UDC 517.512.6

MATHEMATICS

N. P. KORNEICHUK

UPPER BOUNDS OF BEST APPROXIMATIONS ON CLASSES OF DIFFERENTIABLE PERIODIC FUNCTIONS IN THE METRICS \(C\) AND \(L\)

(Presented by Academician I. M. Vinogradov on 27 X 1969)

Let \(C_{2\pi}\) be the set of functions continuous on the whole axis with period \(2\pi\), let \(X\) be the metric \(C\) or \(L\), and

\[ E_n(f)_X=\inf_{T_n}\|f-T_n\|_X \qquad (n=0,1,2,\ldots) \]

be the best approximation of a function \(f\in C_{2\pi}\) by trigonometric polynomials \(T_n(x)\) of order \(\le n\). We consider the problem of finding the exact upper bound

\[ E_n(\mathfrak{M})_X=\sup_{f\in\mathfrak{M}} E_n(f)_X, \]

where \(\mathfrak{M}\) is a certain class of functions.

Denote by \(W^rH_\omega\) \((r=0,1,2,\ldots;\ W^0H_\omega=H_\omega)\) the class of \(r\)-times continuously differentiable functions \(f(x)\in C_{2\pi}\) for which

\[ |f^{(r)}(x')-f^{(r)}(x'')|\le \omega(|x'-x''|) \qquad (f^{(0)}=f), \]

where \(\omega(t)\) is a given modulus of continuity. For \(\omega(t)=Kt^\alpha\) \((0\le t\le \pi,\ 0<\alpha\le 1)\) we shall write \(W^rKH^\alpha\). Note that the class \(W^{r-1}KH^1\) for \(r=1,2,\ldots\) coincides with the class \(W^rK\) of functions \(f\in C_{2\pi}\) for which the \((r-1)\)-st derivative is absolutely continuous and \(|f^{(r)}(x)|\le K\) almost everywhere.

For a given modulus of continuity \(\omega(t)\) and all \(n=1,2,\ldots\) and \(r=0,1,2,\ldots\), consider the functions \(f_{nr}(\omega,x)\) of period \(2\pi/n\) with mean value over a period equal to zero, whose derivative of order \(r\) is odd and is defined by the equalities

\[ f_{nr}^{(r)}(\omega,x)= \begin{cases} \frac12\,\omega(2x), & 0\le x\le \pi/2n,\\[4pt] \frac12\,\omega(2\pi/n-2x), & \pi/2n\le x\le \pi/n. \end{cases} \]

It is easy to verify that if \(\omega(t)\) is a convex upward modulus of continuity, then \(f_{nr}(\omega,x)\in W^rH_\omega\).

The principal content of the present note is the following.

Theorem 1. Whatever convex upward modulus of continuity \(\omega(t)\) may be, for all \(r=0,1,2,\ldots\) the equalities

\[ E_{n-1}(W^rH_\omega)_X = E_{n-1}(f_{nr}(\omega,x))_X = \|f_{nr}(\omega,x)\|_X \qquad (n=1,2,\ldots), \tag{1} \]

hold, where \(X\) is the metric \(C\) or \(L\).

We note that, in a number of special cases, the values of the upper bounds \(E_n(W^rH_\omega)_X\) were obtained earlier. It was proved by J. Favard \((^1)\), and also by N. I. Akhiezer and M. G. Krein \((^2)\), that

\[ E_{n-1}(W^rKH^1)_C = E_{n-1}(W^{r+1}K)_C = \frac{4K}{\pi n^{r+1}} \sum_{\nu=0}^{\infty} \frac{(-1)^{\nu r}}{(2\nu+1)^{r+2}} \tag{2} \]

\[ (r=0,1,2,\ldots;\ n=1,2,\ldots). \]

Best approximation in the mean on classes of functions was first considered in a work of S. M. Nikol’skii \((^3)\), where, by means of general theo-

... concerning approximation in a Banach space, a number of definitive results in the metric \(L\) have been obtained. Exact estimates of approximation in the mean are contained in the papers \((^{4,5})\) and \((^{6})\), where the values of \(E_n(W^rK)_L\) were computed \((r=1,2,\ldots)\).

