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UDC 517.5
MATHEMATICS
S. V. SMIRNOV, M. K. POTAPOV
SADDLE POINTS OF FUNCTIONALS OF UNIFORM APPROXIMATION
(Presented by Academician A. N. Kolmogorov on 19 II 1966)
§ 1. Interchanging the operations of taking the upper and lower bounds. In approximation theory one can establish several simple, but sufficiently general, theorems analogous to the well-known von Neumann theorem \((^{1,2})\) on a saddle point.
Let \(F(x,y;g)\), \(x \in X\), \(y \in Y\), be a real-valued single-valued function defined on the Cartesian product \(R=X\times Y\) and for every \(g\), an element of some set \(G\). Consider \(\varphi(x,y)\), a given real-valued single-valued function of two variables, defined on \(R=X\times Y\).
Denote
\[
\mathcal E(y)=\inf_{g\in G}\sup_{x\in X}|\varphi(x,y)-F(x,y;g)|,
\tag{1,1}
\]
where \(\mathcal E(y)\) may also be infinite.
Now suppose that, instead of \(g\), in \(F(x,y,g)\) an arbitrary single-valued function \(g(y)\) is substituted, defined on the set \(Y\) and taking values \(g(y)\in G\). As a result one obtains the function \(F(x,y;g(y))\), depending on the choice of a mapping of the set \(Y\) into the set \(G\).
Let \(\{g(y)\}\) be the set of all mappings mentioned above. Denote
\[
\mathcal E_1=\inf_{\{g(y)\}}\sup_{(x,y)\in R}|\varphi(x,y)-F(x,y;g(y))|.
\tag{1,2}
\]
Theorem 1. If \(\mathcal E(y)\), as a function of \(y\), is bounded on \(Y\), then \(\mathcal E_1\) has a finite value and
\[
\mathcal E_1=\mathcal E=\sup_{y\in Y}E(y).
\tag{1,3}
\]
Proof. \(\mathcal E_1\) is finite. Indeed, for any finite \(\mathcal E(y)\) and arbitrary \(\varepsilon>0\) there exists a \(\bar g\in G\), depending on the choice of \(\varepsilon\) and, generally speaking, on \(y\), such that
\[
\sup_{x\in X}|\varphi(x,y)-F(x,y;\bar g)|<\mathcal E(y)+\varepsilon .
\tag{1,4}
\]
From the last inequality it follows that
\[
\sup_{y\in Y}\left\{\sup_{x\in X}|\varphi(x,y)-F(x,y;\bar g)|\right\}
=
\sup_{(x,y)\in R}|\varphi(x,y)-F(x,y;\bar g(y))|
\le
\le \sup_{y\in Y}\mathcal E(y)+\varepsilon=\mathcal E+\varepsilon .
\tag{1,5}
\]
Since \(\bar g=\bar g(y)\) is a mapping of the set \(Y\) into \(G\), the inequality is valid:
\[
\sup_{(x,y)\in R}|\varphi(x,y)-F(x,y;\bar g(y))|
\ge
\inf_{\{g(y)\}}\sup_{(x,y)\in R}|\varphi(x,y)-F(x,y;g(y))|=\mathcal E_1 .
\tag{1,6}
\]
From (1,5) and (1,6) it follows that
\[
\mathcal E+\varepsilon\ge \mathcal E_1 .
\tag{1,7}
\]
For the same \(\varepsilon>0\) as above, the function \(g_\varepsilon^*(y)\) can be chosen so that
\[ \sup_{(x,y)\in R}\left|\varphi(x,y)-F\left(x,y;g_\varepsilon^*(y)\right)\right|<\mathscr E_1+\varepsilon. \tag{1.8} \]
From the definition of the least upper bound it follows that, for any fixed \(y_0\in Y\),
\[ \sup_{(x,y)\in R}\left|\varphi(x,y)-F\left(x,y;g_\varepsilon^*(y)\right)\right| \ge \sup_{x\in X}\left|\varphi(x,y_0)-F\left(x,y_0;g_\varepsilon^*(y_0)\right)\right| \ge \mathscr E(y_0). \tag{1.9} \]
Inequalities (1.8) and (1.9) give \(\mathscr E_1+\varepsilon>\mathscr E(y_0)\), which implies
\[ \mathscr E_1+\varepsilon\ge \mathscr E. \tag{1.10} \]
In view of the arbitrariness of \(\varepsilon>0\), inequalities (1.7) and (1.10) mean that the numbers \(\mathscr E\) and \(\mathscr E_1\) coincide, which proves the theorem.
