D. L. BERMAN

ON SOME LINEAR POLYNOMIAL OPERATIONS IN THE COMPLEX DOMAIN

(Presented by Academician S. N. Bernstein on 23 V 1967)

1°. Let \(\widetilde C\) be the space of all continuous \(2\pi\)-periodic functions \(f(x)\) with norm \(\|f\|_{\widetilde C}=\max |f(x)|\). Let \(U_{n,m}(f)\) be a linear operation from \(\widetilde C\) into \(\widetilde C\) having the following properties: 1) for any \(f\in \widetilde C\), \(U_{n,m}(f)\) is a trigonometric polynomial of order not exceeding \(n+m\), where \(n\) and \(m\) are positive integers; 2) if \(f\) is a trigonometric polynomial of order not exceeding \(n\), then \(U_{n,m}(f)=f\). We denote the set of all such \(U_{n,m}\) by \(\Omega_{n,n+m}\). It is obvious that the Vallée-Poussin sum

\[ \sigma_{n,m}(f)=\frac{1}{m+1}\sum_{k=m}^{n+m}s_k(f), \tag{1} \]

where \(s_k(f)\) is a partial sum of the Fourier series, belongs to \(\Omega_{n,n+m}\). Put

\[ \widetilde U_{n,m}(f)=\frac{1}{2\pi}\int_0^{2\pi}(U_{n,m}(f_t))_{-t}\,dt, \]

where \(f_t(x)=f(x+t)\). In \((^1)\) it was proved that for any \(U_{n,m}\in\Omega_{n,n+m}\) and any \(f\in\widetilde C\) the equality

\[ \widetilde U_{n,m}(f)=s_n(f)+\frac{1}{2}\sum_{k=n+1}^{n+m}(A_k\cos kx+B_k\sin kx), \tag{2} \]

holds, where the numbers \(A_k\) and \(B_k\) are computed by the formulas

\[ A_k=a_k(c_k^{(1)}+s_k^{(2)})+b_k(s_k^{(1)}-c_k^{(2)}),\qquad B_k=a_k(c_k^{(2)}-s_k^{(1)})+b_k(s_k^{(2)}+c_k^{(1)}). \]

Here \(a_k\) and \(b_k\), \(c_k^{(1)}\) and \(c_k^{(2)}\), \(s_k^{(1)}\) and \(s_k^{(2)}\) are the Fourier coefficients of order \(k\), respectively, for the functions \(f(x)\), \(U_{n,m}(\cos kx)\), \(U_{n,m}(\sin kx)\). With the aid of formula (2) and its modifications and generalizations, various problems in the theory of linear polynomial operations were solved in the \(2\pi\)-periodic real case \((^{2-4})\). In the present article the results from \((^1)\) are extended to the complex domain.

2°. Let \(D\) be an arbitrary finite domain with simply connected complement \(D_1\) and rectifiable boundary \(\Gamma\). By \(w=\Phi(z)\) denote the function mapping \(D_1\) conformally onto the domain \(|w|\ge 1\) of the \(w\)-plane under the condition \(\Phi(\infty)=\infty\). Let \(z=\psi(w)\) be the inverse function. By \(A(D)\) we denote the set of all functions \(f(z)\) continuous in the closed domain \(\overline D\) and analytic inside \(D\). For \(f\in A(D)\) we define the norm by the equality \(\|f\|=\max_{z\in\overline D}|f(z)|\). For real \(t\) and \(f\in A(D)\) define the operations

\[ f_t(z)=\frac{1}{2\pi i}\int_\Gamma \frac{f\{\psi[\Phi(\zeta)e^{it}]\}}{\zeta-z}\,d\zeta; \tag{3} \]

\[ f_{t^\wedge}(z)=\frac{f_t(z)+f_{-t}(z)}{2}. \tag{4} \]

We shall call transformation (3) a shift of the function \(f\) by \(l\), and transformation (4) an even shift of the function \(f\) by \(t\). Operations (3) and (4) were considered in another connection by V. K. Dzyadyk \((^{5,6})\).

Let \(F_k\) denote the Faber polynomial of degree \(k\) generated by the domain \(D\) \((^{7-9,14})\). The equalities

\[ (F_k)_t=F_k e^{ikt}, \qquad (F_k)_t \sim \cos kt, \tag{5} \]

are easily proved; they are consequences of an identity of V. K. Dzyadyk \((^{5,6})\).

