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MATHEMATICS
N. P. KORNEICHUK
ON THE BEST UNIFORM APPROXIMATION OF DIFFERENTIABLE FUNCTIONS
(Presented by Academician S. L. Sobolev on 22 VI 1961)
We introduce the following notation. \(H_\omega\) is the class of continuous periodic functions \(f(x)\), with period \(2\pi\), whose modulus of continuity
\[ \omega(f;t)=\sup_{|x'-x''|\leq t}|f(x')-f(x'')| \]
does not exceed the prescribed modulus of continuity \(\omega(t)\). In particular, for \(\omega(t)=Kt^\alpha\) \((0<\alpha\leq 1;\ 0\leq t\leq \pi)\) the class \(H_\omega\) is the class \(KH^{(\alpha)}\) of periodic functions satisfying on the whole axis the Lipschitz condition of order \(\alpha\) with constant \(K\). \(W^{(r)}H_\omega\) is the class of periodic functions \(f(x)\), with period \(2\pi\), for which the derivative of order \(r\), \(f^{(r)}(x)\), belongs to the class \(H_\omega\). \(W^{(r)}KH^{(\alpha)}\) is the class of functions \(f(x)\) for which \(f^{(r)}(x)\in KH^{(\alpha)}\). \(E_n(f)\) is the best uniform approximation of the periodic function \(f\) by trigonometric polynomials of degree not exceeding \(n\).
In my note \((^4)\) it was shown that if the function \(\omega(t)\) is convex upward, then
\[ \sup_{f\in H_\omega} E_n(f)=\frac{1}{2}\omega\left(\frac{\pi}{n+1}\right)\quad (n=0,1,2,\ldots). \tag{1} \]
Here, under the same assumption concerning \(\omega(t)\), the exact value is given for the upper bound of the best approximations on the class \(W^{(1)}H_\omega\). Namely, the following holds:
Theorem. If \(\omega(t)\) is a convex upward modulus of continuity, then
\[ \sup_{f\in W^{(1)}H_\omega} E_n(f)=\frac{1}{4}\int_{0}^{\frac{\pi}{n+1}}\omega(t)\,dt \quad (n=0,1,\ldots). \tag{2} \]
The proof of relation (2) is based on the following lemma:
Lemma. Let \(f\in H_\omega\), where \(\omega(t)\) is a convex upward modulus of continuity. Whatever the number \(K>0\), in the class \(KH^{(1)}\) there is a function \(\varphi_0\) such that, for any \(a\) and \(b\),
\[ \left|\int_a^b [f(x)-\varphi_0(x)]\,dx\right| \leq \frac{1}{2}\max_{x\geq 0}\int_0^x |\omega(t)-Kt|\,dt. \tag{3} \]
The scheme of the proof of the lemma is as follows. Let \(g(x)\) be a polygonal periodic function inscribed in \(f\). The graph of \(g(x)\) on the period \([0,2\pi]\) consists of a finite number of rectilinear segments (links) \(l_1,l_2,\ldots,l_p\), whose slopes we shall denote respectively by \(\gamma_1,\gamma_2,\ldots,\gamma_p\).
