Mathematics

D. L. Berman

CLASSIFICATION OF POLYNOMIAL OPERATIONS

(Presented by Academician S. N. Bernstein on 2 February 1962)

\(1^\circ\). Let \(\widetilde C\) denote the space of all continuous \(2\pi\)-periodic functions \(f(x)\) with norm
\[ \|f(x)\|=\max_{0\le x<2\pi}|f(x)|; \]
let \(\Pi_n\) denote the set of all trigonometric polynomials of order \(\le n\); let \(M_n\) denote the set of all linear operations from \(\widetilde C\) to \(\widetilde C\) that map \(\widetilde C\) into \(\Pi_n\). Let \(U_n\in M_n\). Introduce the operator
\[ \widetilde U_n(f,x)=\frac{1}{2\pi}\int_0^{2\pi} U_n(f_t,x-t)\,dt, \]
where \(f_t(x)=f(x+t)\). It is easy to see that \(\widetilde U_n\) also belongs to \(M_n\).

\(2^\circ\). Let \(U_n^{(i)}\in M_n,\ i=1,2\). We say that \(U_n^{(1)}\) and \(U_n^{(2)}\) belong to one and the same class if \(\widetilde U_n^{(1)}=\widetilde U_n^{(2)}\). Thus the set \(M_n\) is partitioned into classes. The set of all classes will be denoted by \(\Sigma_n\). It is noteworthy that, under this classification, the partial sum of the Fourier series \(S_n(f,x)\) and the Lagrange interpolation polynomial \(L_n(f,x)\), constructed for an arbitrary system of nodes
\[ 0\le x_0<x_1<\cdots<x_{2n}<2\pi, \]
belong to one and the same class, since \(\widetilde L_n=\widetilde S_n\) \((^1)\). It is easy to see that the classes are pairwise disjoint.

Theorem 1. Between the set of all classes \(\Sigma_n\) and the set of all trigonometric polynomials of order not exceeding \(n\), \(\Pi_n\), there is a one-to-one correspondence. This mapping is an isomorphism if the composition of elements in \(\Sigma_n\) and \(\Pi_n\) is defined in the corresponding manner.

We give the main points of the proof. Let \(T_n\in\Pi_n\). Introduce the convolution \(\sigma_n\) with kernel \(T_n\):
\[ \sigma_n=\sigma_n(f,T_n,x)=\int_0^{2\pi} f(x+t)T_n(t)\,dt. \tag{1} \]
Obviously, \(\sigma_n\in M_n\), and therefore there exists a unique class \(K_n\in\Sigma_n\) containing \(\sigma_n\). To the polynomial \(T_n\) we assign the class \(K_n\). We denote this mapping of \(\Pi_n\) into \(\Sigma_n\) by \(\varphi\). Now map \(\Sigma_n\) into \(\Pi_n\). Let \(K'_n\) be any element of \(\Sigma_n\). To the class \(K'_n\) we assign the polynomial
\[ T'_n(t)=\widetilde U_n(D_n,-t), \]
where \(U_n\) is an arbitrary operator from \(K'_n\). Here \(D_n\) denotes the Dirichlet kernel of order \(n\). We denote the indicated mapping of \(\Sigma_n\) into \(\Pi_n\) by \(\psi\). We now show that \(\varphi\) is a one-to-one mapping. Let \(\varphi(T_n)=K_n\). According to the definition of \(\varphi\), this means that the convolution \(\sigma_n\in K_n\). One can prove that, for any \(U_n\in K_n\), the equalities
\[ \widetilde U_n=\widetilde \sigma_n=\sigma_n \]
hold. Note also that
\[ \widetilde\sigma_n(D_n,-t) =\int_0^{2\pi}D_n(-t+t_1)T_n(t_1)\,dt_1 =T_n(t). \]
Therefore, in accordance with the definition of the mapping \(\psi\), we have that

