UDC 517.522.3

MATHEMATICS

V. Ya. ARSENIN

ON OPTIMAL SUMMATION OF FOURIER SERIES WITH APPROXIMATE COEFFICIENTS

(Presented by Academician A. N. Tikhonov on 28 III 1968)

In paper \((^1)\), following the regularization method developed in \((^2)\), a solution is given to the problem of finding an approximate value at a point \(x_0 \in [a,b]\) of a function \(f(x)\), defined on the interval \([a,b]\), from the approximate values (in the \(l_2\) metric) of its Fourier coefficients \(\{c_n\}\) with respect to the orthonormal system of eigenfunctions \(\{u_n(x)\}\) of the boundary-value problem:

\[ u'' - q^2(x)u + \lambda u = 0,\qquad u(a)=u(b)=0,\qquad 0\le q^2(x)\le M. \tag{*} \]

Let \(a_n\) be the exact values of the Fourier coefficients of the function \(f(x)\), and let \(\gamma_n\) be their errors such that

\[ \sum_{n=1}^{\infty}\gamma_n^2 \le \delta^2. \]

Then \(c_n=a_n+\gamma_n\). As the approximate value of the function \(f(x)\), in \((^1)\) one takes the sum of the series

\[ f_\alpha(x)=\sum_{n=1}^{\infty} r(n,\alpha)c_nu_n(x) \]

with regularizing factors \(r(n,\alpha)=1/(1+\alpha\lambda_n)\), where \(\lambda_n\) are the eigenvalues of the boundary-value problem \((*)\), and \(\alpha\) is the regularization parameter \((\alpha>0)\).

A. N. Tikhonov posed the problem of finding an optimal, in some sense, summation method, i.e. of choosing the regularizing factors \(r(n,\alpha)\). In the present article, a solution of this problem is given in various formulations close to Wiener’s \((^3)\) and to the formulations contained in \((^{4-8})\).

Let, in a finite closed domain \(\overline D\) of \(n\)-dimensional Euclidean space \(R_n\), there be given a complete orthonormal (with weight \(\rho(x)>0\)) system of functions \(\{u_n(x)\}\), \(x=(x_1,x_2,\ldots,x_n)\), and a continuous function \(\hat f(x)\) in \(\overline D\), representable by its Fourier series with respect to the system \(\{u_n(x)\}\),

\[ \hat f(x)=\sum_{n=1}^{\infty} a_nu_n(x),\qquad \text{where }\quad a_n=\int_D \rho(x)\hat f(x)u_n(x)\,dx. \]

Suppose that we know approximate values of the Fourier coefficients of this function \(\hat f(x)\), \(c_n=a_n+\gamma_n\), with small (in \(l_2\)) errors \(\gamma_n\),

\[ \sum_{n=1}^{\infty}\gamma_n^2 \le \delta^2. \]

It is required to find an approximate value of the function \(\hat f(x)\) from the approximate values of its Fourier coefficients \(\{c_n\}\). We shall solve this problem by the regularization method.

Consider the functional

\[ M^m=\sum_{n=1}^{\infty}(f_n-c_n)^2+\Omega[f_n], \]

where

\[ f_n=\int_D \rho(x)f(x)u_n(x)\,dx,\qquad \Omega[f_n]=\sum_{n=1}^{\infty} f_n^2m_n, \]

\(\{m_n\}\) is a sequence of positive numbers whose order of growth as \(n\to\infty\) is not less than

\(n^{1+\varepsilon}\), where \(\varepsilon>0\). We shall denote by \(\mathfrak{M}\) the class of all such sequences corresponding to different values of \(\varepsilon\). The functional \(\Omega[f]\) will be called stabilizing. If, as the system \(\{u_n(x)\}\), one takes the eigenfunctions of the boundary-value problem \((*)\) and sets \(m_n=\alpha\lambda_n\), where \(\lambda_n\) are the eigenvalues of problem \((*)\), then one obtains the stabilizing functional used in paper \((^1)\).