Suppose that \(\gamma_k>K\). Then there exist points \(t'_k<x_k<t''_k\) such that the segment \([t'_k,t''_k]\) contains within itself the abscissas of the endpoints of the link \(l_k\), and,
if we set
\[
\tau_k(x)=g(x_k)+K(x-x_k),\qquad \Delta_k(x)=g(x)-\tau_k(x),
\]
\[
\int_a^b \Delta_k(x)\,dx=I_k(a,b),
\tag{4}
\]
then the following relations will be satisfied:
\[ I_k(t'_k,x_k)=\min_{x\leq x_k} I_k(x,x_k)<0,\qquad I_k(x_k,t''_k)=\max_{x\geq x_k} I_k(x_k,x)>0, \]
\[ I_k(t'_k,t''_k)=0. \]
Similarly, if \(\gamma_k<-K\), then we find an interval \([t'_k,t''_k]\) containing the abscissas of the endpoints of \(l_k\), and a point \(x_k\in(t'_k,t''_k)\) such that, replacing \(K\) by \(-K\) in the notation (4), we shall have
\[ I_k(t'_k,x_k)=\max_{x\leq x_k} I_k(x,x_k)>0,\qquad I_k(x_k,t''_k)=\min_{x\geq x_k} I_k(x_k,x)<0, \]
\[ I_k(t'_k,t''_k)=0. \]
From the line segments \(\tau_k(x)\), constructed in the indicated way for the links with angular coefficients exceeding \(K\) in absolute value, and from the remaining links of the broken line \(g(x)\), it is not difficult to construct a periodic polygonal function \(\varphi(x)\in KH^{(1)}\) such that
\[ \max_{a,b}\left|\int_a^b [g(x)-\varphi(x)]\,dx\right| \leq 2\max_k |I_k(t'_k,x_k)|. \tag{5} \]
Now let us note that the following assertion is valid: whatever the function \(f\in H_\omega\), where \(\omega(t)\) is convex upward, if on the interval \([a,b]\) for the function
\[ \Delta(x)=f(x)-f(c)-K(x-c)\qquad (a<c<b) \]
the relations
\[ -\int_a^c \Delta(x)\,dx=\int_c^b \Delta(x)\,dx>0, \]
are satisfied, then
\[ M(f)=-\int_a^c \Delta(x)\,dx+\int_c^b \Delta(x)\,dx \leq \frac12\int_0^{b-a}[\omega(t)-Kt]\,dt. \tag{6} \]
Indeed, without loss of generality one may assume that \(\Delta(x)<0\) on \((a,c)\) and \(\Delta(x)>0\) on \((c,b)\). Then, defining the function \(\rho(x)\) by the equality
\[ \int_x^{\rho(x)} \Delta(t)\,dt=0\qquad (a\leq x\leq c,\ c\leq \rho(x)\leq b), \]
one can obtain the estimates
\[ M(f)\leq \frac12\int_a^c |\Delta(\rho(t))-\Delta(t)|(1-\rho'(t))\,dt\leq \]
\[ \leq \frac12\int_a^c \{\omega[\rho(t)-t]-K[\rho(t)-t]\}(1-\rho'(t))\,dt = \frac12\int_0^{b-a}[\omega(t)-Kt]\,dt. \]
If the polygonal function \(g(x)\) is inscribed in \(f \in H_\omega\) and \(\omega(t)\) is convex upward, then, as was already noted in the paper \((^4)\), \(g \in H_\omega\).
Thus, from (5), using (6), we find that
\[ \max_{a,b}\left|\int_a^b [g(x)-\varphi(x)]\,dx\right| \le \frac12 \max_{x\ge 0}\int_0^x [\omega(t)-Kt]\,dt . \]
It is known that in any continuous function \(f(x)\) one can inscribe a polygonal function \(g(x)\) so that \(\max_x |f(x)-g(x)|<\varepsilon\), where \(\varepsilon\) may be prescribed arbitrarily small. Therefore, if \(f\) satisfies the conditions of the lemma, then it is easy to prove the existence of a function \(\varphi_0 \in KH^{(1)}\), for which, for any \(a\) and \(b\), inequality (3) is fulfilled.
