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MATHEMATICS
D. L. BERMAN
EXTREMAL PROBLEMS IN THE THEORY OF POLYNOMIAL OPERATORS
(Presented by Academician A. N. Kolmogorov, 21 I 1961)
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The present note is devoted to extremal problems in the theory of polynomial operators for the space \(\widetilde C\) of \(2\pi\)-periodic continuous functions with norm \(\|f\|=\max_{0\le x<2\pi}|f(x)|\). In what follows \(k\) is an integer, \(k\ge 0\); \(n\) and \(m\) are natural numbers; \(f^{(k)}(x)\) is the \(k\)-th derivative of \(f(x)\); polynomials are everywhere understood to be trigonometric.
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Theorem 1. Let \(\Omega_n^{(k)}\) be the set of all linear operators \(U_n(f,x)\) from \(\widetilde C\) into \(\widetilde C\) having the property
\[ U_n(f,x)=f^{(k)}(x), \tag{1} \]
if \(f(x)\) is a polynomial of order \(\le n\). Put \(\rho_n^{(k)}=\inf_{U_n\in\Omega_n^{(k)}}\|U_n\|\). The equalities \(\rho_n^{(k)}=n^k,\ k=0,1,2,\ldots\) are valid. For any \(k\) one can indicate such an operation \(\overline U_n\in\Omega_n^{(k)}\) that \(\|\overline U_n\|=\rho_n^{(k)}\).
Proof. For definiteness let us consider the case of even \(k\). According to property (1) of the operators \(U_n(f,x)\),
\[ U_n(\cos n\theta)=\pm n^k\cos n\theta. \]
Therefore \(\|U_n\|\ge n^k\). Hence \(\rho_n^{(k)}\ge n^k\). Now consider the operator
\[ U_n(f,x)= \sum_{j_1,j_2,\ldots,j_k} f(x+x_{j_1}+x_{j_2}+\cdots+x_{j_k})r_{j_1}r_{j_2}\cdots r_{2j_k}, \]
\[ 1\le j_s\le 2n;\quad s=1,2,\ldots,k, \tag{2} \]
where
\[ r_{j_s}=(-1)^{j_s-1}\frac{1}{2n\sin^2 x_{j_s}/2}; \qquad x_{j_s}=\frac{2j_s-1}{2n}\pi. \]
With the aid of the well-known identity of Riesse \((^1)\) it is easy to verify that the operator (2) belongs to \(\Omega_n^{(k)}\). Since \((^1)\)
\[ \sum_{j_s=1}^{2n}|r_{j_s}|=n,\qquad s=1,2,\ldots,k, \]
the norm of the operator (2) is equal to \(n^k\). Therefore, for \(k\ge 1\), the operator (2) is extremal. If \(k=0\), then the identity operator is extremal.
Along with the set of operators \(\Omega_n^{(k)}\), introduce the set \(\Omega_{n,n}^{(k)}\) of linear operators \(U_{n,n}(f,x)\) from \(\widetilde C\) into \(\widetilde C\) having the properties: 1) for any \(f\in\widetilde C\), \(U_{n,n}(f,x)\) is a polynomial of order \(\le n\); 2) if \(f(x)\) is a polynomial of order \(\le n\), then \(U_{n,n}(f,x)=f^{(k)}(x)\).
Put
\[ \rho_{n,n}^{(k)}=\inf_{U_{n,n}\in\Omega_{n,n}^{(k)}}\|U_{n,n}\|. \]
Since \(\Omega_{n,n}^{(k)}\subset \Omega_n^{(k)}\), it follows that \(\rho_{n,n}^{(k)}\ge \rho_n^{(k)}\). A natural question arises about the asymptotic estimate of the ratio \(\rho_{n,n}^{(k)}/\rho_n^{(k)}\) as \(n\to\infty\). The solution of this question is given by Theorem 2.
