Full Text
Reports of the Academy of Sciences of the USSR
- Vol. 132, No. 2
MATHEMATICS
R. M. TRIGUB
APPROXIMATION OF FUNCTIONS WITH A GIVEN MODULUS OF SMOOTHNESS ON THE EXTERIOR OF AN INTERVAL AND A HALF-LINE
(Presented by Academician V. I. Smirnov on January 15, 1960)
A. F. Timan proved the following theorem \((^7)\):
If a function \(f(x)\), defined on \([-1,1]\), has there a continuous \(r\)-th derivative, then for every natural number \(n\) there exists an ordinary polynomial \(P_n(x)\) of degree not exceeding \(n\), satisfying, for every \(x \in [-1,1]\), the inequality
\[ |f(x)-P_n(x)| \leq C_r \left(\frac{\sqrt{1-x^2}}{n}+\frac{1}{n^2}\right)^r \omega^{(r)}\left(\frac{\sqrt{1-x^2}}{n}+\frac{1}{n^2}\right), \]
where
\[
\omega^{(r)}(h)=\omega(f^{(r)};h)
=\sup_{|x_1-x_2|\leq h}|f^{(r)}(x_1)-f^{(r)}(x_2)|,
\quad x_1,x_2\in[-1,1],
\]
is the modulus of continuity of the \(r\)-th derivative; \(C_r\) does not depend on \(x\) or \(n\).
This theorem (unlike the well-known Jackson theorem for a finite interval) admits an inversion to the same extent as Jackson’s theorem in the periodic case: the inverse theorems to it \((^8)\); the case \(\omega^{(r)}(t)=t^\alpha\) \((0<\alpha<1)\), see \((^4)\), make it possible to judge structural properties of functions on the whole interval, and for some moduli of continuity give a complete inversion of this theorem. V. K. Dzyadyk \((^5)\) generalized the indicated theorem, replacing the modulus of continuity by the modulus of smoothness*
\[ \omega_2(h)= \sup_{|x_1-x_2|\leq 2h} \left|f(x_1)-2f\left(\frac{x_1+x_2}{2}\right)+f(x_2)\right|, \quad x_1,x_2\in[-1,1]. \]
Yu. A. Brudnyi \((^3)\) obtained theorems similar to A. F. Timan’s theorem in the case of approximation by entire functions on the exterior of an interval and on a half-line. The inverse theorems to them also belong to him \((^3)\).
In the present work the indicated results of Yu. A. Brudnyi are generalized. Let the function \(f(x)\) have on
\[
E=(-\infty,-1]\cup[1,\infty)
\]
\(r\) uniformly continuous and bounded derivatives, and let
\[
\omega_2^{(r)}(h)=\omega_2(f^{(r)};h)
\]
be the modulus of smoothness of the \(r\)-th derivative, while \(B_\sigma\) is the class of entire functions of degree not exceeding \(\sigma\), bounded on the real axis. Then the following holds:
Theorem 1. For every \(\sigma \geq 1\) there exists an entire function \(G_\sigma(f;x)\in B_\sigma\) such that, for every \(x\in E\),
\[ |f(x)-G_\sigma(f;x)| \leq C_r \left(\frac{\sqrt{x^2-1}}{|x|\sigma}+\frac{1}{\sigma^2}\right)^r \omega_2^{(r)} \left(\frac{\sqrt{x^2-1}}{|x|\sigma}+\frac{1}{\sigma^2}\right), \]
where \(C_r\) does not depend on \(x\) or \(\sigma\).
The proof of the theorem is based on the following lemmas.
Lemma 1. If \(\omega_2(h)\ne 0\) for \(h\ne 0\), then for \(h\leq 1\)
\[ \omega^2(h)\leq C\omega_2(h); \tag{1} \]
\[ \omega(h^2)\leq C\omega_2(h), \tag{2} \]
* The same result was obtained by Freud \((^9)\).
where \(C\) does not depend on \(h\);
\[
\omega(h)=\sup_{\substack{x\\0\le \delta \le h}} |f(x+\delta)-f(x)|.
\]
Lemma 2. If \(f(x)\) and \(g(x)\) are bounded and \(\omega_2(f;h)\ne 0\) for \(h\ne 0\), then
\[
\omega_2(fg;h)\le C[\omega_2(f;h)+\omega_2(g;h)].
\]
where \(C\) does not depend on \(h\).
Lemma 3. A function \(f(x)\) bounded on \(E\) can be extended to the whole axis with preservation (up to a constant) of \(\omega_2(h)\) and \(\omega(h)\). The exception is formed by the functions
\[
f_{a,b}(x)=a,\quad x\ge 1;\qquad f_{a,b}=b,\quad x\le -1
\]
when \(a\ne b\).
