Full Text
D. L. BERMAN
ON THE THEORY OF LINEAR POLYNOMIAL OPERATIONS
(Presented by Academician S. N. Bernstein on 8 II 1963)
$1^\circ$. Denote by $C$ the space of all continuous functions $f(x)$ defined on the segment $[-1,1]$, with norm
\[
\|f\|=\max_{-1\le x\le 1}|f(x)|.
\]
Denote by $\Pi_n$ the set of all algebraic polynomials of degree $\le n$. Denote by $\mathfrak M_n$ the set of all linear operations $V_n$ from $C$ into $C$ that map $C$ into $\Pi_n$. The following theorem is known.
Theorem 1. In order that the sequence $\{V_n(f)\}_{n=1}^{\infty}$, $V_n\in\mathfrak M_n$, satisfy, for every $f\in C$, the relation
\[
\|V_n(f)-f\|\to 0,\qquad n\to\infty;
\tag{1}
\]
it is necessary and sufficient that the following conditions hold: 1) for every polynomial relation (1) is fulfilled; 2) the norms of the operators $V_n$ are bounded in the aggregate
\[
\|V_n\|\le C,\qquad n=1,2,\ldots,
\]
where the constant $C$ does not depend on $n$.
Verification of condition 2) is usually associated with great difficulties. Therefore one tries to replace it by other conditions. Thus, for example, if $V_n$ is a method of summation of a Fourier–Chebyshev series,
\[
V_n(f)=V_n(f,x,\lambda)=\frac{a_0\lambda_0^{(n)}}{2}
+\sum_{k=1}^{n} a_k\lambda_k^{(n)}T_k(x),
\tag{2}
\]
\[
a_k=\frac{2}{\pi}\int_0^\pi f(\cos t)\cos kt\,dt,
\]
then condition 2) is replaced by the condition (${ }^{1,2}$)
\[
\int_0^\pi\left|\frac{\lambda_0^{(n)}}{2}
+\sum_{k=1}^{n}\lambda_k^{(n)}\cos kt\right|\,dt\le C,\qquad n=1,2,\ldots .
\]
Not every operator from $\mathfrak M_n$ is an operator of the form (2). Therefore the question arises of replacing, in the case of arbitrary linear polynomial operations $V_n$, condition 2) by other conditions which, together with condition 1), are necessary and sufficient for relation (1) to hold for every $f\in C$. In the present note this question is studied in the nonperiodic case.
$2^\circ$. Denote by $C^\wedge$ the space consisting of all even $2\pi$-periodic functions $\varphi(\theta)$ with norm
\[
\|\varphi\|=\max_{0\le \theta\le \pi}|\varphi(\theta)|.
\]
Between the spaces $f\in C$ and $\varphi\in C^\wedge$ one can establish a one-to-one correspondence according to the formulas
\[
\varphi(\theta)=f(\cos\theta),\qquad 0\le \theta\le \pi;
\]
\[
f(x)=\varphi(\arccos x),\qquad -1\le x\le 1,
\]
where $f\in C$ and $\varphi\in C^\wedge$. It is clear that $\|f\|=\|\varphi\|$. As is known (1), $\varphi$ is called the function induced by the function $f$. To preserve the connection in notation between the function itself and the induced function, we shall use
will be denoted by the sign \(\hat{\ }\). Thus,
\[ \hat f(\theta)=f(\cos\theta),\qquad f(x)=\hat f(\arccos x). \]
Following Faber \((^3)\) and Fejér \((^4)\), we define the shift \(T_h\) of a function \(\varphi\in C^{\hat{\ }}\) by \(h\) according to the formula
\[ T_h\varphi=T_h^\theta\varphi=\frac{\varphi(\theta+h)+\varphi(\theta-h)}{2}. \tag{3} \]
The operator (3) has the following obvious properties: 1) from \(\varphi\in C^{\hat{\ }}\) it follows that, for every \(-\infty<h<\infty\), \(T_h\varphi\in C^{\hat{\ }}\); 2) \(\|T_h\varphi\|\leq \|\varphi\|\); 3) if \(\varphi\) is a cosine polynomial of order \(n\), then \(T_h^\theta\varphi\) is also a cosine polynomial of order \(n\), both with respect to \(\theta\) and with respect to \(h\).
By the shift of a function \(f\in C\) we mean the shift of the induced function. Thus,
\[ T_h f=T_h^\theta f=\frac{\hat f(\theta+h)+\hat f(\theta-h)}{2}. \]
\(3^\circ\). Let \(V_n\in\mathfrak M_n\). Introduce the operator
\[ \widetilde V_n^{\hat{\ }}(f,\theta)=\frac{1}{\pi}\int_0^{2\pi} T_h^\theta V_n^{\hat{\ }}(T_h^\theta f)\,dh, \tag{4} \]
where \(h\) is considered as a parameter, and \(\theta\) as the independent variable. The operator \(\widetilde V_n\) has the following properties: 1) if \(V_n\in\mathfrak M_n\), then \(\widetilde V_n\in\mathfrak M_n\); 2) \(\|\widetilde V_n\|\leq 2\|V_n\|\).
Theorem 2. Let \(V_n\in\mathfrak M_n\); then, for every \(f\in C\), the equality
\[ \widetilde V_n^{\hat{\ }}(\hat f,\theta)=\frac{1}{\pi}\int_0^\pi \hat f(h)\widetilde V_n^{\hat{\ }}(T_h^\theta D_n)\,dh, \tag{5} \]
holds, where \(D_n(t)\) is the Dirichlet kernel of order \(n\).