Let \(U_{n,m}\) be a linear operation from \(A(D)\) into \(A(D)\) having the following properties: 1) for any \(f\in A(D)\), \(U_{n,m}(f)\) is a polynomial of degree \(\le n+m\); 2) if \(f\) is a polynomial of degree \(\le n\), then \(U_{n,m}(f)=f\). We shall also denote the set of all such \(U_{n,m}\) by \(\Omega_{n,n+m}\). The most important example of an operation of type \(U_{n,m}\) is furnished by the sum (1), where now

\[ s_n(f)=\sum_{k=0}^{n} a_k F_k(z), \tag{6} \]

\[ a_k=\frac{1}{2\pi}\int_{-\pi}^{\pi} f\!\left[\psi(e^{it})\right] e^{-ikt}\,dt. \tag{7} \]

Theorem 1. Let \(f\in A(D)\) and \(U_{n,m}\in \Omega_{n,n+m}\); then

\[ \frac{1}{2\pi}\int_{0}^{2\pi} \bigl(U_{n,m}(f_t)\bigr)_{-t}\,dt = s_n(f)+\sum_{k=n+1}^{n+m} a_k\alpha_{k,k}F_k, \tag{8} \]

where \(a_k\) is determined according to (7) and

\[ \alpha_{k,j} = \frac{1}{2\pi}\int_{0}^{2\pi} U_{n,m}(F_k)\bigl(\psi(e^{it})\bigr)e^{-ijt}\,dt. \tag{9} \]

We outline the proof. Denote the left-hand side of equality (8) by \(\widetilde U_{n,m}(f)\). One may verify that for any \(f\in A(D)\),
\(\widetilde U_{n,m}(f)=\widetilde U_{n,m}(s_{n+m}(f))\), where \(s_n(f)\) is defined according to (6). It is clear that

\[ \widetilde U_{n,m}(s_{n+m}(f)) = \widetilde U_{n,m}(s_n(f)) + \widetilde U_{n,m}(s_{n+m}(f)-s_n(f)). \tag{10} \]

According to (6) and (5), we have

\[ (s_n(f))_t=\sum_{k=0}^{n} a_k F_k e^{ikt}. \]

Since \(U_{n,m}\in\Omega_{n,n+m}\), by virtue of (5) we obtain

\[ U_{n,m}\!\left[(s_n(f))_t\right] = \sum_{k=0}^{n} a_k F_k e^{ikt}. \]

Consequently,

\[ \{U_{n,m}[(s_n(f))_t]\}_{-t}=s_n(f). \]

Therefore

\[ \widetilde U_{n,m}(s_n(f))=s_n(f). \tag{11} \]

Since

\[ U_{n,m}(F_k)=\sum_{j=0}^{n+m}\alpha_{k,j}F_j, \]

where \(\{\alpha_{k,j}\}\) are determined according to (9), by virtue of (5) we have

\[ \widetilde U_{n,m}(F_k)=\alpha_{k,k}F_k. \tag{12} \]

Let us now note that

\[ U_{n,m}(s_{n+m}(f)-s_n(f)) = \sum_{k=n+1}^{n+m} a_k \widetilde U_{n,m}(F_k). \]

Therefore, from (12) it follows that

\[ \widetilde U_{n,m}\bigl(s_{n+m}(f)-s_n(f)\bigr) = \sum_{k=n+1}^{n+m} a_k \alpha_{k,k} F_k . \tag{13} \]

From (10), (11), and (13), (8) follows.

In the same way one proves

Theorem 2. Let \(f \in A(D)\) and \(U_{n,m}\in \Omega_{n,n+m}\). Then

\[ \frac{1}{2\pi}\int_0^{2\pi} [U_{n,m}(f_t^\wedge)]_{t^\wedge}\,dt = \frac12 s_n(f)+\frac{a_0}{2} +\frac12 \sum_{j=n+1}^{n+m} a_j\alpha_{j,j}F_j, \tag{14} \]

where \(\{a_k\}\) and \(\{\alpha_{j,j}\}\) are the same numbers as in Theorem 1.

\(3^\circ.\) Put

\[ \chi(z)= \sum_{k=m+1}^{n-m} \frac{F_k(z)-F_{2n+2m+2-k}(z)}{\,n+m+1-k\,}, \]

where \(F_k\) is the Faber polynomial of degree \(k\). It is known that, in the trigonometric case, similar polynomials were first considered by Fejér \((^{10})\). Let the equation of the contour \(\Gamma\) of the domain \(D\) have the form \(\xi=\xi(s)\), where \(s\) is the length of the arc measured from some fixed point of the contour. We shall assume that \(\xi''(s)\) exists, and moreover \(\xi''(s)\in \mathrm{Lip}\,\alpha\) for some \(\alpha\), \(0<\alpha<1\). The following holds.

Theorem 3. Let the contour \(\Gamma\) satisfy the stated conditions; then there exists a constant \(A\) such that, for all \(z\in \overline D\), the inequality holds

\[ |\chi(z)|\leq A. \]

Denote by \(A'(D)\) the subset of \(A(D)\) consisting of all functions analytic in the closed domain \(\overline D\). For what follows we need a lemma.

Lemma. Let a function \(\psi(w)\), mapping conformally the domain \(|w|>1\) onto the domain \(D_1\), have at all points of the circle \(|w|=1\) a finite continuous nonzero derivative \(\psi'(w)\), satisfying a Lipschitz condition. Then the operator (4), considered as a mapping from \(A'(D)\) into \(A'(D)\), is linear, and for any \(f\in A'(D)\) and all \(t\), \(-\infty<t<\infty\), the inequality \(\|f_t^\wedge\|\leq C\|f\|\) holds, where the constant \(C\) depends neither on \(f\) nor on \(t\).

