Reports of the Academy of Sciences of the USSR
1968. Vol. 179, No. 5

MATHEMATICS

M. Aleksidze

ON THE APPROXIMATE SOLUTION OF SOME INFINITE SYSTEMS OF EQUATIONS

(Presented by Academician I. N. Vekua on 6 VI 1967)

Variational methods for determining the coefficients \(a_j\) of an expansion of the solution of functional equations in some system \(\{\varphi_j\}\) lead \((^1)\) to the following infinite system of equations:

\[ \sum_{j=1}^{\infty} (A_1\varphi_j, A_2\varphi_k)a_j = (A_3\varphi, A_4\varphi_k) \quad (k=1,2,\ldots), \tag{1} \]

where \(A_i\) \((i=1,2,3,4)\) are positive or positive-definite \((^1)\) operators, and \(\varphi\) is a certain known function. If the sequences \(\{A_1\varphi_j\}\) and \(\{A_2\varphi_j\}\) are biorthonormal, or if \(A_1 \equiv A_2\) the sequence \(\{A_1\varphi_j\}\) is orthonormal, then for the coefficients \(a_j\) we obtain

\[ a_j = (A_3\varphi, A_4\varphi_k). \tag{2} \]

In some cases, however, the system \(\{A_1\varphi_j\}\) is not strongly minimal, and its preliminary orthonormalization encounters great difficulties \((^3)\). Therefore the question arises of the approximate solution of system (1). By an approximate solution of system (1) we shall mean the vector \(\bar a^{(N)}(a_1^{(N)}, \ldots, a_N^{(N)})\), which satisfies the system

\[ A^{(N)}\bar a^{(N)} = B^{(N)} + \varepsilon \tag{3} \]

or

\[ \sum_{j=1}^{N} (A_1\varphi_j, A_2\varphi_k)a_j^{(N)} = (A_3\varphi, A_4\varphi_k) + \varepsilon_k \quad (k=1,2,\ldots,N), \]

where \(\varepsilon(\varepsilon_1,\ldots,\varepsilon_N)\) is the residual vector. For the vector \((a^{(N)}-\bar a^{(N)})\) to be small, where \(a^{(N)}\) is the solution of system (3) for \(\varepsilon \equiv 0\) (\(a_i^{(N)}\) are the coefficients of the best, in a certain sense, expansion of the solution of the functional equation in the system \(\{\varphi_i\}\)), it is necessary that the expression \(\|(A^{(N)})^{-1}\varepsilon\|\) be small. However, in the case of unreliable systems (which are not a Bari basis \((^4)\)), the norm of the vector \((a^{(N)}-\bar a^{(N)})\) may be arbitrarily large and, nevertheless, the difference between the function \(\psi\) being expanded and its approximation \(\sum_{i=1}^{N} a_i^{(N)}\varphi_i\) may, in some metric, be less than any sufficiently small number \(\varepsilon > 0\):

\[ \left\|\psi - \sum_{i=1}^{N} a_i^{(N)}\varphi_i\right\| < \varepsilon. \tag{4} \]

In the latter case the corresponding matrix is ill-conditioned, and in its neighborhood there is a singular matrix. As shown in \((^5)\), without regularization we may then obtain strongly differing solutions, distinct from the normal solution.

Therefore, by an \(\varepsilon\)-approximate solution of system (1) we shall call a vector \(\bar a^{(N)}\) satisfying inequality (4).

In the present note we indicate one new method for obtaining approximate solutions of the system (1). The essence of the new method is that, in the case of certain systems \(\{\varphi_i\}\) (the corresponding conditions for the systems will be formulated below), in order to obtain an approximate solution of the system (1) for large \(N\), it is sufficient to carry out a single Seidel iteration, taking the zero vector as the initial approximation.

Let there be a Hilbert space \(H\), in which the scalar product \([u,v]\) is defined as follows:

\[ [u,v]=(A_1u,A_2v)=(A_1v,A_2u), \]

and let the best approximation, in the metric \(H\), to the function \(\psi\) be sought by a generalized polynomial \(\sum_{j=1}^{N} a_j\varphi_j\). From the orthogonality of the difference \(\left(\psi-\sum_{j=1}^{n} a_j\varphi_j\right)\) to an arbitrary function \(\varphi_j\), we obtain, for determining the coefficients \(a_j\), the system

\[ \left[\psi-\sum_{j=1}^{N} a_j\varphi_j,\varphi_k\right]=0,\qquad k=1,2,\ldots,N, \]

or

\[ \sum_{j=1}^{N} a_j(A_1\varphi_j,A_2\varphi_k)=(A_1\psi,A_2\varphi_k). \]

If the operator \(A_1\) is the product \(A_1=A_5A_4A_6\), where \(A_5\) satisfies the condition

\[ (A_5A_4\varphi,A_2\varphi_i)=(A_4\varphi,A_5A_2\varphi_i), \]

and the functional equation

\[ A_6\psi=\varphi \]

is given, then, denoting \(A_5A_2=A_3\), for determining the coefficients \(a_j\) we obtain the system

\[ \sum_{j=1}^{N} a_j(A_1\varphi_j,A_2\varphi_k)=(A_3\varphi_k,A_4\varphi). \]