The function \(\tilde f(x)\) realizing the minimum of the functional \(M^m\) in the class of functions continuous in \(\overline D\) (the class \(C_D\)) will be taken as the approximate value of the function \(\hat f\). It is easy to see that the Fourier coefficients of the function \(\tilde f(x)\) are equal to

\[ \tilde f_n=\frac{c_n}{1+m_n}, \quad \text{i.e. } \tilde f(x)=\sum_{n=1}^{\infty} c_n r(n)u_n(x), \quad \text{where } r(n)=\frac{1}{1+m_n}. \]

Thus, the summation method is determined by the choice of the sequence \(\{m_n\}\). This summation method is stable in the sense that small (in \(l_2\)) changes of the coefficients correspond to a small change of \(\tilde f(x)\) (in the metric of \(C_D\)).

To prove this, it is enough first of all to note that the set of functions \(f\in C_D\) for which \(\Omega[f]\le d\) is compact in \(C_D\) for any \(d\ge 0\). Define an operator \(A\) acting from \(C_D\) into \(l_2\): to a function \(f\in C_D\) we assign the sequence of its Fourier coefficients with respect to the system \(\{u_n(x)\}\) with weight \(\rho(x)\). This mapping is continuous and one-to-one. Consequently, the inverse mapping is also continuous. This proves the assertion on the stability of the summation method.

The errors in the Fourier coefficients, i.e. \(\gamma_n\), are random numbers, about which we shall assume:

1) \(\{\gamma_n\}\) is a sequence of uncorrelated random numbers.
2) The mathematical expectations \(E\gamma_n=\overline{\gamma_n}=0\) for all \(n\). Under these conditions the approximate values of the Fourier coefficients are also random numbers, and \(c_n^2=a_n^2+\gamma_n^2\). The variances of the random variables \(\gamma_n\) and \(c_n\) are the same and are equal to \(\sigma_n^2=\overline{\gamma_n^2}\). The function \(\tilde f(x)\), realizing the minimum of the functional \(M^m\) for a fixed sequence \(\{m_n\}\), is a random function.

Let \((\Delta f)_m=\tilde f(x)-\hat f(x)\), where \(\hat f(x)=\sum_{n=1}^{\infty} a_nu_n(x)\). As a measure of the deviation of \(\tilde f(x)\) from \(\hat f(x)\) one may take: a) \(\overline{(\Delta f)_m^2}\) or b)

\[ \int_D \rho(x)\overline{(\Delta f)_m^2}\,dx. \]

Putting \(m_n=\alpha\xi_n\) \((n=1,2,\ldots)\), where \(\alpha>0,\ \xi_n>0\), we obtain

\[ \tilde f(x)=f_\alpha(x)=\sum_{n=1}^{\infty}\frac{c_n}{1+\alpha\xi_n}u_n(x). \]

In this case

\[ \Omega[f]=\alpha\Omega_1[f]=\alpha\sum_{n=1}^{\infty} f_n^2\xi_n \]

and \((\Delta f)_m=\Delta f_\alpha\); \(\alpha\) is the regularization parameter.

The following formulations of problems are natural:

\(1_{\mathrm{A}}\). For a fixed sequence \(\{\xi_n\}\), find such a value \(\alpha_0\) of the regularization parameter \(\alpha\) for which

\[ \overline{(\Delta f_{\alpha_0})^2}=\min_\alpha \overline{(\Delta f_\alpha)^2} \]

at a fixed point \(x_0\).

\(1_{\mathrm{B}}\). For a fixed sequence \(\{\xi_n\}\), find such a value \(\alpha=\alpha_0\) for which

\[ \int_D \rho(x)\overline{(\Delta f_{\alpha_0})^2}\,dx = \min_\alpha \int_D \rho(x)\overline{(\Delta f_\alpha)^2}\,dx. \]

The summation of the Fourier series determined by such a value of \(\alpha\), for a fixed stabilizing functional

\[ \Omega_1[f]=\sum_{n=1}^{\infty} f_n^2\xi_n \]

will be called \(\alpha\)-optimal.

