MATHEMATICS

A. F. TIMAN

ON A CONSTRUCTIVE PRINCIPLE OF DUALITY IN THE CLASS OF CONTINUOUS FUNCTIONS DECREASING MONOTONICALLY TO ZERO AND CONVEX ON THE SEMI-AXIS

(Presented by Academician S. N. Bernstein on 31 X 1962)

Let \(\omega(t)\) be a modulus of continuity defined on the semi-axis \(0 \le t < \infty\), i.e., a continuous, nondecreasing, semiadditive function for which \(\omega(0)=0\) (see \((^1)\), Ch. III). For any nonnegative value of the constant \(M\) consider the class \(MH_\omega\) of all functions \(g(x)\) defined on the semi-axis \(0 \le x < \infty\) and satisfying the condition

\[ |g(x_1)-g(x_2)| \le M\omega(|x_1-x_2|). \tag{1} \]

Denote by \(E_M^\omega(f)\) the best uniform approximation on \([0,\infty)\) of a bounded function \(f(x)\) by functions \(g(x)\) satisfying condition (1), i.e., the quantity

\[ E_M^\omega(f)=\inf_{g\in MH_\omega}\ \sup_{0\le x<\infty}|f(x)-g(x)|. \tag{2} \]

It is obvious that \(E_M^\omega(f)\), as a function of \(M\), defined on the positive semi-axis \(0 \le M < \infty\), is always nonincreasing.

Assuming the modulus of continuity \(\omega(t)\) to be convex and unbounded as \(t\to\infty\), in this note we shall give an exhaustive characterization of the best approximation \(E_M^\omega(f)\) in the class of bounded monotone functions \(f(x)\), convex (or concave) on the semi-axis.

The following theorem of the type of the well-known duality theorems holds.

Theorem. For any convex (upward) and unbounded on \([0,\infty)\) modulus of continuity \(\omega(t)\), the class of all continuous functions \(f(x)\), monotonically decreasing to zero and convex (downward) on the semi-axis \(0 \le x < \infty\), coincides with the class of their best approximations \(E(M)=E_M^\omega(f)\).

This theorem, showing that for any convex and unbounded modulus of continuity \(\omega(t)\) a nonnegative and nonincreasing function \(E(M)\) on the semi-axis \(0 \le M < \infty\) is the best uniform approximation \(E_M^\omega(f)\) of some bounded monotone and convex on \([0,\infty)\) function \(f(x)\) if and only if it is continuous and convex, in particular, holds for \(\omega(t)=t\), i.e., when considering the best uniform approximation \(E'_M(f)\) by absolutely continuous functions whose derivative is almost everywhere bounded by one or another number \(M\ge 0\).

It should be noted that the requirement of unboundedness of the modulus of continuity \(\omega(t)\), which plays an essential role in the proof, is necessary in order that the duality principle formulated in the theorem hold.

One can indicate a continuous function \(E(M)\), monotonically decreasing to zero and convex on the semi-axis \(0 \le M < \infty\), which for no bounded function \(f(x)\), and for no modulus of continuity \(\omega(t)\) bounded on \([0,\infty)\), will be the best approximation \(E_M^\omega(f)\). It suffices

take, for example, the function

\[ E(M)= \begin{cases} 1-\sqrt{M}, & 0\leq M\leq 1,\\ 0, & M\geq 1. \end{cases} \]

It is easy to see that if \(\omega(t)\) is bounded as \(t\to\infty\), then, whatever bounded function \(f(x)\) we take, for sufficiently small values of \(M\) the inequality

\[ E_M^\omega(f)\geq E_0^\omega(f)-\frac12 M\sup_{t>0}\omega(t) \]

will hold.

Hence it is clear that either \(E_0^\omega(f)\neq 1\), or, for all sufficiently small positive \(M\), one will have \(E_M^\omega(f)>E(M)\).

The proof of the fact that, if a nonnegative continuous and nonincreasing on \([0,\infty)\) function \(f(x)\) is convex, then its best approximation \(E_M^\omega(f)\) is a continuous convex function of \(M\), tending to zero as \(M\to\infty\), under the condition \(f(0)=0\), is connected with consideration of the function

\[ \Omega(x)=\min\{f(x),\,M\omega(x)\} \tag{3} \]

and of the properties of the difference \(f(x)-\Omega(x)\)*.

To prove the converse assertion, namely that every continuous function \(f(M)\) monotonically decreasing to zero is the best uniform approximation \(E_M^\omega(\varphi)\) of some other function \(\varphi(x)\) of the same kind, one establishes the existence of a sequence of monotone and convex functions \(\varphi_n(x)\) \((n=0,1,2,\ldots)\) possessing the property that, for any \(q=0,1,\ldots,2^n\cdot n\),

\[ E_{q/2^n}^{\omega}(\varphi_n)=f\!\left(\frac{q}{2^n}\right). \tag{4} \]

A subsequent application of Helly’s selection principle, taking into account the special properties of the functions \(\varphi_n(x)\), leads to a function \(\varphi(x)\) for which \(E_M^\omega(\varphi)=f(M)\) for every \(M\geq 0\).

The theorem stated remains valid if, instead of the class of all functions \(g(x)\) satisfying condition (1), one considers that part of it which contains only functions convex on \([0,\infty)\).

Received
29 X 1962

REFERENCES

1. A. F. Timan, Theory of approximation of functions of a real variable, Moscow, 1960.
2. A. F. Timan, DAN, 140, No. 2, 307 (1961).

* In this connection see (1).