Theorem 2. Let \(k\) be a fixed integer satisfying \(k\ge 0\). Then the equality
\[ \lim_{n\to\infty}\left(\frac{\rho_{n,n}^{(k)}}{\rho_n^{(k)}}:\frac{4}{\pi^2}\ln n\right)=1. \]
Let us now consider the set \(\Omega_{n,n+m}^{(\Phi)}\) of linear operators \(U_{n,n+m}(f,x)\) from \(\widetilde C\) to \(\widetilde C\), possessing the following properties: 1) for every \(f\in \widetilde C\), \(U_{n,n+m}(f,x)\) is a polynomial of order \(\leqslant (n+m)\); 2) if \(T\) is a polynomial of order not higher than \(n\), then
\[ U_{n,n+m}(T,x)=\sigma^{(\Phi)}(T,x),\qquad \sigma^{(\Phi)}(f,x)=\int_{0}^{2\pi} f(x+t)\Phi(t)\,dt, \tag{3} \]
where \(\Phi(t)\) is a given polynomial of order \(\leqslant (n+m)\).
If, as the polynomial \(\Phi(t)\), one takes the Vallée-Poussin kernel \(V_{n,m}(t)\),
\[ V_{n,m}(t)=\frac{1}{\pi(m+1)}\sin\left(n+\frac{m+1}{2}\right)t\sin\frac{m+1}{2}t:\sin^2\frac{t}{2}, \]
then equality (3) takes the form
\[ U_{n,n+m}(T,x)=T(x) \tag{4} \]
for every polynomial \(T(x)\) of order not higher than \(n\). The most important operation of the form \(U_{n,n+m}(f,x)\) with property (4) is the Vallée-Poussin partial sum
\[ \sigma_{n,m}^{(V)}(f)=\frac{1}{m+1}\sum_{k=n}^{n+m} S_k(f), \]
where \(S_k(f)\) is the partial sum of order \(k\) of the Fourier series of the function \(f(x)\). Among operations of the form \(U_{n,n+m}(f,x)\) with property (4) are also the well-known interpolation processes of S. N. Bernstein \((2)\).
Theorem 3. Let
\[
\rho_{n,n+m}^{(\Phi)}=\inf_{U_{n,n+m}\in\Omega_{n,n+m}^{(\Phi)}} \|U_{n,n+m}\|.
\]
Then the equality holds
\[ \rho_{n,n+m}^{(\Phi)} = \inf_{\alpha_k,\beta_k} I_{\Phi}(\alpha_1,\alpha_2,\ldots,\alpha_m,\beta_1,\beta_2,\ldots,\beta_m), \]
where
\[ I_{\Phi}(\alpha_1,\alpha_2,\ldots,\alpha_m,\beta_1,\beta_2,\ldots,\beta_m)= \]
\[ =\int_{0}^{2\pi} \left| S_n(\Phi,t)+ \sum_{j=1}^{m}\bigl(\alpha_j\cos(n+j)+\beta_j\sin(n+j)t\bigr) \right|\,dt. \tag{5} \]
If the integral (5) attains its minimum for \(\alpha_j=\alpha_j^{(0)}\), \(\beta_j=\beta_j^{(0)}\), \(j=1,2,\ldots,m\), then the extremal operation is computed by the formula
\[ \overline U_{n,n+m}(f,x)= \int_{0}^{2\pi} f(x+t) \left[ S_n(\Phi,t)+ \sum_{j=1}^{m}\bigl(\alpha_j^{(0)}\cos(n+j)t+\beta_j^{(0)}\sin(n+j)t\bigr) \right]dt. \tag{6} \]
That is,
\[ \|\overline U_{n,n+m}\|=\rho_{n,n+m}^{(\Phi)}. \]
We outline the proof of this theorem. In (3) it is proved that for every \(U_{n,n+m}\in\Omega_{n,n+m}^{(\Phi)}\)
\[ \frac{1}{2\pi}\int_{0}^{2\pi} U_{n,n+m}[f(z+t),x-t]\,dt = \sigma^{(\Phi)}(S_n(f),x)+ \]