§ 2. Interchange of the operations of taking the maximum and the minimum. By the functional of uniform approximation of a function of two variables \(\varphi(x,y)\) we shall mean the functional
\[ A[g(y)]=\max_{(x,y)\in R}\left|\varphi(x,y)-F(x,y;g(y))\right|, \tag{2.1} \]
depending on the choice of a mapping of the set \(Y\) into \(G\). This functional, for given \(\varphi(x,y)\) and \(F(x,y;g)\), may fail to be defined on the whole set \(\{g(y)\}\). Let \(\Gamma\) be the maximal subset of \(\{g(y)\}\) on which the functional (2.1) is defined.
Put, by definition,
\[ \inf A=\inf_{\{g(y)\}} A[g(y)]=\inf_{g(y)\in\Gamma} A[g(y)], \tag{2.2} \]
if \(\Gamma\) is nonempty; in the opposite case let \(\inf A=\infty\).
If there exists \(\bar g(y)\in\Gamma\), i.e., if there exists a mapping \(g=\bar g(y)\) such that \(\inf A=A[\bar g(y)]\), then we denote by
\[ \overline{\mathscr E}_1=\min A[g(y)] \]
the minimum of the functional \(A[g(y)]\).
Theorem 2. If, on the set \(R=X\times Y\), the function \(\varphi(x,y)\), for every fixed \(y\in Y\) as a function of the single variable \(x\), can be best uniformly approximated by means of the given function \(F(x,y;g)\), \(g\in G\), and
\[ \overline{\mathscr E}(y)=\min_{g\in G}\max_{x\in X}\left|\varphi(x,y)-F(x,y;g)\right| \tag{2.3} \]
attains its greatest value \(\overline{\mathscr E}\) on the set \(Y\), then, considering the value \(\bar g(y)\) realizing \(\overline{\mathscr E}(y)\) as a function of \(y\), one may assert that this function \(\bar g(y)\) realizes the minimum \(\overline{\mathscr E}_1\) of the functional \(A[g(y)]\). Moreover, the equality
\[ \overline{\mathscr E}_1=\overline{\mathscr E} \tag{2.4} \]
holds.
Proof. Directly from the definition of the lower bound of the functional \(A[g(y)]\) it follows that
\[ \overline{\mathscr E} = \max_{y\in Y}\overline{\mathscr E}(y) = \max_{y\in Y}\left\{\max_{x\in X}\left|\varphi(x,y)-F\left(x,y;\bar g(y)\right)\right|\right\} = A[\bar g(y)]\ge \inf A. \tag{2.5} \]
From Theorem 1 it follows that
\[ \overline{\mathscr E} = \inf_{\{g(y)\}}\sup_{(x,y)\in R}\left|\varphi(x,y)-F(x,y;g(y))\right|. \tag{2.6} \]
If in the right-hand side of equality (2.6) one replaces \(\{g(y)\}\) by its subset \(\Gamma\), then it becomes the inequality
\[ \overline{\mathscr E}\leq \inf_{g(y)\in\Gamma} A = \inf_{g(y)\in\Gamma}\max_{(x,y)\in R} \left|\varphi(x,y)-F(x,y;g(y))\right|. \tag{2.7} \]
From (2.5) and (2.7) it follows that the minimum of the functional \(A[g(y)]\), equal to \(A[\overline{g}(y)]\), is indeed attained; moreover, equality (2.4) is in fact valid, which proves the theorem.
The essence of the theorem just proved lies in the possibility, under the conditions of the theorem, of interchanging the operations of taking a maximum and a minimum when computing the minimax of a certain functional.
Writing this functional explicitly, the result obtained can be interpreted as a saddle-point theorem.
