UDC 517.51+517.52

MATHEMATICS

E. P. DOLZHENKO

UNIFORM APPROXIMATIONS BY RATIONAL FUNCTIONS (ALGEBRAIC AND TRIGONOMETRIC) AND GLOBAL FUNCTIONAL PROPERTIES

(Presented by Academician A. N. Kolmogorov on 18 V 1965)

Without dwelling on the history of the question (it may be found in (5)), let us recall some definitions and notation.

Let \(\Phi(u)\) be increasing, continuous, and convex downward on \([0,\infty)\), \(\Phi(0)=0\). We say that \(\Phi\) has the \(\Delta_2\)-property, \(\Phi\in\Delta_2\), if for all \(u\geq 0\) we have \(\Phi(2u)\leq C\Phi(u)\), where \(C=C(\Phi)=\mathrm{const}\). The \(\Phi\)-variation of a function \(f(x)\) \((x\in[a,b],-\infty\leq a<b\leq+\infty)\) is defined as follows (see (1)):

\[ V_\Phi(f)=V_\Phi(f,[a,b])=\sup\left\{\sum_{k=1}^{n}\Phi\left(\left|f(x_k)-f(x_{k-1})\right|\right)\right\}, \tag{1} \]

where the least upper bound is taken over all \(\{x_k\}: a=x_0<x_1<\cdots<x_n=b,\ n=1,2,\ldots\).

If \(V_\Phi(f)<\infty\), then \(f\in V_\Phi\). If \(kf\in V_\Phi\) for some \(k=\mathrm{const}>0\), then \(f\in V_\Phi^*\). Obviously, \(V_\Phi\subset V_\Phi^*\) and \(V_\Phi=V_\Phi^*\) for \(\Phi\in\Delta_2\). Note that any function \(f\) continuous on \([a,b]\) belongs to some \(V_\Phi^*\). If, following Musielak and Orlicz (1), one introduces the norm

\[ \|f\|_\Phi=\|f\|_{\Phi,[a,b]}=\inf\{k:k>0,\ V_\Phi(f/k)\leq 1\} \tag{2} \]

and regards \(f_1=f_2\) when \(f_1(x)-f_2(x)\equiv\mathrm{const}\), then \(V_\Phi^*\) becomes a complete linear normed space (with the usual addition of functions and multiplication of a function by a number) ((1), Sec. 3.21), and, if \(\|f-f_n\|_\Phi\to0\), then \(V_\Phi(k(f-f_n))\to0\) for some \(k=\mathrm{const}>0\) ((1), Sec. 3.11).

A function \(f(x)\) \((x\in[a,b])\) is called \(\Phi\)-absolutely continuous, \(f\in AC_\Phi\), if for every \(\varepsilon>0\) there is a \(\delta>0\) such that \(\sum \Phi(|f(\beta_i)-f(\alpha_i)|)<\varepsilon\) whenever \(\sum \Phi(\beta_i-\alpha_i)<\delta\), \(a\leq\alpha_1<\beta_1\leq\alpha_2<\beta_2\leq\cdots\leq b\). \(f\in AC_\Phi^*\) if \(kf\in AC_\Phi\) for some \(k=\mathrm{const}>0\). Obviously, \(AC_\Phi\subset V_\Phi\), \(AC_\Phi^*\subset V_\Phi^*\), \(AC_\Phi\subset AC_\Phi^*\), and \(AC_\Phi=AC_\Phi^*\) for \(\Phi\in\Delta_2\). \(AC_\Phi^*\) is a closed subspace of the space \(V_\Phi^*\) ((1), Sec. 3.21).