\[ +\frac{1}{2}\sum_{k=n+1}^{n+m}(A_k\cos kx+B_k\sin kx), \tag{7} \]
where \(A_k=\delta_k b_k+\gamma_k a_k\); \(B_k=\delta_k b'_k-\gamma_k a_k\); \(a_k\) and \(b_k\) are the Fourier coefficients of \(f(x)\); \(\delta_k\) and \(\gamma_k\) are numbers depending only on \(U_{n,n+m}\). Since \(\sigma^{(\Phi)}(S_n(f),x)=S_n(\sigma^{(\Phi)}(f),x)\), equality (7) takes the form
\[ \frac{1}{2\pi}\int_{0}^{2\pi} U_{n,n+m}[f(z+t),x-t]\,dt= \]
\[ = S_n(\sigma^{(\Phi)}(f),x)+\frac12\sum_{k=n+1}^{n+m}(A_k\cos kx+B_k\sin kx). \]
From this it is easy to derive that
\[ \|U_{n,n+m}\|\ge \inf_{\alpha_k,\beta_k} I_\Phi(\alpha_1,\ldots,\alpha_m,\beta_1,\ldots,\beta_m). \]
Thus,
\[ \rho^{(\Phi)}_{n,n+m}\ge \inf_{\alpha_k,\beta_k} I_\Phi(\alpha_1,\ldots,\alpha_m,\beta_1,\ldots,\beta_m). \tag{8} \]
If the integral (5) attains its minimum for \(\alpha_j=\alpha_j^{(0)}\), \(\beta_j=\beta_j^{(0)}\), \(j=1,2,\ldots,m\), then for the operation \(\overline U_{n,n+m}(f,x)\), defined by (6), we have
\[ \|\overline U_{n,n+m}\|=I_\Phi(\alpha_1^{(0)},\ldots,\alpha_m^{(0)},\beta_1^{(0)},\ldots,\beta_m^{(0)}). \tag{9} \]
From (8) and (9) theorem 3 follows.
Theorem 4. Let \(\Omega^{(k)}_{n,n+m}\) be the set of operators \(U_{n,n+m}(f,x)\) from \(\widetilde C\) to \(\widetilde C\) possessing the properties: 1) for every \(f\in\widetilde C\), \(U_{n,n+m}(f,x)\) is a polynomial of order \(\le (n+m)\); 2) if \(f(x)\) is a polynomial of order \(\le n\), then \(U_{n,n+m}(f,x)=f^{(k)}(x)\). Put
\[
\rho^{(k)}_{n,n+m}=\inf_{U_{n,n+m}\in\Omega^{(k)}_{n,n+m}}\|U_{n,n+m}\|.
\]
Then
\[ \rho^{(k)}_{n,n+m}=\inf_{\alpha_k,\beta_k} I(\alpha_1,\alpha_2,\ldots,\alpha_k,\beta_1,\beta_2,\ldots,\beta_k), \]
where*
\[ I=I(\alpha_1,\alpha_2,\ldots,\alpha_k,\beta_1,\beta_2,\ldots,\beta_k)= \]
\[ =\frac1\pi\int_0^{2\pi}\left|D_n^{(k)}(t)+\sum_{j=1}^{m}\bigl(\alpha_j\cos(n+j)t+\beta_j\sin(n+j)t\bigr)\right|\,dt. \tag{10} \]
If the integral (10) attains its least value for \(\alpha_j=\alpha_j^{(1)}\), \(\beta_j=\beta_j^{(1)}\), \(j=1,2,\ldots,m\), then the extremal operation is
\[ \overline U=\overline U_{n,n+m}(f,x)= \]
\[ =\frac1\pi\int_0^{2\pi} f(x+t)\left[D_n^{(k)}(t)+\sum_{j=1}^{m}\bigl(\alpha_j^{(1)}\cos(n+j)t+\beta_j^{(1)}\sin(n+j)t\bigr)\right]\,dt. \]
This theorem follows directly from theorem 3, if one observes that the set of operators \(\Omega^{(k)}_{n,n+m}\) is a special case of the set of operators \(\Omega^{(\Phi)}_{n,n+m}\), when \(\Phi(t)=\dfrac1\pi D_n^{(k)}(t)\).