Having completed the proof of the lemma, we note that, although in the present paper we shall not need this, a somewhat more general assertion holds, namely
\[ \sup_{f\in H_\omega}\ \inf_{\varphi\in KH^{(1)}}\ \max_{a,b} \left|\int_a^b [f(x)-\varphi(x)]\,dx\right| = \frac12 \max_{x\ge 0}\int_0^x [\omega(t)-Kt]\,dt . \]
We now pass to the proof of the theorem. Let \(f\in W^{(1)}H_\omega\). Then \(f'\in H_\omega\), and, by the lemma, for any \(K>0\) there exists a function \(\varphi_0\in KH^{(1)}\) such that
\[ \left|\int_{a_0}^{b_0} [f'(x)-\varphi_0(x)]\,dx\right| = \max_{a,b}\left|\int_a^b [f'(x)-\varphi_0(x)]\,dx\right| \le \frac12 \delta(K), \]
where, for brevity, we have denoted
\[ \delta(K)=\max_{x\ge 0}\int_0^x [\omega(t)-Kt]\,dt . \]
Set
\[ \psi_0(x)=\int_c^x \varphi_0(t)\,dt, \]
where the value of \(c\) has been chosen from the condition
\[ \int_{a_0}^c (f'(t)-\varphi_0(t))\,dt = \int_c^{b_0} (f'(t)-\varphi_0(t))\,dt . \]
Then
\[ \max_x |f(x)-\psi_0(x)| = \left|\int_c^{t_0} [f'(t)-\varphi_0(t)]\,dt\right| \le \frac14 \delta(K). \]
Since \(\psi_0\in W^{(1)}KH^{(1)}\), by \((^{1,2})\)
\[ E_n(\psi_0)\le K\frac{\pi^2}{8(n+1)^2}, \]
and therefore
\[ E_n(f)\le \max_x |f(x)-\psi_0(x)|+E_n(\psi_0) \le \frac14 \delta(K)+K\frac{\pi^2}{8(n+1)^2}. \]
If we take \(K=K_n=\dfrac{n+1}{\pi}\omega\left(\dfrac{\pi}{n+1}\right)\), then
\[ \delta (K_n)=\int_0^{\frac{\pi}{n+1}}[\omega(t)-K_n t]\,dt, \]
and we obtain
\[ E_n(f)\leq \frac14\int_0^{\frac{\pi}{n+1}}\omega(t)\,dt . \tag{7} \]
It remains to indicate, in the class \(W^{(1)}H_\omega\), a function for which equality holds in (7). Such a function is, for example, the function \(f_0(x)\), whose derivative \(f_0'\) has period \(\dfrac{2\pi}{n+1}\), is odd, and is defined on
\[ \left[-\frac{\pi}{2(n+1)},\frac{\pi}{2(n+1)}\right] \]
by the equalities
\[ f_0'(x)= \begin{cases} \dfrac12\,\omega(2x), & \left(0\leq x\leq \dfrac{\pi}{2(n+1)}\right),\\[6pt] -\dfrac12\,\omega(-2x), & \left(-\dfrac{\pi}{2(n+1)}\leq x\leq 0\right). \end{cases} \]
The theorem is proved. Let us note the most important special case, when \(\omega(t)=Kt^\alpha\) \((0\leq t\leq \pi)\).
Corollary. For all \(0<\alpha\leq 1\),
\[ \sup_{f\in W^{(1)}KH^{(\alpha)}} E_n(f) = \frac{K}{4(1+\alpha)} \left(\frac{\pi}{n+1}\right)^{1+\alpha} \qquad (n=0,1,2,\ldots). \tag{8} \]
Let \(W^1KH^{(\alpha)}_{[-1,1]}\) be the class of functions whose derivative satisfies, on the interval \([-1,1]\), a Lipschitz condition of order \(\alpha\) \((0<\alpha\leq 1)\) with constant \(K\), and let \(E_n(f;-1,1)\) be the best approximation of the function \(f\) by algebraic polynomials of degree \(n\) on this interval. Then, taking into account the limiting equality proved by S. N. Bernstein \((^3)\), from (8) we immediately find that
\[ \lim_{n\to\infty}(n+1)^{1+\alpha} \sup_{f\in W^{(1)}KH^{(\alpha)}_{[-1,1]}} E_n(f;-1,1) = \frac{K\pi^{1+\alpha}}{4(1+\alpha)} \qquad (0<\alpha\leq 1). \]
Dnepropetrovsk State University
named after the 300th Anniversary of the Reunification of Ukraine with Russia
Received
16 VI 1961
REFERENCES
- N. I. Akhiezer, M. G. Krein, DAN, 15, 107 (1937).
- J. Favard, Bull. de Sci. Math. 61, 209 (1937).
- S. N. Bernstein, DAN, 57, 3 (1947).
- N. P. Korneichuk, DAN, 140, No. 4 (1961).