, \tag{2} \]
then the method of local smoothing as applied to trigonometric polynomials is called the method of \(\sigma\)-multipliers [2]. Applying transformation (1) to the trigonometric polynomial \(T_n(x)\) in its canonical form, we obtain
\[ \overline{T}_n(x)=\frac{a_0}{2}+\sum_{k=1}^{n}\frac{\sin\frac{hk}{2}}{\frac{hk}{2}}\left(a_k\cos kx+b_k\sin kx\right) \tag{3} \]
for \(h\) satisfying (2).
Retaining the previous formulation of the problem [1], for the unavoidable error of the second kind we obtain the expression
\[ \bar{\sigma}_2^2=\frac{1}{2}\sum_{k=1}^{n} \frac{\sin^2 \frac{kh}{2}}{\left(\frac{kh}{2}\right)^2} \left(M[\alpha_k^2]+M[\beta_k^2]\right) \tag{4} \]
or, in final form,
\[ \bar{\sigma}_2^2=\frac{d^2}{\pi^2}(n-1)\sum_{k=1}^{2n} y_k^2 . \tag{5} \]
In deriving relation (5), the equality
\[ \sum_{k=1}^{n}\sin^2 \frac{hk}{2}=\frac{1}{2}(n-1), \tag{6} \]
was used, whose proof is based on the well-known formula
\[ \sum_{m=1}^{n} e^{ihm}=\frac{1-e^{ihn}}{1-e^{ih}} . \tag{7} \]
Considering the sequence of readings \(y_k\) \((k=0, 1\ldots, 2n)\), bounded by the constant \(M\), we obtain the inequality
\[ \bar{\sigma}_2^2<\frac{d^2}{\pi^2}(n-1)(2n+1)M^2 . \tag{8} \]
For sufficiently large \(n\), inequality (8) can be simplified:
\[ \bar{\sigma}_2<\frac{\sqrt{2}}{\pi}\,dMn . \tag{9} \]
It is interesting to compare the values of the unavoidable errors for the unsmoothed and smoothed series. The ratio of the errors is
\[ \frac{\sigma_2}{\bar{\sigma}_2} = \frac{\pi}{\sqrt{3}}\cdot \sqrt{\frac{n(n+1)}{(n+1)(2n+1)}} \tag{10} \]
or, for large \(n\),
\[ \frac{\sigma_2}{\bar{\sigma}_2} = \frac{\pi}{\sqrt{6}} = 1.28\ldots \tag{11} \]
From the last relation it is evident that, in the second case, the unavoidable error \(\bar{\sigma}_2\) for large \(n\) is somewhat smaller than the unavoidable error for the unsmoothed series. For small \(n\), which is what is most often encountered in approximation practice, the reduction in error is more substantial. In particular, for \(n=2;\ 3;\ 4\) the corresponding values \(\sigma_2/\bar{\sigma}_2\) are \(1.98;\ 1.68;\ 1.56\). The insignificance of the gain in reducing the error for the smoothed series for large \(n\) is due to the fact that,
that, as \(n\) increases, the smoothing interval \(h \to 0\), and the smoothing effect becomes smaller and smaller. On the whole, it should be noted that smoothing the approximating series by means of \(\sigma\)-multipliers does not solve the problem of stability of the approximation process under variation of the sequence of nodes within bounded limits.
With a large number of ordinates, when \(h\) is sufficiently small, the smoothing interval can be taken larger than \(h\). Let, for example, the interval of local averaging be \(\delta = mh\), where \(m\) is an integer and \(h\) is determined according to (2). Then transformation (1) takes the form
\[ \overline{\mathrm{T}}_{n}(x)=\frac{1}{mh}\int_{-\frac{h}{2}m}^{\frac{h}{2}m}\mathrm{T}_{n}(x-\xi)\,d\xi \tag{12} \]
and the ratio
\[ \frac{\overline{\sigma}_{2}}{\sigma_{2}}=\frac{\pi}{\sqrt{6}}\cdot m, \tag{13} \]
i.e., it turns out to be \(m\) times greater than the initial value. Thus, increasing the interval of local averaging makes the approximation process more stable. However, it is necessary that \(m\) remain significantly smaller than \(N=2n+1\); otherwise the distortion of the approximating series caused by the local averaging will be significant. An estimate of this distortion can be found in [2].
REFERENCES
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I. E. Naats. Approximation in the mean by trigonometric polynomials at approximate nodes (in the present collection).
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K. Lanczos. Practical Methods of Applied Analysis. Fizmatgiz, Moscow, 1961.