N. P. KORNEICHUK

ON THE EXISTENCE OF A LINEAR POLYNOMIAL OPERATOR GIVING THE BEST APPROXIMATION ON A CLASS OF FUNCTIONS

(Presented by Academician A. N. Kolmogorov on 3 XI 1961)

1. Let \(C_{2\pi}\) be the space of continuous functions of period \(2\pi\) with norm \(\|f\|=\max_x |f(x)|\), and let the operator \(U_n\), for fixed \(n=0,1,2,\ldots\), assign to each function \(f\in C_{2\pi}\) a trigonometric polynomial \(U_n(f,x)\) of degree not exceeding \(n\). Let \(\mathfrak M\) be some set of functions in the space \(C_{2\pi}\). We consider the question of the existence of a linear operator \(U_n\) such that

\[ \sup_{f\in\mathfrak M}\|U_n(f,x)-f(x)\|=\sup_{f\in\mathfrak M}E_n(f), \tag{1} \]

where \(E_n(f)\) is the best uniform approximation of the function \(f\in C_{2\pi}\) by trigonometric polynomials of degree not exceeding \(n\).

It is known \({}^{(1-4)}\) that if \(\mathfrak M\) is the class \(W^{(r)}M\) \((r>0)\) of functions \(f\) for which \(|f^{(r)}(x)|\le M\), then for any \(n=0,1,\ldots\) there exists a linear polynomial operator \(U_n\) for which (1) holds; that is, on this class there exists a best linear method of approximation. For integral \(r\) such a method is furnished, with a suitable choice of the multipliers \(\lambda_k^{(n)}\), by the sequence of polynomials

\[ U_n(f,x)=U_n(f,x,\lambda)=\frac{a_0}{2}+\sum_{k=1}^{n}\lambda_k^{(n)}(a_k\cos kx+b_k\sin kx), \tag{2} \]

\[ n=0,1,2,\ldots, \]

where \(a_0,a_k,b_k\) are the Fourier coefficients of the function \(f\).

At the same time \({}^{(5)}\), on the class \(H_\omega\) of functions of the space \(C_{2\pi}\) whose modulus of continuity \(\omega(f,t)\) does not exceed a given concave upward modulus of continuity \(\omega(t)\), the polynomials (2), for no choice of \(\lambda_k^{(n)}\), can realize equality (1), provided only that \(\omega(t)\) is nonlinear on the interval \(\left[0,\frac{\pi}{n+1}\right]\). In particular, this will be the case if \(\omega(t)=Kt^\alpha\) \((0<\alpha<1,\ 0\le t\le \pi)\).

We prove here that, for the class \(H_\omega\) under the stated assumptions on \(\omega(t)\), the answer to the question posed remains negative, whatever the linear operators \(U_n\) may be.

Let us first note that if some operator \(U_n\) realizes equality (1) on the set \(\mathfrak M\), then, for any function \(f_0\in\mathfrak M\) such that

\[ E_n(f_0)=\sup_{f\in\mathfrak M}E_n(f), \]

the polynomial \(U_n(f_0,x)\) must identically coincide with the polynomial of best approximation of the function \(f_0\) (by virtue of the uniqueness of the latter).

Let \(f_0\) be an arbitrary function from \(H_\omega\) \((\omega(t)\) concave upward), and let \(x_1,x_2,\ldots,x_{2n+2}\) be points of the interval \([0,2\pi]\) at which the difference between \(f_0(x)\) and its polynomial of best approximation \(T_n(f_0,x)\) takes alternately the values \(\pm E_n(f_0)\). Denote by \(\theta\) the largest of the intervals \((x_k,x_{k+1})\) \((k=1,2,\ldots,2n+2;\ x_{2n+3}=x_1+2\pi)\); let it be, for-

for example, \((x_m, x_{m+1})\), and by \(\delta\) its length. The continuous function \(f_1(x)\), coinciding with \(f_0(x)\) outside \(\theta\) and linear on \(\theta\), belongs to the class \(H_\omega\). Let us next consider the function

\[ \psi(x)= \begin{cases} f_0(x), & (x\notin\theta),\\ \inf\limits_{t\in\theta}\left[f_0(t)+\omega\left(|x-t|\right)\right], & (x\in\theta). \end{cases} \]

It is not hard to verify that \(\psi\in H_\omega\), \(\psi(x)\) is convex upward on the interval \(\theta\), and, if \(\omega(t)\) is nonlinear on the segment \([0,\delta]\), then \(\psi(x)-f_1(x)>0\) for all \(x_m<x<x_{m+1}\).