§ 3. Saddle-point theorem. Let the conditions of Theorem 2 be satisfied. Denote
\[ B[g(y);y]=\max_{x\in X}\left|\varphi(x,y)-F(x,y;g(y))\right|; \tag{3.1} \]
for a fixed arbitrary \(y_0\in Y\), this is the functional of uniform approximation of the function \(\varphi(x,y_0)\) by means of \(F(x,y_0;g(y_0))\), \(x\in X\), where \(g(y_0)\) ranges over the set \(G\). As is known, finding the minimum of \(B[g(y_0);y_0]\) in the set \(G\) is equivalent to finding the best approximation of \(\varphi(x,y_0)\) by means of \(F(x,y_0;g(y_0))\) on the set \(X\).
By virtue of the conditions of Theorem 2, the functional (3.1) is defined however the function \(g(y)\in\{g(y)\}\) is chosen and for any \(y_0\in Y\). Consequently, it is defined on the Cartesian product \(\{g(y)\}\times Y\), and hence, a fortiori, on the set \(D=\Gamma\times Y\subseteq\{g(y)\}\times Y\).
As has already been proved, there exists a minimum \(\mathscr E_1\) of the functional \(A[g(y)]\), realized at \(g=\overline{g}(y)\); moreover, as follows from (3.1) and (2.4),
\[ \mathscr E_1 = \min_{g(y)\in\Gamma} A[g(y)] = A[\overline{g}(y)] = \]
\[ = \max_{y\in Y}\max_{x\in X} \left|\varphi(x,y)-F(x,y;\overline{g}(y))\right| = B[\overline{g}(\widetilde y);\widetilde y], \tag{3.2} \]
where \(\widetilde y\in Y\) is the element of the set \(Y\) at which the maximum
\[ \overline{\mathscr E} = \max_{y\in Y}\min_{g\in G}\max_{x\in X} \left|\varphi(x,y)-F(x,y;g)\right| \tag{3.3} \]
of \(\overline{\mathscr E}(y)\), coinciding with \(\mathscr E_1\), is attained.
Theorem 3. Under the conditions of Theorem 2, the pair \((\overline{g}(y),\widetilde y)\) is a saddle point of the functional \(B[g(y);y]\) in the set \(D=\Gamma\times Y\).
Proof. By definition of \(\overline{\mathscr E}\) one may write
\[ \overline{\mathscr E} = \max_{y\in Y}\min_{\{g(y)\}} B[g(y);y] = \min_{\{g(\widetilde y)\}} B[g(\widetilde y);\widetilde y]. \tag{3.4} \]
Indeed, by virtue of the construction of the set \(\{g(y)\}\) (see § 1), the set \(\{g(y_0)\}\), for any fixed \(y_0\in Y\), is mapped one-to-one onto \(G\).
From (2.5) and (2.7) it follows that equality (3.4) may be written, replacing the set \(\{g(y)\}\) by the set \(\Gamma\):
\[ \overline{\mathscr E} = \max_{y\in Y}\min_{g(y)\in\Gamma} B[g(y);y] = \min B[g(\widetilde y);\widetilde y], \tag{3.5} \]
in the latter case, when computing \(g(\widetilde y)\), only \(g(y)\in\Gamma\) are taken.
The equalities (3.2) and (3.5) imply
\[ B[\underline{g}(\tilde y);\tilde y]\geq \overline{\mathscr E}, \tag{3.6} \]
\[ B[\overline{g}(y);y]\leq \overline{\mathscr E}_1, \tag{3.7} \]
where \(g(y)\in\Gamma,\ y\in Y\).
In view of the coincidence of \(\overline{\mathscr E}\) and \(\overline{\mathscr E}_1\), the last inequalities mean (3) that the pair \((\overline{g}(y);\tilde y)\) is indeed a saddle point.
Results analogous to Theorems 1–3 are also valid in other metrics, for example, in the space \(L_p\).
Theorems on the interchange of the operations of taking the maximum and minimum, or the upper and lower bounds, simplify the search for the best approximation. In particular, they were applied by the authors (4, 5) to the best uniform approximation of a function of two variables by means of nomograms.
Received
19 II 1966
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