IIA. In the class \(\mathfrak M\) of sequences of positive numbers \(\{m_n\}\), find a sequence \(\{m_n'\}\) for which \(\overline{(\Delta f)_{m'}^2}=\min_{\mathfrak M}\overline{(\Delta f)_m^2}\) at the fixed point \(x_0\).

IIB. In the class \(\mathfrak M\) of sequences \(\{m_n\}\) of positive numbers, find a sequence \(\{m_n'\}\) for which
\[ \int_D \rho(x)\overline{(\Delta f)_{m'}^2}\,dx = \min_{\mathfrak M}\int_D \rho(x)\overline{(\Delta f)_m^2}\,dx . \]
The summation of a Fourier series determined by such a sequence \(\{m_n'\}\) will be called optimal. This formulation of the problem is analogous to the Wiener problem on optimal filtering \((^3)\).

Solution of Problems IB and IIB. Since
\[ \Delta f_\alpha=\sum_{n=1}^{\infty}\frac{\gamma_n-\alpha a_n\xi_n}{1+\alpha\xi_n}u_n(x), \]
we have
\[ \int_D \rho(x)\overline{(\Delta f_\alpha)^2}\,dx = N(\alpha) = \sum_{n=1}^{\infty} \frac{\sigma_n^2+\alpha^2\xi_n^2\left(\overline{c_n^2}-\sigma_n^2\right)} {(1+\alpha\xi_n)^2}. \]

From the minimum condition for \(N(\alpha)\) we find the solution of problem IB, i.e. the equation for determining \(\alpha_0\):
\[ \sum_{n=1}^{\infty}\frac{\sigma_n^2\xi_n}{(1+\alpha\xi_n)^3} = \alpha \sum_{n=1}^{\infty} \frac{\xi_n^2\left(\overline{c_n^2}-\sigma_n^2\right)} {(1+\alpha\xi_n)^3}. \]

To solve problem IIB one must find the minimum of the function \(\varphi(m_1,m_2,\ldots,m_n,\ldots)\) of the variables \(m_1,m_2,\ldots,m_n,\ldots\), equal to
\[ \varphi(m_1,m_2,\ldots,m_n,\ldots) = \int_D \rho(x)\overline{(\Delta f)_m^2}\,dx = \sum_{n=1}^{\infty} \frac{\sigma_n^2+\left(\overline{c_n^2}-\sigma_n^2\right)m_n^2} {(1+m_n)^2}. \]
The minimum is attained for \(m_n=m_n'=\sigma_n^2/\left(\overline{c_n^2}-\sigma_n^2\right)\). Consequently,
\(1/(1+m_n)=1-\sigma_n^2/\overline{c_n^2}\). Thus the optimal (in the sense of IIB) summation has the form:
\[ \widetilde f_{\mathrm{op}}(x) = \sum_{n=1}^{\infty} \left(1-\frac{\sigma_n^2}{c_n^2}\right)c_nu_n(x). \]

To solve problems IA and IIA we shall assume that \(\hat f(x)\) is a realization of a random process such that:

3) The sequence \(\{a_n\}\), determining the function \(\hat f(x)\), is a sequence of uncorrelated random numbers.

4) The sequences \(\{\gamma_n\}\) and \(\{a_n\}\) are uncorrelated with each other.

Condition 3) is satisfied, for example, in the case when \(\hat f(x)\) is a realization of a periodic stationary random process.