Denote by \(x_0\) the midpoint of the interval \((x_m,x_{m+1})\); note further that the function \(g(x)\), equal to \(f_0(x)\) outside \(\theta\), taking at the point \(x_0\) the value \(\psi(x_0)\), and linear on the intervals \([x_m,x_0]\) and \([x_0,x_{m+1}]\), also belongs to \(H_\omega\).

If \(E_n(f_0)=\sup\limits_{f\in H_\omega} E_n(f)\), then \(E_n(g)=E_n(f_1)=E_n(f_0)\), and for the functions \(f_0\), \(f_1\), and \(g\) the polynomial of best approximation is one and the same polynomial \(T_n(f_0,x)\).

Suppose now that for the operator \(\overline U_n\), for some fixed \(n\), the equality

\[ \sup_{f\in H_\omega}\left\|\overline U_n(f,x)-f(x)\right\|=\sup_{f\in H_\omega}E_n(f) \tag{3} \]

is fulfilled. Then, in view of the remark made above,

\[ \overline U_n(g,x)\equiv \overline U_n(f_1,x)\equiv T_n(f_0,x). \tag{4} \]

Let us show that the operator \(\overline U_n\) cannot be linear. Indeed, let \(g(x_0)-f_1(x_0)=\beta\), where \(\beta>0\). Put

\[ \varphi(x)=\frac{1}{\beta}\,\omega\!\left(\frac{1}{2}\delta\right)\,[g(x)-f_1(x)]. \]

It is clear that \(\varphi\in H_\omega\), \(\max_x|\varphi(x)|=\omega\!\left(\frac{1}{2}\delta\right)\), and, if we assume \(\overline U_n\) linear, then, by (4), \(\overline U_n(\varphi,x)\equiv0\).

Consequently, if \(\omega(t)\) is nonlinear on \(\left[0,\dfrac{\pi}{n+1}\right]\), then, taking into account the exact value of the upper bound of the best approximations on the class \(H_\omega\) \((^5)\), we obtain

\[ \left\|\overline U_n(\varphi,x)-\varphi(x)\right\| =\omega\!\left(\frac{1}{2}\delta\right) \geq \omega\!\left(\frac{\pi}{2(n+1)}\right) >\frac{1}{2}\omega\!\left(\frac{\pi}{n+1}\right) =\sup_{f\in H_\omega}E_n(f), \]

in contradiction with (3).

We note that the proof of the nonexistence of a linear operator \(\overline U_n\) realizing equality (3) can be carried out without relying on the relation

\[ \sup_{f\in H_\omega}E_n(f)=\frac{1}{2}\omega\!\left(\frac{\pi}{n+1}\right). \]

One could, for example, show that the assumption of the linearity of \(\overline U_n\) implies that \(\overline U_n(f,x)\equiv0\) for any \(f\in H_\omega\), which also contradicts (3).

If \(\omega(t)\) is convex upward and, for \(0\leq t\leq\gamma\) \((0<\gamma<\pi)\), \(\omega(t)=Kt\), then for sufficiently large \(n\)

\[ \sup_{f\in H_\omega}E_n(f)=\sup_{f\in KH^{(1)}}E_n(f), \]

where \(KH^{(1)}=W^{(1)}K\) is the class of functions \(f\in C_{2\pi}\) satisfying on the whole axis the Lipschitz condition of degree 1 with constant \(K\), and equality (3) will be realized by linear operators of the form (2).

Thus, the following assertion holds:

Theorem. In order that, for fixed \(n\), some linear polynomial operator \(\overline U_n\) on the class \(H_\omega\), where \(\omega(t)\) is convex upward, realize equality (3), it is necessary and sufficient that the function \(\omega(t)\) be linear on the interval \(\left[0,\dfrac{\pi}{n+1}\right]\).

