Reports of the Academy of Sciences of the USSR
1970. Volume 195, No. 3

UDC 513.831

MATHEMATICS

V. A. VINOKUROV

ON ONE NECESSARY CONDITION FOR TIKHONOV REGULARIZABILITY

(Presented by Academician A. N. Tikhonov on 17 IV 1970)

Suppose we have a mapping (A) of a metric space into a metric space (U), with (Az_1 \ne Az_2) if (z_1 \ne z_2). The problem of approximate solution of the equation

[
Az = u
\tag{1}
]

is called Tikhonov-regularizable ((^1)) if there exists a one-parameter family of mappings (R_\delta: U \to Z,\ 0 < \delta \le 1), for which

[
\lim_{\delta \to 0}\ \sup_{{u \in U:\ \rho(u,Az)\le \delta}} \rho(z,R_\delta u)=0,\quad \forall z \in Z.
\tag{2}
]

Theorem. If problem (1) is Tikhonov-regularizable and (N) is an everywhere dense set in (U), then
[
\bigcup_{n=1}^{\infty} {R_{1/n}N}
]
is an everywhere dense set in (Z).

Proof. For an arbitrary point (z \in Z) and any natural (n), take (u_n \in N) such that (\rho(Az,u_n)\le 1/n). By the regularizability condition (2),
[
\lim_{n\to\infty} \rho(z,R_{1/n}u_n)=0,
]
i.e. the sequence ({R_{1/n}u_n}) converges to the element (z) of the space (Z). Since (u_n \in N), the theorem is proved.

Corollary. If problem (1) is Tikhonov-regularizable and in the metric space (U) there exists an infinite everywhere dense set of cardinality (\tau), then in the metric space (Z) there also exists an everywhere dense set of cardinality (\tau). In particular, separability of (Z) follows from separability of (U).

From this, examples of nonregularizable problems easily follow.

1. For the embedding operator of the space of measurable bounded functions on the segment (M_{[0,1]}) into (L_{2[0,1]}), problem (1) is not Tikhonov-regularizable.

2. Let (Z) be a nonseparable metric space of monotonically increasing functions on ([0,1]), the distance between which is measured in the metric of the space (M_{[0,1]}); let (U) be an arbitrary separable metric space (for example (L_2)). Then for any mapping (A: Z \to U), regularization is impossible, i.e. it is impossible to construct approximate solutions (z_\delta \in Z) whose deviation from the exact solution (z \in Z) would tend to zero in the metric of the space (M_{[0,1]}). Note that (A) may be a linear continuous or a completely discontinuous operator.

3. The integral equation

[
\int_0^1 k(x,t)\,dg(t)=u(x),
]

considered from (M_{[0,2\pi]}) into (L_{2[0,2\pi]}), is nonregularizable for any kernel (k(x,t)).

1. The integral equation

[
Az=\int_0^{2\pi} k(x-t)z(t)\,dt=u(x),
]

considered from (M_{[0,\,2\pi]}) into (L_{2[0,\,2\pi]}), where (k(t)) is a (2\pi)-periodic even function satisfying the condition

[
\int_{0}^{2\pi} k(t)\cos nt\,dt \ne 0
\qquad (n=0,1,2,\ldots),
]

is not regularizable. With a suitable choice of (k(t)) (for example, (k(t)\in L_{2[0,\,2\pi]})) the linear operator (A) will be completely continuous.

The last result shows that A. B. Bakushinskii’s question (see ({}^{2}), p. 71) on the possibility of Tikhonov regularization of a linear bounded or completely continuous operator from a Banach space into a Banach space, under the condition of uniqueness of the solution of equation (1), must be answered in the negative. However, the question of regularizability remains open for a separable Banach space (Z).

From the corollary to the theorem, together with A. B. Bakushinskii’s result on the regularizability of problem (1), if (Z) is a uniformly convex Banach space, (A) is a linear bounded operator, (A^{-1}0={0}), and (U) is a Banach space, we obtain an interesting corollary for linear operators in a Banach space: if there exists a linear continuous mapping (A) of a uniformly convex Banach space (Z) into a separable Banach space (U), (A^{-1}0={0}), then (Z) is separable.

If there exists a completely continuous linear mapping (A) of a uniformly convex Banach space (Z) into a normed space, such that (A^{-1}0={0}), then (Z) is separable. This follows from the separability of the range of a completely continuous operator.

Moscow State University
named after M. V. Lomonosov

Received
8 IV 1970

CITED LITERATURE

({}^{1}) A. N. Tikhonov, DAN, 153, No. 1, 49 (1963). ({}^{2}) A. B. Bakushinskii, in the collection Computational Methods and Programming, issue 12, Moscow, 1969, p. 56.