For an arbitrary convex modulus of continuity \(\omega(t)\), the values of the upper bounds \(E_n(W^rH_\omega)_C\) for \(0\le r\le 3\) are given in \((^{7-10})\), and the proof was based on exact estimates, found by the author, of intermediate approximation of functions of the class \(W^rH_\omega\) by functions from \(W^rKH^1\), on the optimal choice of the constant \(K\), and on equalities (2).

It was natural, however, to try to solve the problem of computing the upper bounds \(E_n(W^rH_\omega)_C\) and \(E_n(W^rH_\omega)_L\) by a “direct” method, i.e., using only properties of best approximations in the metrics \(C\) and \(L\). The author found a proof of the equalities (1) for \(X=C\) and \(L\) and for all \(r=0,1,2,\ldots\), based on the relations obtained by S. M. Nikol’skii \((^3)\):

\[ E_{n-1}(f)_C=\sup_{h\in H_L^n}\int_0^{2\pi} f(t)h(t)\,dt,\qquad E_{n-1}(f)_L=\sup_{h\in H_M^n}\int_0^{2\pi} f(t)h(t)\,dt, \tag{3} \]

where \(H_L^n\) \((H_M^n)\) are the sets of functions \(h(t)\) of period \(2\pi\), orthogonal to all trigonometric polynomials of degree \(n-1\) and such that
\(\|h\|_L\le 1\) \((\|h\|_M=\sup_x |h(x)|\le 1)\). An essential role here was played by the investigation—of independent interest—of the properties of functions that are periodic integrals of \(h(t)\), i.e., of the form

\[ g(x)=\frac{1}{\pi}\int_0^{2\pi}\sum_{k=1}^{\infty} \frac{\cos[k(t-x)+r\pi/2]}{k^r}\,h(t)\,dt \qquad (r=1,2,\ldots), \tag{4} \]

where \(h\in H_L^n\) or \(h\in H_M^n\). (The sets of such functions will be denoted respectively by \(W^rH_L^n\) and \(W^rH_M^n\).) In order to formulate the results obtained in this direction, we introduce the following definition.

We shall call a function \(\varphi(x)\), continuous on the whole axis, simple if it is equal to zero outside some interval \((\alpha,\beta)\), \(|\varphi(x)|>0\) for \(x\in(\alpha,\beta)\), and for every \(y\), \(0<y<\max_x|\varphi(x)|\), the equation \(|\varphi(x)|-y=0\) has exactly two roots on \((\alpha,\beta)\). If \(g\in C_{2\pi}^1\) (i.e., \(g\in C_{2\pi}\) and has a continuous first derivative), and \(g(x_0)=0\), but \(g\ne 0\), then \(g(x)\) can be represented on \([x_0,x_0+2\pi]\) in the form of a finite or countable sum

\[ g(x)=\sum_k \varphi_k(x)\qquad (x_0\le x\le x_0+2\pi), \]

where \(\varphi_k\) are simple functions such that \(|\varphi_k(x)|>0\) on the interval \((\alpha_k,\beta_k)\subset (x_0,x_0+2\pi)\), \(\varphi_k=0\) outside \((\alpha_k,\beta_k)\), and

\[ \sum_k \|\varphi_k\|_C=\frac12\,\bigvee_0^{2\pi}(g),\qquad \sum_k \|\varphi_k\|_L=\|g\|_L. \]

Let \(\bar\varphi_k(x)\) be the rearrangement of the functions \(|\varphi_k(x)|\) in decreasing order, i.e., the function inverse to \(x=M(y)\), where \(M(y)\) is the measure of the set on which \(|\varphi_k(x)|\ge y\) (see, for example, \((^{11})\), p. 332). We shall put \(\bar\varphi_k(x)=0\) for \(x>\beta_k-\alpha_k\). Denoting

\[ \Phi(g,x)=\sum_k \bar\varphi_k(x)\qquad (0\le x\le 2\pi), \]

to each function \(g(x)\) we uniquely assign a function \(\Phi(g,x)\), moreover
\[ \Phi(g(t+a),x)=\Phi(g(t),x) \]
for any \(a\).

Of importance in the study of the properties of functions of the form (4) is

Theorem 2. If \(g\in C_{2\pi}^{3}\), \(g(0)=0\), then
\[ |\Phi'(g,x)|\leq {1\over 4}\int_0^x \Phi(g'',t)\,dt \tag{5} \]
everywhere on \((0,2\pi)\), where \(\Phi'(g,x)\) exists. The inequality is sharp on the set \(C_{2\pi}^{3}\).