We say that \(f\in L_\Phi[a,b]=L_\Phi\) if the function \(\Phi(|f(x)|)\) is summable on \([a,b]\), and \(f\in L_\Phi^*[a,b]=L_\Phi^*\) if \(kf\in L_\Phi\) for some \(k=\mathrm{const}>0\). The norm in \(L_\Phi^*\) is defined as follows (see (3)):

\[ \|f\|_{L_\Phi[a,b]}=\inf\left\{k:k>0,\ \int \Phi(|f|/k)\,dx\leq 1\right\}. \tag{3} \]

\(L_\Phi^*\) (the Orlicz space) is a complete normed linear space. If \(\Phi\in\Delta_2\), then \(L_\Phi=L_\Phi^*\). If \(\Phi(u)=u^p,\ p\geq1\), then \(L_\Phi=L^p\), and \(\|f\|\) in the metric \(L_\Phi^*\) coincides with \(\|f\|\) in \(L^p\).

* For example, with \(\Phi(u)=\tilde{\omega}^{-1}(u/3)\), where \(\tilde{\omega}\) is a regularization of the modulus of continuity of the function \(f\); see (4), Sec. 3.3.

For \(\delta>0\) (and \(f_h(x)=f(x+h)\)) the quantity

\[ \omega_\Phi(\delta,f)=\omega_\Phi(\delta,f[a,b])=\sup_{|h|\leq\delta}\|f-f_h\|_{L\Phi([a,b]\cap[a-h,b-h])} \tag{4} \]

will be called the \(\Phi\)-modulus of continuity of the function \(f\) (on \([a,b]\)). \(R_n=R_n[f]=R_n[f,[a,b]]\) denotes the least deviation (in the uniform metric) of the function \(f\) from real rational functions (algebraic or trigonometric) of degree (order) not exceeding \(n\)* (see (5)); \(r_n(x)=r_n(x,f)\) is a rational function of order \(\leq n\) for which \(\max |f(x)-r_n(x)|=R_n\). In the case of rational trigonometric functions we always assume \(0\leq a<b\leq 2\pi\).

\(1^\circ\). Rate of approximation and \(\Phi\)-variation.

Lemma 1. For every rational (algebraic or trigonometric) function \(r(x)\) we have:

\[ V_\Phi(r)\leq 2n\Phi(2M) \tag{5a} \]

\[ \|r\|_\Phi\leq 4M/\Phi^{-1}(1/n), \tag{5b} \]

where \(n\) is the order of the function \(r\), \(M=\max\{|r(x)|:x\in[a,b]\}\).

Proof. From the increase of \(\Phi\) it follows that the sum (1) will not decrease if to the partition points \(x_k\) one adds a point \(y\), \(x_{k-1}<y<x_k\), at which \(r(y)\in[r(x_{k-1}),r(x_k)]\).

From the convexity downward of \(\Phi\) it follows that this sum will not decrease if one removes a partition point \(x_k\) for which \(r(x_k)\in[r(x_{k-1}),r(x_{k+1})]\).

Thus,

\[ V_\Phi(r)=\sum \Phi(|r(y_i)-r(y_{i-1})|), \]

where the \(y_i\) are some of the points \(y\in[a,b]\) at which \(r\) attains local extrema (\(y_0=a<y_1<\cdots<y_k=b\), \(k\leq 2n\)). Hence we obtain (5a). From this and (2) we have: \(\|r\|_\Phi\leq 2M/\Phi^{-1}(1/2n)\). Taking into account that \(\Phi^{-1}\) is convex upward and increasing, we obtain (5b).

Remark. The preceding arguments show that if the graph of the function \(y=f(x)\) can be divided into \(n\) intervals of monotonicity, then estimate (5) remains valid (\(r=f\)). If, in addition, \(f\) is continuous, then the right-hand sides in inequalities (5) should be halved.

