D. L. BERMAN

ON THE IMPOSSIBILITY OF CONSTRUCTING A LINEAR POLYNOMIAL OPERATOR GIVING AN APPROXIMATION OF THE ORDER OF THE BEST APPROXIMATION

(Presented by Academician A. N. Kolmogorov on 13 II 1958)

Mathematics

1°. For definiteness we shall first consider the space \(\widetilde C\) of all continuous \(2\pi\)-periodic functions \(f(x)\) with norm

\[ \|f\|_{\widetilde C}=\max_{0\le x<2\pi}|f(x)|. \]

Put \(f_t(x)=f(x+t)\).

It is known that for every \(f\in\widetilde C\) there exists a unique polynomial of best approximation of order \(n\), which we shall denote by \(r_n(f,x)\). The polynomial \(r_n(f,x)\) may be regarded as an operator that assigns to each \(f\in\widetilde C\) its polynomial of best approximation of order \(n\). The operator \(r_n(f,x)\) is nonlinear.

At present there is as yet no method for computing polynomials of best approximation; therefore the following question is natural:

Does there exist a linear operator \(U_n(f,x)\) possessing the properties:

1) \(U_n(f,x)\) maps \(\widetilde C\) into \(\widetilde C\);

2) for every \(f\in\widetilde C\), \(U_n(f,x)\) is a polynomial of order \(\le n\);

3) for every \(f\in\widetilde C\),

\[ \|f(x)-U_n(f,x)\|=O(E_n), \]

where \(E_n(f)=E_n=\|f(x)-r_n(f,x)\|\)?

In the present note it is proved that this question must be answered in the negative.

2°. Theorem 1. There does not exist a linear operation \(U_n(f,x)\) satisfying conditions 1)—3).

Proof. Suppose that there exists an operation \(U_n(f,x)\) satisfying conditions 1)—3). Then from condition 3), taking into account that \(E_n(f)=0\) when \(f\) is a trigonometric polynomial of order not exceeding \(n\), it follows that

\[ U_n(f,x)\equiv f(x), \tag{1} \]

when \(f\) is a trigonometric polynomial of order not exceeding \(n\).

But it is known \((^1)\) that for every linear operation satisfying conditions 1)—2) and equality (1), the equality

\[ \frac{1}{2\pi}\int_0^{2\pi} U_n(f_t,x-t)\,dt=S_n(f,x), \tag{2} \]

holds, where \(S_n(f,x)\) is the partial sum of order \(n\) of the Fourier series of the function \(f(x)\).

It is easy to see that

\[ \frac{1}{2\pi}\int_0^{2\pi} (f_t,\ x-t)\,dt=f(x), \tag{3} \]

where \((f,x)\) is the value of the function \(f\) at the point \(x\).

From (2) and (3) it follows that

\[ \frac{1}{2\pi}\int_0^{2\pi} \bigl(U_n(f_t)-f_t,\ x-t\bigr)\,dt = S_n(f,x)-f(x). \tag{4} \]

Since the operator \(U_n(f,x)\) satisfies condition 3), there exists a constant \(C\) such that

\[ \|U_n(f_t)-f_t\|\leq C E_n(f_t),\qquad -\infty<t<\infty. \tag{5} \]

We note that

\[ E_n(f_t)=E_n(f),\qquad -\infty<t<\infty. \]

Therefore inequality (5) takes the form

\[ \|U_n(f_t,x)-f_t(x)\|\leq C E_n(f),\qquad -\infty<t<\infty, \]

and then from equality (4) it follows that

\[ |S_n(f,x)-f(x)|\leq C E_n(f),\qquad -\infty<x<\infty. \tag{6} \]

Inequality (6) contradicts the classical result of du Bois-Reymond from the theory of Fourier series. Theorem 1 is proved.

An analogous theorem can be proved in the case of the space \(C\) of all functions \(f(x)\) continuous on the segment \([-1,1]\), with norm \(\|f\|=\max_{-1\leq x\leq 1}|f(x)|\).