Let, for each \(n=1,2,\ldots\), for \(0\leq x\leq \pi/n\),
\[ \Phi_{n1}(x)={1\over 2},\qquad \Phi_{nr}(x)={1\over 2}\int_0^{\pi/n-x}\Phi_{n,r-1}(t)\,dt \quad (r=2,3,\ldots). \]

It is easy to verify that for \(r\geq 2\)
\[ \Phi_{nr}(x)=\Phi(g_{nr},x)\quad (0\leq x\leq \pi/n), \]
where \(g_{nr}(x)\) is the \((r-1)\)-st periodic integral of the function
\[ g_{n1}(x)={1\over 4n}\operatorname{sign}\sin nx. \]

With the aid of inequality (5) one proves

Theorem 3. If \(g\in W^{r}H_L^{n}\) \((g\in W^{r}H_M^{n})\) \((n=1,2,\ldots;\ r=2,3,\ldots)\), then \(\Phi'(g,x)\geq \Phi'_{nr}(x)\) (respectively \(\Phi'(g,x)\geq 4n\Phi_{n,r+1}(x)\)) at all points of the interval \((0,\pi/n)\) at which \(\Phi'(g,x)\) exists.

Theorem 2 makes it possible to obtain an exact upper estimate for the integral
\[ \int_0^{2\pi} f(t)g(t)\,dt, \]
where \(f\in H_\omega\), and \(g\in W^{r}H_L^{n}\) or \(g\in W^{r}H_M^{n}\), for \(r=1,2,\ldots\).

In the case \(r=0\), an analogous result is obtained by passing to Steklov functions. Taking into account relations (3), this gives Theorem 1. We note that the values of the upper bounds \(E_{n-1}(W^{r}H_\omega)_X\) can be written with the aid of the functions \(\Phi_{nr}\):
\[ E_{n-1}(H_\omega)_C=\Phi_{n1}\!\left({\pi\over n}\right)\omega\!\left({\pi\over n}\right), \]
\[ E_{n-1}(W^{r}H_\omega)_C ={1\over 2}\int_0^{\pi/n}\omega(t)\Phi_{nr}\!\left({\pi\over n}-t\right)\,dt \quad (r=1,2,\ldots), \]
\[ E_{n-1}(W^{r}H_\omega)_L =2n\int_0^{\pi/n}\omega(t)\Phi_{n,r+1}\!\left({\pi\over n}-t\right)\,dt \quad (r=0,1,\ldots). \]

In conclusion we note that equalities (1) give upper estimates for the widths of the classes \(W^{r}H_\omega\) in the spaces of continuous and summable functions. (In the case \(\omega(t)=Kt\), \(r=0,1,2,\ldots\), and for convex \(\omega(t)\) with \(r\leq 3\), the widths of these classes in the space of continuous functions with the uniform metric were computed in \((^{12},\,^{10})\).)

Dnepropetrovsk State University
named after the 300th anniversary of the reunification of Ukraine with Russia

Received
21 X 1969

CITED LITERATURE

\({}^{1}\) J. Favard, Bull. Sci. Math., 61, 209 (1937). \({}^{2}\) N. I. Akhiezer, M. G. Krein, DAN, 15, 107 (1937). \({}^{3}\) S. M. Nikol’skii, Izv. AN SSSR, Ser. Mat., 10, 207 (1946). \({}^{4}\) L. V. Taikov, DAN, 163, 301 (1965). \({}^{5}\) L. V. Taikov, Tr. Matem. Inst. im. V. A. Steklova AN SSSR, 88, 61 (1967). \({}^{6}\) S. P. Turovets, Dokl. AN UkrSSR, No. 5, 417 (1968). \({}^{7}\) N. P. Korneichuk, DAN, 140, 748 (1961). \({}^{8}\) N. P. Korneichuk, DAN, 141, 304 (1961). \({}^{9}\) N. P. Korneichuk, Izv. AN SSSR, Ser. Mat., 27, 29 (1963). \({}^{10}\) N. P. Korneichuk, DAN, 150, 1218 (1963). \({}^{11}\) G. G. Hardy, J. E. Littlewood, G. Pólya, Inequalities, Moscow, 1948. \({}^{12}\) V. M. Tikhomirov, UMN, 15, no. 3 (93), 81 (1960).