MATHEMATICS

Corresponding Member of the Academy of Sciences of the USSR A. N. TIKHONOV

ON STABLE METHODS FOR SUMMING FOURIER SERIES

Let a function (\bar f(x)) ((a \leqslant x \leqslant b)) and an orthonormal system ({u_n(x)}) of eigenfunctions of the boundary-value problem be given:
[
L(u)=\frac{d^2u}{dx^2}-q^2(x)u=-\lambda u\quad (a\leqslant x\leqslant b),\quad
u(a)=u(b)=0\quad (0\leqslant q^2(x)\leqslant M).
]

In the present paper we shall consider a method for the approximate determination of the function (\bar f(x)) at a point (x_0) from (\tilde f={\tilde f_n}), the approximate values of the Fourier coefficients of the function (\bar f(x)) in the metric (l_2): (\tilde f_n=\bar f_n+\Delta_n), where (\bar f={\bar f_n}) are the exact values of the Fourier coefficients of the function (\bar f(x)), and (\Delta={\Delta_n}) are the errors
[
\left(|\Delta|_{l_2}=\left(\sum \Delta_n^2\right)^{1/2}\leqslant \delta\right).
]

We call a method stable if, from the approximate data of the problem and the known order of their accuracy (\delta) (in the corresponding metric), it gives an approximate solution of the problem with any degree of accuracy (\varepsilon), provided (\delta) is sufficiently small.

The proposed method is stable in the sense of this definition*. We note that the problem under consideration, for the prescribed metric for (\tilde f={\tilde f_n}), is an ill-posed problem, and the ordinary summation of the Fourier series
[
\tilde f(x)=\sum_{n=1}^{\infty}\tilde f_n u_n(x)
]
is not a stable method for solving the problem.

Following the regularization method ((^1,\ ^2)), developed for the solution of ill-posed problems, we consider the functional
[
M^\alpha[f(x),\tilde f]=
\sum_{n=1}^{\infty}\left[\int_a^b f(x)u_n(x)\,dx-\tilde f_n\right]^2
+\alpha\int_a^b\left[f'(x)^2+q^2(x)f^2(x)\right]\,dx,
]
where (\alpha=r\delta^2,\ r=\mathrm{const}). The function (\tilde f^{\,\alpha}(x)) that realizes the minimum of this functional in the class of functions vanishing at the points (a) and (b) satisfies the conditions
[
\alpha(f''-q^2 f)-{f-\tilde f(x)}=0
\quad\text{or}\quad
L_0(f)\equiv f''-q_0^2 f=\tilde f(x),
\tag{1}
]
[
f(a)=f(b)=0,
]
where (q_0^2(x)=q^2(x)+\dfrac{1}{\alpha}), and (\tilde f(x)) is the function from (L_2) determined by the Fourier coefficients (\tilde f={\tilde f_n}).

Multiplying this equation by (u_n(x)), integrating and using the boundary conditions, we obtain
[
\alpha\lambda_n\tilde f_n^{\,\alpha}+{\tilde f_n^{\,\alpha}-\tilde f_n}=0
\quad\text{or}\quad
\tilde f_n^{\,\alpha}=\frac{\tilde f_n}{1+\alpha\lambda_n},
]

* By the definition of stability of a method, (\delta) depends not only on (\varepsilon), but also on (\bar f(x)).

so that

[
\widetilde f^{\alpha}(x)=\sum_{n=1}^{\infty}\widetilde f_n\,\frac{1}{1+\alpha\lambda_n}\,u_n(x)\qquad(\alpha=r\delta^2).
\tag{2}
]

Following (1, 2), one can prove that the series (2) represents a stable method for solving the problem posed, under sufficiently restrictive conditions on (\bar f(x)). We shall prove, by means of a direct estimate, that the following holds.

Theorem. For any function (\bar f(x)) with integrable square ((a\le x\le b)), continuous at the point (x_0)* ((a<x_00) there exists such a (\delta_0) that

[
\left|\widetilde f^{\alpha}(x_0)-\bar f(x_0)\right|<\varepsilon,
\quad\text{if}\quad
\left[\sum_{n=1}^{\infty}(\widetilde f_n-\bar f_n)^2\right]^{1/2}<\delta\le\delta_0.
]

This theorem holds not only for (\alpha=r\delta^2), but also for any dependence (\alpha(\delta)) satisfying the conditions
[
\frac{1}{p(\delta)}\delta^4<\alpha(\delta)<p(\delta),
]
where (p(\delta)\to0) as (\delta\to0).