Theorem 1. Always

\[ \|f\|_\Phi\leq 100\sum_{n=0}^{\infty}\lambda_nR_n,\quad \text{where}\quad \lambda_{n-1}=\left[n\Phi^{-1}\left(\frac1n\right)\right]^{-1} \leq[\Phi^{-1}(1)]^{-1}. \tag{6} \]

Proof. Obviously,

\[ \|f\|_\Phi\leq \|r_1-r_0\|_\Phi+\sum_{k=1}^{\infty}\|r_{2^k}-r_{2^{k-1}}\|_\Phi. \]

By Lemma 1,

\[ \|r_1-r_0\|_\Phi\leq 8R_0/\Phi^{-1}(1); \]

\[ \|r_{2^k}-r_{2^{k-1}}\|_\Phi \leq 4\cdot 2R_{2^{k-1}}/\Phi^{-1}(1/3\cdot 2^{k-1})\leq \]

\[ \leq 48R_{2^{k-1}}/\Phi^{-1}(2^{-(k-2)}) \leq 96\sum_{2^{k-2}+1}^{2^{k-1}} R_{n-1}\left[n\Phi^{-1}\left(\frac1n\right)\right]^{-1}. \tag{7} \]

Theorem 2. If

\[ \sum R_n[f]\,[n\Phi^{-1}(1/n)]^{-1}<\infty, \]

then \(f\in AC_\Phi^*\). If, moreover, \(\rho_n(x)\) are rational functions of order respectively not exceeding \(n\), for which (with some \(C=\mathrm{const}\))

\[ \max\{|f(x)-\rho_n(x)|:x\in[a,b]\}\leq CR_n[f], \]

then \(\|f-\rho_n\|_\Phi\to 0\) and \(V_\Phi(k(f-\rho_n))\to 0\) for some \(k=\mathrm{const}>0\) \((n\to\infty)\).

\(*\) \(r(x)=P(x)/Q(x)\) is a rational algebraic (trigonometric) function of order \(\leq n\), if \(P\) and \(Q\) are algebraic (respectively trigonometric) polynomials of order \(\leq n\).

Proof. Choose natural \(p\) from the condition \(2^{p-1}\le n<2^p\). From the inequality

\[ \|f-\rho_n\|_{\Phi}\le \sum_{p+1}^{\infty}\|\rho_{2^q}-\rho_{2^{q-1}}\|_{\Phi}+\|\rho_{2^p}-\rho_n\|_{\Phi} \]

and estimates of type (7), we find that \(\|f-\rho_n\|_{\Phi}\to 0\), whence \(f\in AC_{\Phi}^{*}\) (obviously, \(\rho_n\in AC_{\Phi}\)).

The case \(\Phi(r)=r\) was considered by the author in (5). In this case the condition of the theorem cannot be weakened. The constructions preceding Theorem 2 of paper (5) show that, in the general case, if \(a_0\ge a_1\ge \cdots>0\) and \(\sum \Phi(a_n)=\infty\), there always exists a function \(f\), not belonging to \(V_{\Phi}^{*}\), for which \(R_n[f]\le a_n\).

Thus, if \(R_n[f]\le C n^{-1/p-\varepsilon}\) \((p\ge 1;\ C,\varepsilon>0)\), then \(f\in AC_{\Phi}\) with \(\Phi(u)=u^p\).* However, the condition \(R_n[f]\le C n^{-1/p}\) is insufficient even for \(f\) to belong to \(V_{\Phi}^{*}\).

\(2^\circ\). Rate of approximation and the integral modulus of continuity.

Lemma 2a. For any function \(f(x)\) \((-\infty\le a\le x\le b\le +\infty)\), for \(0<\delta<b-a\) we have:

\[ 1)\quad \int_a^{b-\delta}\Phi(|f(x+\delta)-f(x)|)\,dx\le 2V_{\Phi}(f)\delta; \tag{8} \]

\[ 2)\quad \omega_{\Phi}(\delta,f)\le \Omega/\Phi^{-1}((b-a)^{-1}), \tag{9} \]

where \(\Omega\) is the total oscillation of \(f\) on \([a,b]\); \(\Omega=\max f-\min f\).