Of particular interest are the special cases of theorem 4 when \(k=0\) and \(k=1\). Let us first consider the case \(k=0\). Since \(D_n(t)\) is an even function, \(\rho^{(0)}_{n,n+m}\) is computed by the formula
\[ \rho^{(0)}_{n,n+m}=\frac2\pi\inf_{\alpha}\int_0^\pi\left|D_n(t)+\sum_{j=1}^{m}\alpha_j\cos(n+j)t\right|\,dt. \tag{11} \]
\[ \text{* }D_n(t)\text{ is the Dirichlet kernel.} \]
If the integral on the right-hand side of (11) attains its minimum for \(\alpha_j=\widetilde{\alpha}_j,\ j=1,2,\ldots,m\), then the extremal operation is found from the formula
\[ \overline{U}=\frac{1}{\pi}\int_0^{2\pi} f(x+t)\left[D_n(t)+\sum_{j=1}^m \alpha_j \cos (n+j)t\right]\,dt . \tag{12} \]
Thus, among all linear operations from \(\widetilde{C}\) into \(\widetilde{C}\), taking functions from \(\widetilde{C}\) into polynomials of order \((n+m)\) and preserving polynomials of order \(n\), the operation (12) has the least norm.
As is known, the numbers \(\widetilde{\alpha}_j\) can be found from the equalities
\[ \int_0^\pi \operatorname{sign}\left[D_n(t)+\sum_{j=1}^m \widetilde{\alpha}_j \cos (n+j)t\right]\cos (n+i)t\,dt=0, \qquad i=1,2,\ldots,m . \]
From this, by direct verification, we see that, when \(n\) is a multiple of \((m+1)\), the extremal operation coincides with the Vallée-Poussin partial sum \(\sigma^{(V)}_{n,m}(f)\) (for this, see also (4)). Incidentally, this result follows directly from inequality (14) of note \((^3)\), if one takes into account the simple observation that, when \(n\) is a multiple of \((m+1)\), the expansion of the function \(\operatorname{sign} V_{n,m}(t)\) in a Fourier series contains no terms with \(\cos jt\), where \(n+1\le j\le n+m\). Let us now consider the case when \(k=1\) and \(m=n-1\). With the help of Theorem 4 we obtain the theorem:
Theorem 5. Among all linear operators \(U_{n,2n-1}(f,x)\) from \(\widetilde{C}\) into \(\widetilde{C}\), taking functions from \(\widetilde{C}\) into polynomials of order \((2n-1)\) and possessing the property that for every polynomial of order \(\le n\) the equality \(U_{n,2n-1}(f,x)=f'(x)\) holds, the operator
\[ \overline{U}_{n,2n-1}(f,x)= \frac{1}{\pi}\int_0^{2\pi} f(x+t)\,\sin nt\left(\frac{\sin nt/2}{\sin t/2}\right)^2 dt \tag{13} \]
has the least norm. Thus:
\[ \rho^{(1)}_{n,2n-1}=\|\overline{U}_{n,2n-1}\| =2nF_{n-1}(|\sin nt|,0), \]
where \(F_{n-1}(f,x)\) is the Fejér mean of order \((n-1)\).
As is known \((^5)\), the operator (13) was used by A. Zygmund to prove S. N. Bernstein’s inequality for the modulus of the derivative of a trigonometric polynomial. As for the extremal property of this operator, expressed by Theorem 5, it apparently has not been noted until now. In the proof of the above-mentioned S. N. Bernstein inequality by means of the operator (13), the constant obtained was twice its exact value. From Theorem 5, in particular, it follows that among the operators \(\Omega^{(1)}_{n,2n-1}\) there is no such operator by means of which one can prove S. N. Bernstein’s inequality with the exact constant.
Remark. The question of extremality of the Vallée-Poussin partial sums is resolved by the following theorem:
For the Vallée-Poussin partial sum \(\sigma^{(V)}_{n,m}(l)\) to have the least norm in the class of operators \(\Omega^{(0)}_{n,n+m}\), it is necessary and sufficient that \(2n\) be a multiple of \((m+1)\).
Received
14 I 1961
REFERENCES
- V. L. Goncharov, Theory of Interpolation and Approximation of Functions, Moscow–Leningrad, 1934.
- S. N. Bernstein, Collected Works, vol. 2, Publishing House of the Academy of Sciences of the USSR, 1954, p. 146.
- L. Berman, DAN, 95, No. 2 (1954).
- V. F. Nikolaev, DAN, 96, No. 1 (1954).
- A. Zygmund, Trigonometric Series, 1939.