The study of the function (\widetilde f^{\alpha}(x)) is based on the fact that this function is the solution of the boundary-value problem

[
L_0(\widetilde f^{\alpha})=(\widetilde f^{\alpha})''-q_0^2\widetilde f^{\alpha}
=-\frac{1}{\alpha}\widetilde f(x),\qquad
q_0^2=q^2+\frac{1}{\alpha},\qquad
\widetilde f^{\alpha}(a)=\widetilde f^{\alpha}(b)=0.
]

We represent this solution in the form

[
\widetilde f^{\alpha}(x)=\frac{1}{\alpha}\int_a^b G(x,s)\,\widetilde f(s)\,ds,
]

where (G(x,s)) is the Green’s function of the boundary-value problem

[
L_0(f)=f''-q_0^2 f=0,\qquad
q_0^2=q^2+\frac{1}{\alpha},\qquad
f(a)=f(b)=0.
\tag{3}
]

Lemma. For the Green’s function of the boundary-value problem (3), the following estimates hold:

[
1^\circ.\quad
\int_{x-\eta}^{x+\eta}G(x,s)\,ds
=\alpha\left(1+O\left(e^{-\eta/\sqrt{\alpha}}\right)\right).
]

[
2^\circ.\quad
\int_a^b G^2(x,s)\,ds
=\alpha^{3/2}\left(1+O\left(e^{-\eta/\sqrt{\alpha}}\right)\right).
]

Here (a<x<b), (\eta<\eta_0=\min(x-a,b-x)).

Consider, along with the boundary-value problem (3), the other boundary-value problems ((3_i)), (i=0,1,2), on the interval (a\le x\le b):

[
L_i(f_i)=f_i''-q_i^2 f_i=0,\qquad
f_i(a)=f_i(b)=0\quad(i=0,1,2),
\tag{(3_i)}
]

where

[
q_1^2=\frac{1}{\alpha}\le q_0^2=q^2(x)+\frac{1}{\alpha}\le q_2^2=M+\frac{1}{\alpha},
]

so that the boundary-value problem (3) coincides with ((3_0)). The Green’s functions of the boundary-value problems ((3_i)) satisfy the inequalities

[
G_2(x,s)\le G_0(x,s)\le G_1(x,s).
]

* The theorem also holds for any point (x_0) where (f(x_0)) is equal to the derivative of its antiderivative.

These inequalities follow from the following observation. The solutions of the boundary-value problems

[
\overline{L}(\overline{u})=\overline{u}''-\overline{q}^{\,2}\overline{u}=-f,\qquad
\overline{\overline{L}}(\overline{\overline{u}})=\overline{\overline{u}}''-\overline{\overline{q}}^{\,2}\overline{\overline{u}}=-f
\qquad(\overline{q}\geqslant \overline{\overline{q}})
]

with boundary conditions of the first kind are such that, for any function (f(x)\geqslant 0),

[
v(x)=\overline{\overline{u}}(x)-\overline{u}(x)
=\int_a^b[\overline{\overline{G}}(x,s)-\overline{G}(x,s)]f(s)\,ds\geqslant 0,
]

whence (\overline{\overline{G}}(x,s)\geqslant \overline{G}(x,s)).

For the functions (G_1(x,s)) and (G_2(x,s)) ((q_1=\mathrm{const}) and (q_2=\mathrm{const})) we have the explicit expressions

[
G_i(x,s)=
\begin{cases}
\dfrac{1}{\sigma_i}\operatorname{sh} q_i(s-a)\operatorname{sh} q_i(b-x), & s\leqslant x,\[6pt]
\dfrac{1}{\sigma_i}\operatorname{sh} q_i(x-a)\operatorname{sh} q_i(b-s), & s\geqslant x,
\end{cases}
\qquad
\sigma_i=q_i\operatorname{sh} q_i l\quad (i=1,2),
]

whence the lemma follows immediately.

Let us turn to the proof of the theorem. Represent the equation for (\widetilde{f}^{\alpha}(x)) in the form of the sum (\overline{f}^{\alpha}(x)+g^{\alpha}(x)), where

[
L_0(\widetilde{f}^{\alpha}(x))=-\frac{1}{\alpha}{\overline{f}(x)+g(x)}
=L_0(\overline{f}^{\alpha}(x))+L_0(g^{\alpha}(x)).
]

Estimate the function (g^{\alpha}(x)):

[
|g^{\alpha}(x)|
=\frac{1}{\alpha}\left|\int_a^b G_0(x,s)g(s)\,ds\right|
\leqslant
\frac{1}{\alpha}\left[\delta^2\alpha^{3/2}\left(1+O(e^{-\eta/\sqrt{\alpha}})\right)\right]^{1/2}
=
\frac{\delta}{\alpha^{1/4}}\left[1+O(e^{-\eta_0/\sqrt{\alpha}})\right].
]

It is not difficult to construct examples of perturbations (g(x)) showing that the order of this estimate with respect to (\alpha) cannot be improved (for the chosen regularizer).