\(3^\circ\). In \((^2)\) it is proved that formula (2) is valid for functional spaces of type \(E\), a particular case of which is the space \(\widetilde C\). Therefore Theorem 1 admits the following generalization:

Theorem 2. Let a functional space of type \(E\) be such that, for at least one function \(f\in E\), the relation

\[ \varlimsup_{n\to\infty}\|f-S_n(f)\|=\infty \]

holds. Then there does not exist a linear operation \(U_n(f,x)\) satisfying the conditions:

1) \(U_n(f,x)\) maps \(E\) into \(E\);
2) for every \(f\in E\), \(U_n(f,x)\) is a polynomial of order \(\leq n\);
3) for every \(f\in E\),

\[ \|U_n(f)-f\|_E=O(E_n(f)), \]

where

\[ E_n(f)=\inf_{T_n\in\Pi_n}\|f-T_n\|_E \]

and \(T_n\) ranges over the set \(\Pi_n\) of all trigonometric polynomials of order \(\leq n\).

The space \(\widetilde L\) of all summable \(2\pi\)-periodic functions with norm

\[ \|f\|=\frac{1}{2\pi}\int_0^{2\pi}|f(t)|\,dt \]

is a functional space of type \(E\). Moreover, it is known that there exist functions \(f\in \widetilde L\) satisfying the condition

\[ \varlimsup_{n\to\infty}\int_0^{2\pi}|f-S_n(f)|\,dx=\infty . \]

Therefore, from Theorem 2 it follows:

Theorem 3. There does not exist a linear operation satisfying the conditions:

1) \(U_n(f,x)\) maps \(\widetilde L\) into \(\widetilde L\);

2) for every \(f\in \widetilde L\), \(U_n(f,x)\) is a trigonometric polynomial of order \(\le n\);

3) for every \(f\in \widetilde L\),

\[ \int_0^{2\pi}|f(x)-U_n(f,x)|\,dx=O(E_n(f)), \]

where

\[ E_n=E_n(f)=\inf_{T_n\in \Pi_n}\frac1{2\pi}\int_0^{2\pi}|f(x)-T_n(x)|\,dx . \]

Remark. It is known \((^3)\) that formula (2) generalizes to the case of functions of many variables. In view of this, Theorem 2 can also be extended to the case of functions of many variables.

\(4^\circ\). It is well known that, for any fixed \(p\) satisfying the inequalities \(0<p<1\), one can construct a linear polynomial operator having properties 1) and 2) and satisfying, for every \(f\in \widetilde C\), the condition

\[ \|f(x)-U_n(f,x)\|=O(E_{[pn]}(f)), \]

where \([pn]\) is the integer part of the number \(pn\) \((^4,^5)\).

Therefore, in connection with Theorem 1 the following problem arises.

Let an arbitrary sequence of positive numbers \(\{p_n\}_{n=1}^{\infty}\) be given, satisfying the inequalities \(0<p_n\le 1\), \(n=1,2,\ldots\), and let \(\lim_{n\to\infty}p_n=1\). The question is whether one can construct a linear polynomial operator \(U_n(f,x)\) having properties 1) and 2) and satisfying, for every \(f\in \widetilde C\), the condition

\[ 3')\quad \|f(x)-U_n(f,x)\|=O(E_{[p_n n]}(f)). \]

The solution of this problem is given by the theorem:

Theorem 4. There does not exist a linear operation \(U_n(f,x)\) satisfying conditions 1)—2) and \(3'\)).

In connection with Theorem 3 the following theorem is appropriate:

Theorem 5. For any fixed \(f\in \widetilde C\) one can construct such a linear polynomial operator \(U_n(f,x)\) of order \(n\) that

\[ |f(x)-U_n(f,x)|\le E_n(f),\qquad n=0,1,2,\ldots,\quad -\infty<x<\infty . \tag{7} \]

This theorem follows directly from Marcinkiewicz’s theorem \((^6)\), according to which, for any \(f\in \widetilde C\), one can find such a matrix of interpolation nodes that the corresponding Lagrange interpolation process satisfies inequality (7).

Novgorod State
Pedagogical Institute

Received
2 I 1958

CITED LITERATURE

\(^1\) D. L. Berman, DAN, 85, No. 1 (1952).
\(^2\) D. L. Berman, DAN, 88, No. 1 (1953).
\(^3\) D. L. Berman, DAN, 91, No. 6 (1953).
\(^4\) S. N. Bernstein, Izv. AN SSSR, OMEN, No. 9, 1151 (1931).
\(^5\) D. L. Berman, DAN, 109, No. 4 (1956).
\(^6\) I. P. Natanson, Constructive Theory of Functions, 1949.