\(\psi(K_n)=T_n\). Suppose now that \(\psi(K_n^{(2)})=T_n^{(2)}\), where \(K_n^{(2)}\in \Sigma_n\) and \(T_n^{(2)}\in \Pi_n\). According to the definition of \(\psi\),

\[ T_n^{(2)}(t)=\widetilde U_n(D_n,-t),\qquad U_n\in K_n^{(2)}. \]

Consider the convolution

\[ \sigma_n^{(2)}(f,x)=\int_0^{2\pi} f(x+t)T_n^{(2)}(t)\,dt . \tag{1′} \]

Since \(\sigma_n^{(2)}=\widetilde\sigma_n^{(2)}=\widetilde U_n\), it follows that \(\sigma_n^{(2)}\in K_n^{(2)}\). According to the definition of \(\varphi\), this means that \(\varphi(T_n^{(2)})=K_n^{(2)}\). Consequently, \(\varphi\) is a one-to-one correspondence.

It is not difficult to verify that each class \(K_n\) contains only one convolution. Therefore the class containing the convolution \(\sigma\) is naturally denoted by \(K_\sigma\). Obviously, if the composition of two elements \(T_1\) and \(T_2\) of \(\Pi_n\) is defined as their sum \(T_1+T_2\), and the composition of two classes \(K_{\sigma_1}\) and \(K_{\sigma_2}\) is defined as the class containing the convolution \(\sigma_1+\sigma_2\), then the mapping \(\varphi\) becomes an isomorphism. It is curious that the mapping \(\varphi\) is an isomorphism also in the case when the composition of two elements \(T_i\in \Pi_n\), \(i=1,2\), is defined as their convolution \(T_1*T_2\), while the composition of the classes \(K_{\sigma_1}\) and \(K_{\sigma_2}\) is defined as the class containing the convolution \(\sigma_2\sigma_1\). Let us consider some class \(K_\sigma\) and pose the question of finding in \(K_\sigma\) an operator with the smallest norm. The answer to this question is given by Theorem 2.

Theorem 2. Among the various operations of the class \(K_\sigma\), the convolution \(\sigma\) has the smallest norm, i.e.

\[ \inf_{U_n\in K_\sigma}\|U_n\|=\|\sigma\|. \tag{2} \]

Proof. According to the property of the operator \(\widetilde U_n\), \(\|\widetilde U_n\|\leq \|U_n\|\) \((^2)\). It has already been noted that for any \(U_n\in K_\sigma\), \(\widetilde U_n=\widetilde\sigma=\sigma\). Therefore (2) is valid. It is useful to compare this theorem with the results in \((^4\text{--}^7)\).

\(3^\circ\). Theorem 2 suggests the validity of the following theorem:

Theorem 3. Let the sequence \(\{U_{n_j}(f,x)\}_{j=1}^{\infty}\), where \(U_{n_j}\in M_{n_j}\), \(j=1,2,\ldots\), converge uniformly for every \(f\in \widetilde C\),

\[ U_{n_j}(f,x)\to f(x),\qquad n_j\to \infty . \tag{3} \]

Then for every \(f\in \widetilde C\) the relation

\[ \widetilde U_{n_j}(f,x)\to f(x),\qquad n_j\to \infty . \tag{3′} \]

holds uniformly.

In proving Theorem 3 we use the following lemma:

Lemma. Let a sequence of linear operations from \(\widetilde C\) into \(\widetilde C\), \(\{U_n(f,x)\}_{n=1}^{\infty}\), satisfy, for every \(f\in \widetilde C\), the uniform relation (3). Then for every \(\varepsilon>0\) there exists an \(N\) such that, for every \(t\in[0,2\pi]\), the inequality

\[ \|U_n(f_t)-f_t\|<\varepsilon,\qquad n\geq N, \tag{4} \]

holds, where \(N\) does not depend on \(t\).