If there exists a finite limit \(\lim_{t\to0+} \frac{\omega(t)}{t}=K>0\), then from the convexity of \(\omega(t)\) it follows that \(\omega(t)\leq Kt\) \((0\leq t\leq \pi)\) and \(H_\omega \subset KH^{(1)}\). Therefore, as is not difficult to see, there exists a sequence of linear operators \(\{\bar U_n\}\) of the form (2) such that

\[ \lim_{n\to\infty} \frac{ \sup_{f\in H_\omega}\|\bar U_n(f,x,\lambda)-f(x)\| }{ \sup_{f\in H_\omega} E_n(f) } = \lim_{n\to\infty} \frac{ \sup_{f\in KH^{(1)}} E_n(f) }{ \sup_{f\in H_\omega} E_n(f) } =1, \tag{5} \]

i.e. in this case one may speak of the existence of an asymptotically best linear method of approximation.

1. Let us consider the analogous problem for one class of functions continuous on an interval. Let \(C_{[-1,1]}\) be the space of functions continuous on the interval \([-1,1]\), with norm \(\|f\|=\max_{-1\leq x\leq 1}|f(x)|\); \(E_n(f;-1,1)\) is the best approximation of a function \(f\in C_{[-1,1]}\) by algebraic polynomials of degree \(\leq n\) on this interval, and \(P_n(f,x)\) is the polynomial of best approximation for \(f\) on \([-1,1]\).

The following assertion holds, which, incidentally, is also valid in the corresponding formulation for the periodic case.

Let \(f\in C_{[-1,1]}\), and suppose that at the points \(x_1,x_2,\ldots,x_{n+2}\) of the interval \([-1,1]\) the difference \(f(x)-P_n(f,x)\) alternately assumes the values \(\pm E_n(f;-1,1)\). Then there will be two neighboring points, for example \(x_m\) and \(x_{m+1}\), such that

\[ |f(x_{m+1})-f(x_m)|\geq 2E_n(f;-1,1). \tag{6} \]

Denote by \(H^{(\alpha)}_{[-1,1]}\) the class of functions \(f\in C_{[-1,1]}\) satisfying on the interval \([-1,1]\) the Lipschitz condition of order \(\alpha\) \((0<\alpha<1)\) with constant 1. It is known\({}^{5}\) that as \(n\to\infty\)

\[ \sup_{f\in H^{(\alpha)}_{[-1,1]}} E_n(f;-1,1) = \frac{\pi^\alpha}{2(n+1)^\alpha} + o\left(\frac1{n^\alpha}\right). \tag{7} \]

Suppose that for some \(n\) a function \(f_0\in H^{(\alpha)}_{[-1,1]}\) realizes the exact upper bound of the best approximations on this class. If \(n\) is sufficiently large, then, by (6) and (7), for the function \(f_0\) there exists an interval \((x_m,x_{m+1})\) between neighboring alternation points with length differing arbitrarily little from

\[ \frac{\pi}{n+1}. \]

Let the operator \(\bar V_n\), mapping the space \(C_{[-1,1]}\) onto the set of algebraic polynomials \(\bar V_n(f,x)\) of degree not exceeding \(n\), be such that

\[ \sup_{f\in H^{(\alpha)}_{[-1,1]}} \|\bar V_n(f,x)-f(x)\| = \sup_{f\in H^{(\alpha)}_{[-1,1]}} E_n(f;-1,1) = E_n(f_0;-1,1). \tag{8} \]

If one assumes \(\bar V_n\) to be linear, then for \(0<\alpha<1\), arguing as in the periodic case, one can construct a function \(\varphi\in H^{(\alpha)}_{[-1,1]}\) such that

\[ \max_{-1\leq x\leq 1}|\varphi(x)| = \left(\frac{\pi}{2(n+1)}\right)^\alpha + o\left(\frac1{n^\alpha}\right) \quad\text{and}\quad \bar V_n(\varphi,x)=0. \]

Taking (7) into account, we conclude that for sufficiently large \(n\)

\[ \|\bar V_n(\varphi,x)-\varphi(x)\|>E_n(f_0;-1,1), \]

i.e. the operator realizing equality (8) cannot be linear.

In conclusion, the author expresses sincere gratitude to S. M. Nikol’skii for posing the problem and for valuable advice.

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

Received
2 XI 1961

REFERENCES

\({}^{1}\) J. Favard, Bull. de 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. matem., 10, 207 (1946).
\({}^{4}\) V. K. Dzyadyk, Izv. AN SSSR, ser. matem., 23, 933 (1959).
\({}^{5}\) N. P. Korneichuk, DAN, 140, 748 (1961).