Lemma 2 is proved with the aid of inequality (1), and Lemma 3 with the aid of Lemma 2.
Let us prove Lemma 1. From the equality
\[
2[f(x+\delta)-f(x)]=[f(x-\delta)-2f(x)+f(x+\delta)]+[f(x+\delta)-f(x-\delta)]
\]
it follows that \(\omega(h)\le \frac12\omega_2(h)+\frac12\omega(2h)\), whence for any natural \(k\)
\[
\omega(h)\le \sum_{\nu=0}^{k}\frac{\omega_2(2^\nu h)}{2^{\nu+1}}+\frac{\omega(2^{k+1}h)}{2^{k+1}} .
\tag{3}
\]
Taking into account that \(\omega_2(2^\nu h)\le 2^{2\nu}\omega_2(h)\), and putting
\[
k=\left[\log_2 \frac{1}{\sqrt{\omega_2(h)}}\right]
\]
(one may assume that \(\omega_2(1)=1\)), we obtain from (3) inequality (1), and putting
\[
k=\left[\log_2 \frac{1}{\sqrt{h}}\right]
\]
and using the monotonicity of \(\omega_2(h)\) and inequality (1), we obtain from (3) inequality (2).
Inequality (2) also follows from Marchaud’s inequality \((^6)\):
\[
\omega_k(h)=\sup_{\substack{x\\0\le \delta\le h}}
\left|\sum_{\nu=0}^{k}(-1)^\nu {k\choose \nu} f(x+\nu\delta)\right|
\le C_k h^k \int_h^1 \frac{\omega_{k+1}(u)}{u^{k+1}}\,du
\quad\text{for } h\le \frac12 .
\]
In proving the theorem kernels \(K(u)\) with the following properties are used: \(K(u)\) is an entire function of degree not exceeding one; \(K(u)=K_p(u)=K(-u)\), \((|u|+1)^p|K(u)|=O(1)\) (\(p\) an integer \(\ge 2\)); \(\int_{-\infty}^{\infty} K(u)\,du=1^*\).
These conditions are satisfied, for example, by kernels of Jackson type
\[
K(u)=\frac{1}{\gamma_p}\left(\frac{\sin \frac{u}{p}}{u}\right)^p,\qquad
\gamma_p=\int_{-\infty}^{\infty}\left(\frac{\sin \frac{u}{p}}{u}\right)^p\,du .
\]
The role of the kernels \(K(u)\) in approximation questions consists in the fact that the expression
\[
\int_{-\infty}^{\infty} f\left(x+\frac{t}{\sigma}\right)K(t)\,dt
\]
is an entire function of degree not exceeding \(\sigma\), even in the case of evenness of \(f(x)\).
Let us now prove the theorem for \(r=0\). It is possible to consider only even and odd functions \(f(x)\). Let \(f(x)=f(-x)\). Consider the single-valued function
\[
\widetilde f(u)=f\left(\sqrt{1+u^2}\right).
\]
We shall show that the desired function \(G_\sigma(f;x)\) is the expression
\[
g_\sigma(\widetilde f;u)=
2\int_{-\infty}^{\infty}\widetilde f\left(u+\frac{t}{\sigma}\right)K(t)\,dt
-
\int_{-\infty}^{\infty}\widetilde f\left(u+\frac{\sqrt2\,t}{\sigma}\right)K(t)\,dt
=
\sum_{\nu=0}^{\infty} C_\nu u^{2\nu},
\]
\[ \text{* } K(u)\text{ is a special case of kernels of Fejér type (see }(^1),\text{ p. 126).} \]
if instead of \(u\) we substitute \(\sqrt{x^2-1}\). Extend \(f(x)\) by Lemma 3. Taking into account that
\[ \tilde f\left(u+\frac{t}{\sigma}\right) = f\left(T+A\frac{t}{\sigma}+B\frac{t^2}{\sigma^2}+C\frac{t^3}{\sigma^3}+\widetilde D\frac{t^4}{\sigma^4}\right), \]
where
\[ T=\sqrt{1+u^2},\qquad A=\frac{u}{\sqrt{1+u^2}},\qquad B=\frac{1}{2(1+u^2)^{3/2}},\qquad C=-\frac{1}{2(1+u^2)^{5/2}}, \]
\(|\widetilde D|<1/2\), and also that