\(4^\circ.\) Theorem 4. For any \(U_{n,m}\in \Omega_{n,n+m}\) the inequality holds

\[ \|U_{n,m}\|\geq C\ln \frac{n}{m+1}, \tag{15} \]

where the constant \(C>0\) does not depend on \(n\) or \(m\).

We outline the proof. By means of general theorems on the possibility of passing to the norm under the integral sign \((^{11})\), and with the aid of the lemma, we obtain that

\[ \|U_{n,m}(\chi)\|\geq \frac{1}{C^2}\|\widetilde U_{n,m}(\chi)\|. \tag{16} \]

We now estimate \(\|U_{n,m}(\chi)\|\). For this we use Theorem 2, taking into account that, in the expansion of the polynomial \(\chi\) in Faber polynomials, there are no terms with indices \(k\) satisfying the inequalities \(n+1\leq k\leq n+m\), i.e., \(a_k=0\), \(n+1\leq k\leq n+m\). Therefore equality (14) takes the form

\[ \widetilde U_{n,m}(\chi,z)=s_n(\chi,z) = \sum_{k=m+1}^{n-m} \frac{F_k(z)}{\,n+m+1-k\,}. \tag{17} \]

One can verify that

\[ |s_n(\chi,z^*)|\geq C\ln\frac{n}{m+1}, \tag{18} \]

where \(C>0\) is a constant and \(z^*\) is the preimage of the point \(w=1\), i.e. \(\Phi(z^*)=1\). From (17) and (18) we infer that

\[ |\widetilde U_{n,m}(\chi,z^*)|\geq C\ln\frac{n}{m+1}. \]

By virtue of Theorem 3 we have

\[ \|\widehat U_{n,m}(\chi)\|\geq C\ln\frac{n}{m+1}. \tag{19} \]

From (16) and (19), (15) follows.

The Banach–Steinhaus theorem and Theorem 4 lead to the theorem

Theorem 5. Whatever the sequence of linear operations \(\{U_{n_i,m_i}\}\), where \(U_{n_i,m_i}\in\Omega_{n_i,n_i+m_i}\), with \(\lim\limits_{i\to\infty}\dfrac{m_i}{n_i}=0\), there always exists an \(f\in A(D)\) such that

\[ \varlimsup_{i\to\infty}\|U_{n_i,m_i}(f)-f\|=\infty. \]

Theorems 4 and 5 in the case of the space \(\widetilde C\) are found in \({}^{12}\).

5°. Let an arbitrary sequence of positive numbers be given, satisfying the inequalities \(0<p_n\leq 1\), \(n=1,2,\ldots\), and let \(\lim\limits_{n\to\infty} p_n=1\). The question is whether it is possible to construct a sequence of linear operations \(\{U_{n,m}\}\), \(U_{n,m}\in\Omega_{n,n+m}\), in such a way that for every \(f\in A(D)\) the equality

\[ \|U_{n,m}(f)-f\|=O(E_{[np_n]}(f)), \tag{20} \]

holds, where \(E_n(f)\) is the best approximation in the domain \(D\) of the function \(f\in A(D)\) by means of a polynomial of degree \(n\), and \([np_n]\) is the integer part of the number \(np_n\).

Theorem 6. There does not exist a sequence of operations \(\{U_{n,m}\}\), \(U_{n,m}\in\Omega_{n,n+m}\), satisfying condition (20).

This theorem for the case of the space \(\widetilde C\) is found in \({}^{13}\).

Leningrad Institute of Soviet Trade
named after F. Engels

Received
23 V 1967

CITED LITERATURE

\({}^{1}\) D. L. Berman, DAN, 95, No. 2 (1954).
\({}^{2}\) D. L. Berman, DAN, 85, No. 1 (1952).
\({}^{3}\) D. L. Berman, DAN, 91, No. 6 (1953).
\({}^{4}\) D. L. Berman, Matem. sbornik, 60(102), No. 3 (1963).
\({}^{5}\) V. K. Dzyadyk, Izv. AN SSSR, ser. matem., 26, 797 (1962).
\({}^{6}\) V. K. Dzyadyk, Collection Contemporary Problems of the Theory of Analytic Functions, “Nauka,” 1966.
\({}^{7}\) G. Faber, Math. Ann., 57, 389 (1903).
\({}^{8}\) A. O. Gelfond, Calculus of Finite Differences, Moscow–Leningrad, 1952.
\({}^{9}\) A. I. Markushevich, Theory of Analytic Functions, Moscow–Leningrad, 1950.
\({}^{10}\) L. Fejer, Gött. Nachr., 66 (1916).
\({}^{11}\) N. Dunford, J. Schwartz, Linear Operators, Moscow, 1962.
\({}^{12}\) V. F. Nikolaev, DAN, 64, No. 4 (1949).
\({}^{13}\) D. L. Berman, DAN, 120, No. 6 (1958).
\({}^{14}\) V. I. Smirnov, N. A. Lebedev, Constructive Theory of Functions of a Complex Variable, “Nauka,” 1964.