Let us turn to the estimate of the function (\overline{f}^{\alpha}(x)). Denote by (\omega(\eta)) the oscillation of the function (\overline{f}(x)) on the interval ((x_0-\eta,\ x_0+\eta)). Using the nonnegativity of the Green’s function (G(x,s)), we have

[
\overline{f}^{\alpha}(x_0)
=
\frac{1}{\alpha}\int_a^b G_0(x,s)\overline{f}(s)\,ds
\leqslant
]

[
\leqslant
[\overline{f}(x_0)+\omega(\eta)]
[1+O(e^{-\eta/\sqrt{\alpha}})]
+
|\overline{f}|_{L_2}
[\alpha^{3/2}(1+O(e^{-\eta_0/\sqrt{\alpha}}))]^{1/2}.
]

and analogously

[
\overline{f}^{\alpha}(x_0)
=
\frac{1}{\alpha}\int_a^b G(x,s)\overline{f}(s)\,ds
\geqslant
[\overline{f}(x_0)-\omega(\eta)]
[1+O(e^{-\eta/\sqrt{\alpha}})]
-
]

[

|\overline{f}|_{L_2}
[\alpha^{3/2}(1+O(e^{-\eta_0/\sqrt{\alpha}}))]^{1/2},
]

i.e.,

[
|\overline{f}^{\alpha}(x_0)-f(x_0)|
\leqslant
\omega(\eta)+O(e^{-\eta/\sqrt{\alpha}})+\frac{\delta}{\alpha^{1/4}}.
]

Comparing the results obtained, we have

[
|\widetilde{f}^{\alpha}(x_0)-\overline{f}(x_0)|
\leqslant
\omega(\eta)+\frac{\delta}{\alpha^{1/4}}+O(e^{-\eta/\sqrt{\alpha}}).
]

Choosing (\eta=\eta(\delta)) so that (\eta(\delta)\to 0) and (\alpha^{-1/2}\eta(\delta)\to\infty) as (\delta\to 0) (for example, (\eta=\alpha^{1/4})), we are convinced of the validity of the theorem. From the estimate given it also follows that if (\overline{f}(x)) is uniformly continuous on the interval ([c,d]) ((a<c<d<b)), then (\widetilde{f}^{\alpha}(x)) on this interval converges to (\overline{f}(x)) uniformly.

and moderate. Series (2) can be represented in the form

[
\widetilde f^{\alpha}(x)=\sum_{n=1}^{\infty}\widetilde f_n r(n,\alpha)u_n(x),\qquad
r(n,\alpha)=\frac{1}{1+\alpha\lambda_n}.
]

One can give many examples of regularizing factors (r(n,\alpha)). The simplest of them has the form

[
r(n,\delta)=
\begin{cases}
1 & \text{for } n\leq n(\delta)=\dfrac{1}{\delta^2}\rho(\delta)\quad (\rho(\delta)\to 0 \text{ as } \delta\to 0),\
0 & \text{for } n>n(\delta).
\end{cases}
]

The corresponding function

[
\widetilde f^{\delta}(x)=\sum_{n=1}^{n(\delta)}\widetilde f_n u_n(x)
]

is a segment of the ordinary Fourier series, so that

[
\bar f^{\delta}(x)=\sum_{n=1}^{n(\delta)}\bar f_n u_n(x),\qquad
|g^{\alpha}(x)|^2\leq \delta^2 n(\delta)c,\qquad
|u_n(x)|\leq c.
]

If (\bar f(x)) is such that its Fourier series converges to it at the point (x_0), then (\widetilde f^{\alpha}(x_0)\to \bar f(x_0)) (as (\delta\to 0)). Thus, for approximate values of the Fourier coefficients one may use the ordinary Fourier series; only the number of terms must depend on the accuracy with which the initial data are specified.

For the regularizing factors (r(n,\alpha)) to yield a solution of the problem, it is sufficient that, as (\delta\to 0), they define a generalized summation method and that

[
\delta^2\sum_{n=1}^{\infty} r^2(n,\alpha)=\rho(\delta)\qquad (\rho(\delta)\to 0 \text{ as } \delta\to 0).
]

The problem of determining a function (f(x)) ((x={x_1,\ldots,x_n})), given in a domain (G) of (n)-dimensional space, from its Fourier coefficients with respect to the eigenfunctions of the equation

[
\Delta u-q^2(x)u=-\lambda u,\qquad u|_S=0\quad (S\text{ is the boundary of }G)
]

is solved analogously (see the remark on regularization for the case of (n)-dimensional space at the end of (1)).

Using regularization of the (n)-th order of smoothness, we obtain methods of uniform approximation of the derivatives (\bar f(x)) from given functions (\widetilde f(x)), approximating (\bar f(x)) in the (L_2) norm.

Received
8 II 1964

REFERENCES

({}^{1}) A. N. Tikhonov, DAN, 151, No. 3, 501 (1963). ({}^{2}) A. N. Tikhonov, DAN, 153, No. 1, 49 (1963).