\[ \begin{aligned} &\left| f(T)-f\left(T+A\frac{t}{\sigma}+B\frac{t^2}{\sigma^2}+C\frac{t^3}{\sigma^3}\right) -f\left(T-A\frac{t}{\sigma}+B\frac{t^2}{\sigma^2}-C\frac{t^3}{\sigma^3}\right)\right.\\ &\left.\quad+\frac12 f\left(T+A\frac{\sqrt2 t}{\sigma}+B\frac{2t^2}{\sigma^2}+C\frac{2\sqrt2 t^3}{\sigma^3}\right) +\frac12 f\left(T-A\frac{\sqrt2 t}{\sigma}+B\frac{2t^2}{\sigma^2}-C\frac{2\sqrt2 t^3}{\sigma^3}\right)\right|\\ &\le \left|2f\left(T+B\frac{t^2}{\sigma^2}\right) -f\left(T+A\frac{t}{\sigma}+B\frac{t^2}{\sigma^2}+C\frac{t^3}{\sigma^3}\right) -f\left(T-A\frac{t}{\sigma}+B\frac{t^2}{\sigma^2}-C\frac{t^3}{\sigma^3}\right)\right|\\ &\quad+\frac12\left|f\left(T+A\frac{\sqrt2 t}{\sigma}+B\frac{2t^2}{\sigma^2}+C\frac{2\sqrt2 t^3}{\sigma^3}\right)\right.\\ &\left.\qquad\quad+f\left(T-A\frac{\sqrt2 t}{\sigma}+B\frac{2t^2}{\sigma^2}-C\frac{2\sqrt2 t^3}{\sigma^3}\right) -2f\left(T+B\frac{2t^2}{\sigma^2}\right)\right|\\ &\quad+\left|f\left(T+B\frac{2t^2}{\sigma^2}\right)-2f\left(T+B\frac{t^2}{\sigma^2}\right)+f(T)\right|\\ &\le C_1\omega_2\left(\frac{|At|}{\sigma}+\frac{|Bt^2|}{\sigma^2}+\frac{|Ct^3|}{\sigma^3}\right), \end{aligned} \]
where \(C_1\) does not depend on \(t\) and \(\sigma\), we obtain
\[ |f(x)-G_\sigma(f;x)|=|\tilde f(u)-g_\sigma(\tilde f;u)|\le \]
\[ \le C_1\int_{-\infty}^{\infty} \omega_2\left(\frac{|At|}{\sigma}+\frac{|Bt^2|}{\sigma^2}+\frac{|Ct^3|}{\sigma^3}\right)|K(t)|\,dt + C_2\int_{-\infty}^{\infty} \omega\left(\frac{t^4}{\sigma^4}\right)|K(t)|\,dt. \]
It remains to take into account that, for \(\lambda>0\),
\(\omega_2(\lambda h)\le(\lambda+1)^2\omega_2(h)\), to apply Lemma 1, and to take \(K(u)=K_8(u)\).
If \(f(x)=-f(-x)\), we apply what has been proved to the function
\[ F(x)=\frac{f(x)}{g_1(x)} \qquad \left(g_1(x)=\int_0^x\frac{\sin^2\frac{t}{2}}{t^2}\,dt\right), \]
taking into account that \(g_1(x)\) is an entire function of degree one, and
\(\omega_2(F)<C\omega_2(f;h)\) (Lemma 2).
Suppose that the theorem has been proved in the case of existence of the \((r-1)\)-st derivative, and first let \(f(x)=f(-x)\). Then there exists an even entire function \(G_\sigma(x)\in B_\sigma\) such that
\[ |f'(x)-G_\sigma(x)| \le C_{r-1}\left(\frac{\sqrt{x^2-1}}{|x|\sigma}+\frac{1}{\sigma^2}\right)^{r-1} \omega_2^{(r)}\left(\frac{\sqrt{x^2-1}}{|x|\sigma}+\frac{1}{\sigma^2}\right). \]
Let
\[ R(x)=\int_0^x [f'(t)-G_\sigma(t)]\,dt \quad\text{for }x\ge1,\qquad R(x)=R(-x). \]
By means of Lagrange’s mean value theorem we obtain
\[ \left|R\left(\sqrt{1+u^2}\right) -\int_{-\infty}^{\infty} R\left(\sqrt{1+\left[u+\frac{t}{\sigma}\right]^2}\right)K(t)\,dt\right| \le \]
\[ \le C_{r-1}C_3 \left(\frac{|u|}{\sqrt{1+u^2}}\frac{1}{\sigma}+\frac{1}{\sigma^2}\right)^r \omega_2^{(r)} \left(\frac{|u|}{\sqrt{1+u^2}}\frac{1}{\sigma}+\frac{1}{\sigma^2}\right) \int_{-\infty}^{\infty}(t^2+1)|K(t)|\,dt. \]
It remains to take \(K(u)=K_4(u)\) and instead of \(u\) substitute \(\sqrt{x^2-1}\). The proof is completed as in the case \(r=0\). Theorem 1 is proved.