We shall determine approximate values \(\bar a_j\) of the coefficients \(a_j\) by means of the Seidel iteration process, taking as the initial approximation the vector \(a^0(0,\ldots,0)\),

\[ \bar a_1=\frac{(A_3\varphi_1,A_4\varphi)}{(A_1\varphi_1,A_2\varphi_1)},\quad \bar a_2=\frac{(A_3\varphi_2,A_4\varphi)-\bar a_1(A_1\varphi_1,A_2\varphi_2)}{(A_1\varphi_2,A_2\varphi_2)},\ldots \]

\[ \ldots,\bar a_k= \left[(A_3\varphi_k,A_4\varphi)-\sum_{j=1}^{k-1}\bar a_j(A_1\varphi_j,A_2\varphi_k)\right]\big/(A_1\varphi_k,A_2\varphi_k). \]

Let us note that if the systems \(\{A_1\varphi_j\}\), \(\{A_2\varphi_j\}\) are biorthonormal, then the coefficients \(\bar a_j\) coincide with the coefficients (2) of the function \(\psi\). Since

\[ (A_1\psi,A_2\varphi_k)=(A_3\varphi_k,A_4\varphi). \]

then for \(\bar a_k\) we obtain

\[ \bar a_k=\left(A_1\left(\psi-\sum_{j=1}^{k-1}\bar a_j\varphi_j\right),A_2\varphi_k\right)\big/(A_1\varphi_k,A_2\varphi_k). \]

Thus, \(\bar a_k\) is the coefficient of the best approximation, in the sense of the Hilbert space \(H\) under consideration, of the difference \((\psi-\)

\[ -\sum_{j=1}^{k-1}\bar a_j\varphi_j)=\varphi^{(k-1)} \]
by the function \(c_k\varphi_k\) (6)

\[ \min_{c_k}\left\|\psi-\sum_{j=1}^{k-1}\bar a_j\varphi_j-c_k\psi_k\right\| = \left\|\varphi-\sum_{j=1}^{k}\bar a_j\varphi_j\right\| = \frac{G\bigl(\varphi^{(k-1)},\varphi_k\bigr)}{G(\varphi_k)}, \tag{5} \]

where \(G(u_1,\ldots,u_n)\) is the Gram determinant of the functions \(u_1,\ldots,u_n\). Since

\[ G\bigl(\varphi^{(k-1)},\varphi_k\bigr)/G(\varphi_k) = \bigl(\|\varphi^{(k-1)}\|\bigr)^2 - \bigl(A_1\varphi^{(k-1)},A_2\varphi_k\bigr)^2/\|\varphi_k\|^2, \]

from expression (5) we obtain

\[ \bigl(\|\varphi^{(k)}\|\bigr)^2 = \bigl(\|\varphi^{(k-1)}\|\bigr)^2 - \bigl(A_1\varphi^{(k-1)},A_2\varphi_k\bigr)^2/\|\varphi_k\|^2 . \tag{6} \]

Thus, the sequence of positive numbers \(\{\|\varphi^{(k)}\|\}\) is monotonically decreasing and, consequently, has a limit. Assuming that the norms of the functions \(\varphi_i\) are bounded in the aggregate both above and below, from (6) we obtain that for any \(\varepsilon>0\) there is an \(N_0\) such that

\[ \sum_{k=N_0}^{\infty}\bigl(A_1\varphi^{(k-1)},A_2\varphi_k\bigr)^2<\varepsilon . \tag{7} \]

But then, for \(s>N_0\),

\[ \begin{aligned} \varepsilon &> \left|\bigl(\|\varphi^{(s)}\|\bigr)^2-\bigl(\|\varphi^{(s+1)}\|\bigr)^2\right| \\ &= \left| \left(A_1\left(\psi-\sum_{j=1}^{s}a_j\varphi_j\right), A_2\left(\psi-\sum_{j=1}^{s}a_j\varphi_j\right)\right) \right. \\ &\qquad\left. - \left(A_1\left(\psi-\sum_{j=1}^{s+1}a_j\varphi_j\right), A_2\left(\psi-\sum_{j=1}^{s+1}a_j\varphi_j\right)\right) \right| \\ &= \left| \left(A_1\left(\psi-\sum_{j=1}^{s}a_j\varphi_j\right), A_2\left(\psi-\sum_{j=1}^{s}a_j\varphi_j\right)\right) \right. \\ &\qquad -2a_{s+1} \left(A_1\varphi_{s+1}, A_2\left(\psi-\sum_{j=1}^{s}a_j\varphi_j\right)\right) -a_{s+1}^2\left(A_1\varphi_{s+1},A_2\varphi_{s+1}\right) \\ &\qquad\left. - \left(A_1\left(\psi-\sum_{j=1}^{s}a_j\varphi_j\right), A_2\left(\psi-\sum_{j=1}^{s}a_j\varphi_j\right)\right) \right| \\ &= \left|-3a_{s+1}^2\left(A_1\varphi_{s+1},A_2\varphi_{s+1}\right)\right|, \end{aligned} \]

and, by virtue of the boundedness of the norm of the functions \(\varphi_j\) from below,

\[ |a_{s+1}|<\varepsilon . \]