Proof. If \(a,b\ne \pm\infty\), put \(x_k=a+k\delta\) for \(k=0,1,\ldots,n-1=[(b-a)/\delta]\) and \(x_n=b-\delta\). Let \(\sum'\) and \(\sum''\) denote, respectively, summation over odd and over even \(k\). Then

\[ \int_a^{b-\delta}\Phi(|f(x+\delta)-f(x)|)\,dx = \sum \int_{x_{k-1}}^{x_k}\le \]

\[ \le \left(\sum' + \sum''\right) \sup_{x_{k-1}\le x\le x_k}\Phi(|f(x+\delta)-f(x)|)\delta \le 2V_{\Phi}(f)\delta, \]

and (8) is proved. The cases \(a=-\infty\), \(b=\infty\) are obtained by passage to the limit. Since always the integral in (8) is \(\le \Phi(\Omega)(b-a)\), (9) follows from (4) and (3). As a consequence of (8) and (5a) we obtain

Lemma 2. Let \(r(x)\) be a rational (algebraic or trigonometric) function of order \(\le n\), whose modulus on \([a,b]\) does not exceed \(M\) \((-\infty\le a<b\le +\infty)\). Then

\[ \int_a^{b-\delta}\Phi(|r(x+\delta)-r(x)|)\,dx\le 4n\Phi(2M)\delta, \tag{10} \]

\[ \omega_{\Phi}(\delta,r)\le 8M/\Phi^{-1}(1/n\delta). \tag{11} \]

Remark. These estimates are, obviously, also valid for any functions \(r=f\) (not necessarily continuous) of the kind discussed in the remark to Lemma 1.

Theorem 3. For any function \(f(x)\) \((-\infty<a\le x\le b<\infty)\), the inequality holds

\[ \omega_{\Phi}(\delta,f)\le C\sum_{0<n<\delta^{-1}}\lambda_n(\delta)R_n,\qquad \lambda_{n-1}(\delta)=[n\Phi^{-1}(1/n\delta)]^{-1}, \tag{12} \]

where \(C=C(\Phi,b-a)\).

* It is easy to see that in this case \(\|f\|_{\Phi}=\sup \left(\sum |f(x_i)-f(x_{i-1})|^p\right)^{1/p}\).

** This lemma generalizes Lemma 2 of B. I. Golubov’s paper (?), in which \(\Phi(u)=u^p\), \(p\ge 1\). Still earlier the case \(\Phi(u)=u\) was noted by P. L. Ul’yanov.

Proof. Let \(2^{-k-1}<\delta\leq 2^{-k}\). Obviously,

\[ \omega_{\Phi}(\delta,f)\leq \omega_{\Phi}(\delta,f-r_{2^k})+ \sum_{p=1}^{k}\omega_{\Phi}(\delta,r_{2^p}-r_{2^{p-1}}) +\omega_{\Phi}(\delta,r_1-r_0). \]

From (9) and (11) we have

\[ \omega_{\Phi}(\delta,f-r_{2^k}) \leq \frac{2R_{2^k}}{\Phi^{-1}((b-a)^{-1})} \leq \frac{4\Phi^{-1}(4)}{\Phi^{-1}((b-a)^{-1})}\cdot 2^{k-1}\frac{R_{2^k}}{2^k\Phi^{-1}((\delta 2^{k-1})^{-1})}, \]

\[ \omega_{\Phi}(\delta,r_{2^p}-r_{2^{p-1}}) \leq \frac{16R_{2^{p-1}}}{\Phi^{-1}((6\cdot 2^{p-2}\delta)^{-1})} \leq 192\,\frac{2^{p-2}R_{2^{p-1}}}{2^{p-1}\Phi^{-1}((2^{p-2}\delta)^{-1})}. \]

Hence we obtain (12).

Corollary. Let \(\omega^{(1)}(\delta,f)\) be the modulus of continuity of the function \(f(x)\) \((x\in [a,b])\) in the metric \(L\) \((L=L_{\Phi}\) for \(\Phi(r)=r)\). Then

\[ \omega^{(1)}\left(\frac{1}{n},f\right)\leq C\cdot \frac{1}{n}\bigl(R_0[f]+R_1[f]+\cdots+R_{n-1}[f]\bigr), \]

where \(C=200(1+b-a)\). In particular, if \(R_n[f]=O(n^{-\alpha})\) \((0<\alpha<1)\), then \(f\) satisfies on \([a,b]\) the Lipschitz–Hölder integral condition of order \(\alpha\): \(f\in \operatorname{Lip}(\alpha,1)\).