The following theorem, converse to Theorem 1, holds:
Theorem 2. Let, for a function \(f(x)\) given on \(E\), for some \(r\ge0\) and a function \(\widetilde\omega_2(h)\ne0\), given on \([0,2]\) and satisfyi-
satisfying there the inequality \(\widetilde\omega_2(\lambda h) \leq (\lambda+1)^2\widetilde\omega_2(h)\) and such that
\[ \int_0^1 \nu(r)\frac{\widetilde\omega_2(u)}{u^3}\,du<\infty \]
(\(\nu(r)=0\) for \(r=0\); \(\nu(r)=1\) for \(r\geq 1\)), for any \(\sigma\geq 1\) there is an entire function \(G_\sigma(x)\in B_\sigma\) such that, for all \(x\in E\),
\[ \left|f(x)-G_\sigma(x)\right| \leq \left(\frac{\sqrt{x^2-1}}{|x|\sigma}+\frac{1}{\sigma^2}\right)^r \widetilde\omega_2\left(\frac{\sqrt{x^2-1}}{|x|\sigma}+\frac{1}{\sigma^2}\right). \]
Then \(f(x)\) has on \(E\) \(r\) uniformly continuous and bounded derivatives, and for \(h\leq 1\)
\[ \omega_2\left(f^{(r)};h\right) \leq C\left\{ h^2\int_h^2 \frac{\widetilde\omega_2(u)}{u^3}\,du + \int_0^h \nu(r)\frac{\widetilde\omega_2(u)}{u}\,du \right\}, \]
where \(C\) does not depend on \(h\).
The proof is carried out by the known method of S. N. Bernstein and is based on the following inequality of Yu. A. Brudnyi \((^3)\):
If, for some nonnegative \(r\), for all \(x\in E\)
\[ |G_\sigma(x)| \leq \left(\frac{\sqrt{x^2-1}}{|x|\sigma}+\frac{1}{\sigma^2}\right)^r \omega\left(\frac{\sqrt{x^2-1}}{|x|\sigma}+\frac{1}{\sigma^2}\right), \]
then
\[ |G'_\sigma(x)| \leq C\left(\frac{\sqrt{x^2-1}}{|x|\sigma}+\frac{1}{\sigma^2}\right)^{r-1} \omega\left(\frac{\sqrt{x^2-1}}{|x|\sigma}+\frac{1}{\sigma^2}\right), \]
where \(\omega(h)\) is any modulus of continuity. From the proof of this inequality it is clear that it remains valid when \(\omega(h)\) is replaced by \(\widetilde\omega_2(h)\), and also for negative \(r\).
Corollary. If the function \(f(x)\) satisfies the conditions of Theorem 2 and
\[ h^2\int_h^2 \frac{\widetilde\omega_2(u)}{u^3}\,du + \int_0^h \nu(r)\frac{\widetilde\omega_2(u)}{u}\,du = O\left[\widetilde\omega_2(h)\right], \]
then
\[ \omega_2\left(f^{(r)};h\right)=O\left[\widetilde\omega_2(h)\right]. \]
Condition (4) is satisfied, for example, by the function \(\widetilde\omega_2(h)=h\). If the function \(f(x)\) is given on \([0,\infty)\), then, as S. N. Bernstein showed (\((^2)\), see also \((^3)\)), the natural apparatus of approximation is provided by entire functions of finite half-degree \(\sigma\), i.e. such entire functions \(H_\sigma(u)\) that \(G_\sigma(u)=H_\sigma(u^2)\) are entire functions of finite degree not exceeding \(\sigma\).
Analogously to Theorem 1, the following theorem is proved.
Theorem 3. If \(f(x)\) has on \([0,\infty)\) \(r\) uniformly continuous and bounded derivatives, then for any \(\sigma\geq 1\) there exists an entire function \(H_\sigma(x)\) of finite half-degree \(\sigma\) such that, for all \(x\in[0,\infty)\),
\[ |f(x)-H_\sigma(x)| \leq C_r\left(\frac{\sqrt{x}}{\sigma}+\frac{1}{\sigma^2}\right)^r \omega_2^{(r)}\left(\frac{\sqrt{x}}{\sigma}+\frac{1}{\sigma^2}\right), \]
where \(\omega_2^{(r)}(h)\) is the modulus of smoothness of the \(r\)-th derivative, and \(C_r\) does not depend on \(x\) and \(\sigma\).
There is a converse theorem analogous to Theorem 2. We note that the proof of Theorem 1 given above is also applicable in the case of approximation by algebraic polynomials on a finite interval.
In conclusion, I express my gratitude to Prof. A. F. Timan for posing the problem and for his guidance.
Dnepropetrovsk State University
named after the 300th anniversary of the reunification of Ukraine with Russia
Received
13 I 1960
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