Under these additional conditions
\[ \left.\overline{(\Delta f_\alpha)^2}\right|_{x=x_0} = \sum_{n=1}^{\infty} \frac{\sigma_n^2+\alpha^2\xi_n^2\left(\overline{c_n^2}-\sigma_n^2\right)} {(1+\alpha\xi_n)^2} u_n^2(x_0) = \varphi(\alpha,x_0). \]

From the condition that \(\varphi(\alpha,x_0)\) be minimal (with respect to \(\alpha\)), we find the equation for determining \(\alpha_0\):
\[ \sum_{n=1}^{\infty} \frac{\sigma_n^2\xi_n}{(1+\alpha\xi_n)^3}u_n^2(x_0) = \alpha \sum_{n=1}^{\infty} \frac{\xi_n^2\left(\overline{c_n^2}-\sigma_n^2\right)} {(1+\alpha\xi_n)^3}u_n^2(x_0). \]

Problem IIA is solved analogously:
\[ m_n'=\sigma_n^2/\left(\overline{c_n^2}-\sigma_n^2\right). \]

Thus, the optimal summation (in the sense of \(\Pi_A\)) has the form

\[ \widetilde f_{\mathrm{op}}(x_0)=\sum_{n=1}^{\infty}\left(1-\frac{\sigma_n^2}{c_n^2}\right)c_nu_n(x_0). \]

In concrete problems one usually knows only a finite number of possible values of each of the random variables \(c_n\). From these samples we find approximate values of the ratios \(\sigma_n^2/\overline{c_n^2}\). Let

\[ \left(\frac{\sigma_n^2}{c_n^2}\right)_{\mathrm{pr}} = \frac{\sigma_n^2}{c_n^2}(1+\beta_n), \quad \text{where } \sum_{n=1}^{\infty}\beta_n^2\leqslant \varepsilon . \]

Since summation with regularizing factors \(1/(1+m_n)\) is stable, small deviations in the determination of the numbers \(\sigma_n^2/c_n^2\) correspond to small deviations of the sum \(\widetilde f(x)_{\mathrm{op}}\). More precisely, if

\[ \widetilde f_{\mathrm{pr}}(x)= \sum_{n=1}^{\infty} \left[ 1-\left(\frac{\sigma_n^2}{c_n^2}\right)_{\mathrm{pr}} \right]c_nu_n(x), \qquad u_n^2(x)\leqslant M \quad (n=1,2,\ldots), \]

then

\[ \left|\widetilde f_{\mathrm{op}}(x)-\widetilde f_{\mathrm{pr}}(x)\right| \leqslant 2M\varepsilon\sum_{n=1}^{\infty}c_n^2 . \]

If, as the system of functions \(\{u_n(x)\}\), one takes the eigenfunctions of the boundary-value problem

\[ \operatorname{div}(k\nabla u)-q^2(x)u+\lambda\rho(x)u=0, \qquad u|_S=0 \quad (\text{or } \partial u/\partial n|_S=0), \]

where \(S\) is the boundary of the domain \(D\) in which the solution is sought, then the functional \(\Omega_1[f]\) may be taken in the form

\[ \Omega_1[f]=\int_D \{k(\nabla f)^2+q^2f^2\}\,dx \]

or in the equivalent form

\[ \Omega_1[f]=\sum_{n=1}^{\infty} f_n^2\lambda_n, \]

where \(\lambda_n\) are the eigenvalues of the indicated boundary-value problem, and \(f_n\) are the Fourier coefficients of the function \(f(x)\) with respect to the system \(\{u_n(x)\}\). Thus, in this case \(m_n=\lambda_n\alpha\). With such a regularizer the problem of summing a Fourier series was considered in the one-dimensional case in (¹).

I express my gratitude to A. N. Tikhonov for a useful discussion.

Received
26 III 1968

REFERENCES

¹ A. N. Tikhonov, DAN, 156, No. 2, 268 (1964).
² A. N. Tikhonov, DAN, 151, No. 3 (1963).
³ N. Wiener, Extrapolation, Interpolation and Smoothing of Stationary Time Series, N. Y., 1949.
⁴ M. M. Lavrent’ev, V. G. Vasil’ev, Siberian Math. Journal, 7, No. 3 (1966).
⁵ V. Ya. Arsenin, V. V. Ivanov, DAN, 182, No. 1 (1968).
⁶ A. B. Bakushinskii, Candidate’s dissertation, Moscow State University, 1967.