From the last inequality we obtain that, for any \(\varepsilon>0\) and integer \(N\), there is an \(N_0\) such that

\[ \varphi^{(s)}=\varphi^{(r)}+\gamma_r^{(s)} \quad (N_0\le s,\ r\le N_0+N), \tag{8} \]

where

\[ \|\gamma_r^{(s)}\|<\varepsilon . \tag{9} \]

Substituting (8) into (7), we obtain

\[ \begin{aligned} \varepsilon &> \sum_{k=N_0}^{\infty} \bigl(A_1\varphi^{(k-1)},A_2\varphi_k\bigr)^2 \ge \sum_{k=N_0}^{N_0+N} \bigl(A_1\varphi^{(k)},A_2\varphi_{k-1}\bigr)^2 \\ &= \sum_{k=N_0}^{N_0+N} \bigl(A_1\varphi^{(r)},A_2\varphi_{k-1}\bigr)^2 + \sum_{k=N_0}^{N_0+N} \bigl(A_1\gamma_r^{(k)},A_2\varphi_{k-1}\bigr)^2 \\ &\qquad +2\sum_{k=N_0}^{N_0+N} \bigl(A_1\varphi^{(r)},A_2\varphi_{k-1}\bigr) \bigl(A_1\gamma_r^{(k)},A_2\varphi_{k-1}\bigr). \end{aligned} \]

But, taking (9) into account, we find that for any \(\varepsilon>0\) and integer \(N\) there exists such an \(N_0\) that

\[ \sum_{k=N_0}^{N_0+N} (A_1\varphi^{(r)},\, A_2\varphi_{k-1})^2 < \varepsilon \qquad (N_0 \le r \le N_0+N). \tag{10} \]

We shall assume that the system \(\{\varphi_i\}\) satisfies the following condition: for any \(\varepsilon>0\), integer \(N\), and \(\psi\in H\), there exist coefficients \(b_k\) \((k=N_0,\ldots,N_0+N)\) such that, for at least one value of \(r\),

\[ \left\|\psi^{(r)}-\sum_{k=N_0}^{N_0+N} b_k\varphi_k\right\|<\varepsilon \qquad (N_0 \le r \le N_0+N). \tag{11} \]

\[ \sum_{k=N_0}^{N_0+N} b_k^2 < M, \tag{12} \]

where \(M\) is a constant independent of \(N_0\) and \(N\).

By virtue of (10), (11), (12), and the Schwarz–Bunyakovsky inequality, we obtain

\[ \left| \sum_{k=N_0}^{N_0+N} b_{k-1}(A_1\varphi^{(r)},\, A_2\varphi_{k-1}) \right| \ll \varepsilon_1 . \]

Taking into account the last two inequalities and the fact that \(\|\varphi^{(r)}\|\le \|\psi\|\) \((r=1,2,\ldots)\), we obtain

\[ \begin{aligned} \left\|\psi-\sum_{j=1}^{r}\bar a_j\varphi_j\right\|^2 &= \|\varphi^{(r)}\|^2 = \left(A_1\varphi^{(r)},\, A_2\sum_{k=N_0}^{N_0+N} b_k\varphi_k\right) \\ &\quad - \left(A_1\varphi^{(r)},\, A_2\left(\sum_{k=N_0}^{N_0+N} b_k\varphi_k-\varphi^{(r)}\right)\right) \ll \\ &\ll \varepsilon_1+\|\varphi^{(r)}\|\left\|\varphi^{(r)}-\sum_{k=N_0}^{N_0+N} b_k\varphi_k\right\| \ll \varepsilon_1+\|\psi\|\varepsilon \ll \varepsilon_2 , \end{aligned} \]

where \(\varepsilon_2\) is an arbitrarily small number.

Thus, the following theorem has been proved.

If the norms of the functions \(\varphi_j\) in the Hilbert space \(H\) are bounded both above and below, the system \(\{\varphi_j\}\) and the function \(\psi\) satisfy conditions (11) and (12), then, for an \(\varepsilon\)-approximate solution of system (1), it is sufficient to carry out a single Zeidel iteration for the system

\[ \sum_{j=1}^{N(\varepsilon)} (A_1\varphi_j,\, A_2\varphi_k)a_j = (A_3\varphi_k,\, A_4\varphi) \qquad (k=1,2,\ldots,N(\varepsilon)), \]

taking as the initial approximation the zero vector \(a^0(0,0,\ldots,0)\).

Computing Center
Academy of Sciences of the Georgian SSR

Received
1 VI 1967

REFERENCES

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