The proof of Theorem 3 is carried out as follows. From the definition of \(\widetilde U_n\) we obtain

\[ \widetilde U_{n_j}(f,x)-f(x)=\frac{1}{2\pi}\int_0^{2\pi}[U_{n_j}(f_t)-f_t,\ x-t]\,dt, \tag{5} \]

where \([f,x]\) is the value of the function \(f\) at the point \(x\). Hence we obtain

\[ \|\widetilde U_{n_j}(f)-f\|\leq \frac{1}{2\pi}\int_0^{2\pi}\|U_{n_j}(f_t)-f_t\|\,dt . \tag{6} \]

The justification of the validity of the transition from (5) to (6) is given in \((^2)\). From inequalities (4) and (6) we obtain that

\[ \|\widetilde U_{n_j}(f)-f\|<\varepsilon,\qquad n_j>N . \]

Since for every \(U_n\in K_\sigma\) one has \(\widetilde U_n=\sigma\), relation \((3')\) can be written in the form

\[ \sigma_{n_j}(f,x)\to f(x),\qquad n_j\to\infty . \]

Thus, Theorem 3 can be formulated in the following equivalent form:

Theorem 4. Let the sequence of operations \(\{U_{n_j}(f,x)\}_{j=1}^{\infty}\) converge uniformly for every \(f\in \widetilde C\). Then one can construct a sequence of convolutions of the form \((1')\), \(\{\sigma_{n_j}\}_{j=1}^{\infty}\), which also converges uniformly for every \(f\in \widetilde C\).

\(4^\circ\). Let a certain fixed class of operators \(K_n\in\Sigma_n\) be given, and let \(U_n\) be a certain fixed operator from \(K_n\). Denote by \(M_{n+m}(U_n)\) the set of all possible linear operators from \(\widetilde C\) into \(\widetilde C\) having the properties: 1) for every \(f\in\widetilde C\), \(U_{n+m}(f)\in\Pi_{n+m}\); 2) if \(f\in\Pi_n\), then \(U_{n+m}(f)=U_n(f)\) \((m\geq 0)\). A special case of such operations was considered in \((^3)\).

Theorem 5. Let \(U_{n+m}\in M_{n+m}(U_n)\); then for every \(f\in\widetilde C\) the equality holds

\[ \widetilde U_{n+m}(f,x)= \int_0^{2\pi} f(x+t)\left[\widetilde U_n(D_n,-t)+ \right. \]

\[ \left. +\frac{1}{2\pi}\sum_{k=n+1}^{n+m}(\alpha_k\cos kt+\beta_k\sin kt)\right]f(x+t)\,dt, \]

where

\[ \alpha_k=s_k^{(2)}+c_k^{(1)}, \]

\[ \beta_k=a_k s_k^{(1)}-c_k^{(2)}. \]

Here \(c_k^{(1)}\) and \(c_k^{(2)}\), \(s_k^{(1)}\) and \(s_k^{(2)}\) are the Fourier coefficients of order \(k\), respectively, for the functions \(U_{n+m}(\cos kz,x)\), \(U_{n+m}(\sin kz,x)\).

With the aid of Theorem 5 one proves

Theorem 6. For every class \(K_n\in\Sigma_n\) the equality holds

\[ \inf_{U_n\in K_n}\ \inf_{U_{n+m}\in M_{n+m}(U_n)} \|U_{n+m}\|= \]

\[ = \inf_{\alpha_k,\beta_k}\int_0^{2\pi} \left|\widetilde U_n(D_n,t)+ \sum_{k=n+1}^{n+m}(\alpha_k\cos kt+\beta_k\sin kt)\right|\,dt . \]

Leningrad Institute of Soviet Trade
named after F. Engels

Received
18 I 1962

CITED LITERATURE

\(^1\) J. Marcinkiewicz, Acta Litt. Sci. Szeged, 8, 127 (1937).
\(^2\) D. L. Berman, Izv. Vyssh. uchebn. zaved., Matematika, 4 (17) (1960).
\(^3\) D. L. Berman, DAN, 95, No. 2 (1954).
\(^4\) S. M. Lozinskii, DAN, 61, No. 2 (1948).
\(^5\) S. M. Lozinskii, DAN, 89, No. 5 (1953).
\(^6\) D. L. Berman, DAN, 85, No. 1 (1952).
\(^7\) D. L. Berman, DAN, 88, No. 1 (1953).