This theorem is close in form to the theorems of S. N. Bernstein, A. F. Timan, and S. B. Stechkin (see \((^6)\), p. 344) for approximations by trigonometric polynomials.

Remarks.

1. The results of this paper remain valid when the interval \([a,b]\) is replaced by an arbitrary measurable set \(E\subset(-\infty,\infty)\) (cf. \((^5)\)), (if instead of the boundedness of \([a,b]\) one requires the boundedness of \(E\)).

2. Theorems 1–3 remain valid if by \(r_n,\rho_n\) one understands functions whose graphs are polygonal lines with the number of links \(\leq n\), or if one understands polynomials in a Haar, Walsh, etc., system (see the remarks to Lemmas 1, 2), and by \(R_n\) the corresponding deviations from \(f\).

3. Let \(E_n[f]\) be the least deviation of \(f\) from polynomials (algebraic or trigonometric) of degree \(\leq n\), and let \(\Phi\) be as at the beginning of this paper. Then for every \(\Phi\) there exists \(f(x)\) \((x\in[0,1])\) with the following properties:
   a) \(R_n[f]\leq E_n[f]\leq CR_n[f]\) \((C=\mathrm{const},\ n\geq 0)\);
   b) \(f\in V_{\Phi}^{*}\), but \(f\notin V_{\Psi}^{*}\), if \(\Psi(u)\) grows near zero essentially more slowly than \(\Phi(u)\) (i.e., \(\Psi(u)/\Phi(ku)\to\infty\) for every fixed \(k>0\) as \(u\to 0\); this guarantees the strict inclusion \(V_{\Psi}^{*}\subset V_{\Phi}^{*}\)).

Thus the classes \(V_{\Phi}^{*}\) capture something characteristic precisely of rational (and not polynomial) approximations. By substituting \(\omega(\delta)=\Phi^{-1}(\delta)\), this assertion follows from the following fundamental result:

Theorem 4. For every modulus of continuity \(\omega(\delta)\)* there exists a function \(f(x)\) with the properties:
a) its ordinary modulus of continuity \(\omega(\delta,f)\) has order \(\omega(\delta)\):

\[ C_1\omega(\delta)\leq \omega(\delta,f)\leq C_2\omega(\delta); \]

b)

\[ R_n[f]\leq E_n[f]\leq CR_n[f],\qquad R_{2.9^k}[f]=E_{2.9^k}[f] \]

\[ (C,\ C_1,\ C_2 \text{ are constants } >0;\ n,\ k=0,1,\ldots). \]

Moscow State University
named after M. V. Lomonosov

Received
16 IV 1965

References

\(^1\) J. Musielak, W. Orlicz, Studia Math., 18, 11 (1959).
\(^2\) B. I. Golubov, Izv. Akad. Nauk SSSR, Ser. Mat., 28, No. 6 (1964).
\(^3\) M. A. Krasnosel’skii, Ya. B. Rutickii, Convex Functions and Orlicz Spaces, Moscow, 1958.
\(^4\) E. P. Dolzhenko, Izv. Akad. Nauk SSSR, Ser. Mat., 28, No. 6 (1964).
\(^5\) E. P. Dolzhenko, Matem. sbornik, 56 (98), 4 (1962).
\(^6\) A. F. Timan, Theory of Approximation of Functions of a Real Variable, Moscow, 1960.

\[ {}^{*}\ \omega(\delta)\text{ is a modulus of continuity if }0=\omega(+0)\leq \omega(\delta)\leq \omega(\delta+\eta)\leq \omega(\delta)+\omega(\eta) \text{ for all }\delta,\eta\geq 0. \]