We indicate the proof of this theorem. It is known that the partial sum of order \(n\) of the Fourier series of the function \(\hat f\) can be represented in the form
\[ S_n(\hat f,\theta)=\frac{1}{\pi}\int_0^\pi \hat f(h)T_h^\theta D_n\,dh. \tag{6} \]
Since \(\widetilde V_n\in\mathfrak M_n\), according to (5) we have
\[ \widetilde V_n^{\hat{\ }}(\hat f)=\widetilde V_n^{\hat{\ }}(S_n(\hat f)). \tag{7} \]
It follows from (6) and (7) that
\[ \widetilde V_n^{\hat{\ }}(\hat f,\theta) =\frac{1}{\pi}\widetilde V_n^{\hat{\ }}\left(\int_0^\pi \hat f(h)T_h^\theta D_n\,dh\right). \tag{8} \]
It is easy to verify the validity of interchanging the operator and the integral. Therefore (5) follows from (8).
Theorem 3. Let \(V_n\in\mathfrak M_n\); then
\[ \|V_n\|\geq \frac{L_n}{2},\qquad L_n=\max_{0\leq\theta\leq\pi}\frac{1}{\pi}\int_0^\pi \left|\widetilde V_n^{\hat{\ }}(T_h^\theta D_n)\right|\,dh. \tag{9} \]
Proof. According to Theorem 2, \(\widetilde V^{\hat{\ }}(\hat f,\theta)\) is an integral operator from \(C^{\hat{\ }}\) into \(C^{\hat{\ }}\) with kernel \(\widetilde V^{\hat{\ }}(T_h^\theta D_n)\). Using a general theorem of functional analysis concerning integral operators, we obtain that
\[ \|\widetilde V_n^{\hat{\ }}\|=L_n. \tag{10} \]
Since \(\|V_n\| \geqslant \|\widetilde V_n^{\wedge}\|/2\), (9) follows from (10).
From Theorem 3 there follows
Theorem 4. In order that the sequence \(\{V_n(f)\}_{n=1}^{\infty}\), \(V_n \in \mathfrak M_n\), satisfy relation (1) for every \(f \in C\), it is necessary that relation (1) hold for every monomial and that
\[ L_n \leqslant C,\qquad n=1,2,\ldots, \]
where the constant \(C\) does not depend on \(n\).
It is curious that the necessary conditions indicated in Theorem 4 are close to sufficient conditions. This is seen from Theorem 5.
Theorem 5. Let the sequence \(\{V_n(f)\}_{n=1}^{\infty}\), \(\mathfrak M \ni V_n\), be such that for every monomial relation (1) holds, and let
\[ L_n \leqslant C,\qquad n=1,2,\ldots, \tag{11} \]
where the constant \(C\) does not depend on \(n\). Then for every \(f \in C\) the equality holds
\[ \lim_{n\to\infty}\left\|\widetilde V_n^{\wedge}(f^{\wedge})-f^{\wedge}-\frac1\pi\int_0^\pi f^{\wedge}(t)\,dt\right\|=0. \tag{12} \]
Proof. From (10) and (11) it follows that the norms of the operators \(\widetilde V_n^{\wedge}\) are uniformly bounded. Therefore, by Theorem 1, it remains only to prove that equality (12) holds for any function \(T_m^{\wedge}(\theta)=\cos m\theta\). Let us first verify that equality (12) holds for \(\cos m\theta\), where \(m>0\).
From the definition of \(\widetilde V_n^{\wedge}\) we have
\[ \widetilde V_n^{\wedge}(\cos m\theta) = \frac1\pi\int_0^{2\pi} T_h^\theta V_n^{\wedge}(\cos m\theta)\cos mh\,dh. \tag{13} \]
It is easy to see that
\[ \cos m\theta = \frac1\pi\int_0^{2\pi} T_h^\theta \cos m\theta \cos mh\,dh. \tag{14} \]
From equalities (13) and (14) we conclude that
\[ \left\|\widetilde V_n^{\wedge}(\cos m\theta)-\cos m\theta\right\| \leqslant 2\left\|V_n^{\wedge}(\cos m\theta)-\cos m\theta\right\|. \]
Since, by assumption, the right-hand side tends to zero as \(n\to\infty\), we have
\[ \lim_{n\to\infty}\left\|\widetilde V_n^{\wedge}(\cos m\theta)-\cos m\theta\right\|=0, \]
and, for \(f^{\wedge}(\theta)=\cos m\theta\), \(m>0\), this equality is equivalent to (12).
Consider now the case when \(f=1\). We have
\[ \widetilde V_n^{\wedge}(1)-2 = \frac1\pi\int_0^{2\pi}\bigl[T_h^\theta V_n^{\wedge}(1)-T_h^\theta 1\bigr]\,dh. \]
Consequently,
\[ \left\|\widetilde V_n^{\wedge}(1)-2\right\| \leqslant 2\left\|V_n^{\wedge}(1)-1\right\|. \tag{15} \]
By the hypothesis of the theorem, as \(n\to\infty\) the right-hand side of inequality (15) tends to zero. Therefore from (15) follows the equality
\[ \lim_{n\to\infty}\left\|\widetilde V_n^{\wedge}(1)-2\right\|=0, \]
which is the particular case of equality (12) when \(f=1\).
Leningrad Institute of Soviet Trade
named after Fr. Engels
Received
8 II 1963
REFERENCES
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- D. L. Berman, DAN, 85, No. 1 (1952).