Abstract
Full Text
UDC 513.88 : 513.83 + 517.512.6
MATHEMATICS
D. L. BERMAN
A GENERALIZED M. RIESZ FORMULA AND EXTENSION OF CONVOLUTION
(Presented by Academician S. N. Bernstein on 31 VIII 1966)
\(1^\circ\). Let \(\Pi_n\) denote the set of all trigonometric polynomials of order \(\leqslant n\). Let
\[ \Phi(t)=\sum_{k=0}^{n} r_k \sin(kt+\alpha_k). \tag{1} \]
The polynomial
\[ \widetilde{\Phi}(t)=r_n+2\sum_{k=0}^{n-1} r_k \cos[(n-k)t+\alpha_n-\alpha_k] \tag{2} \]
will be called the one associated with the polynomial \(\Phi(t)\). By \(\widetilde{C}\) and \(\widetilde{L}_r\) we shall denote, respectively, the space of all \(2\pi\)-periodic continuous functions and the space of all \(2\pi\)-periodic functions summable with the \(r\)-th power. Any of the spaces mentioned will be denoted by the letter \(E\)*. Clearly, \(\Pi_n\) may be regarded as a subspace of the space \(E\). Put
\[ \sigma(f,x)=\int_{0}^{2\pi} f(x+\theta)\Phi(\theta)\,d\theta; \tag{3} \]
\[ \overline{U}(f,x)=\frac{\pi}{2n}\sum_{r=1}^{2n} f\left(x+\varphi_r-\frac{\alpha_n}{n}\right)(-1)^{r-1} \widetilde{\Phi}\left(\varphi_r-\frac{\alpha_n}{n}\right), \qquad \varphi_r=\frac{2r-1}{2n}\pi. \tag{4} \]
In (2) it was established that, for \(t\in\Pi_n\),
\[ \sigma(t,x)=\overline{U}(t,x),\qquad -\infty<x<\infty. \tag{5} \]
Identity (5) will be called the generalized M. Riesz identity, since the M. Riesz formula \(({}^3,{}^4)\) follows from it; with the aid of this formula one obtains a very simple proof of the classical theorem of S. N. Bernstein, according to which
\[ \|t'\|\leqslant n\|t\|. \]
The present note is devoted mainly to the application and generalization of identity (5).
\(2^\circ\). We shall consider the convolution (3) as a linear operation from \(\Pi_n\) into \(\Pi_n\), where \(\Pi_n\) is considered as a subspace of the space \(E\). Denote the norm of this operation by \(\tau_n\). Thus:
\[ \tau_n=\sup_{\|t\|<1,\ t\in\Pi_n}\|\sigma(t)\|. \]
We pose the following problem. Under what conditions does there exist a linear operation \(U\) from \(E\) into \(E\), satisfying the conditions:
\[ \begin{aligned} &1)\quad U(t)=\sigma(t),\quad t\in\Pi_n;\\ &2)\quad \|U\|_{E}^{E}=\tau_n,\quad \text{where } \|U\|_{E}^{E} \text{ is the norm of the operation } U \text{ from } E \text{ into } E. \end{aligned} \tag{6} \]
* Some theorems of this note are valid for the spaces \(E\) from \(({}^1)\).
Such an operation \(U\) we call an extension of the convolution \(\sigma\) from \(\Pi_n\) to \(E\). It is well known that in the case of linear functionals this problem always has a positive solution.
Denote by \(\Omega_n^\Phi\) the set of all linear operations \(U\) from \(E\) into \(E\) satisfying condition (6). Introduce the quantity
\[ \rho_n=\rho_n^\Phi(E)=\inf_{U\in\Omega_n^\Phi}\|U\|. \]
Theorem 1. Let the linear operation \(U_0\) be an extension of the convolution (3) from \(\Pi_n\) to \(E\). Then
\[ \|U_0\|=\rho_n=\tau_n. \tag{7} \]
Proof. Since \(U_0\in\Omega_n^\Phi\), we have \(\|U_0\|\geq \rho_n\). By assumption, \(\|U_0\|_E^E=\tau_n\); hence
\[ \tau_n\geq \rho_n. \tag{8} \]
On the other hand, for any \(U\in\Omega_n^\Phi\),
\[ \|U\|\geq \tau_n, \tag{9} \]
because \(U\) satisfies equality (6). In view of (9), we have
\[ \tau_n\leq \rho_n. \tag{10} \]
From inequalities (8), (10) we conclude that (7) holds.
Theorem 2. If the kernel (1) is nonnegative, \(\Phi(t)\geq0\), \(-\infty<t<\infty\), then the convolution (3) itself is its own extension from \(\Pi_n\) to \(E\). Moreover,
\[ \tau_n=\int_0^{2\pi}\Phi(t)\,dt. \]
Proof. It is obvious that
\[ \tau_n\geq \|\sigma(1)\|=\int_0^{2\pi}\Phi(t)\,dt. \tag{11} \]
On the other hand, by virtue of the nonnegativity \(\Phi(t)\geq0\) *, \(-\infty<t<\infty\),
\[ \|\sigma(f)\|=\left\|\int_0^{2\pi} f(x+t)\Phi(t)\,dt\right\|\leq \int_0^{2\pi}\Phi(t)\,dt\,\|f\|. \]
Therefore
\[ \tau_n\leq \|\sigma\|_E^E\leq \int_0^{2\pi}\Phi(t)\,dt. \tag{12} \]
From (11) and (12) it follows
Theorem 3. In the space \(\widetilde L_2\), for a kernel \(\Phi(t)\) of arbitrary sign, the convolution (3) itself is its own extension from \(\Pi_n\) to \(L_2\). Moreover,
\[ \tau_n=\pi r_{j_0},\qquad r_{j_0}=\max\left\{\max_{j=1,2,\ldots,n} r_j,\ 2r_0|\sin\alpha_0|\right\}. \]
Proof. It is easy to see that
\[ \left\|\sigma\left(\frac{\cos kx}{\|\cos kx\|_{\widetilde L_2}}\right)\right\|= \begin{cases} |2\pi r_0\sin\alpha_0|, & k=0,\\ \pi r_k, & k=1,2,\ldots,n. \end{cases} \]
Therefore
\[ \|\sigma\|_{\widetilde L_2}^{\widetilde L_2}\geq \tau_n\geq \pi r_{j_0}. \tag{13} \]
On the other hand, from Parseval’s equality it follows that
\[ \tau_n\leq \|\sigma\|_{\widetilde L_2}^{\widetilde L_2}\leq \pi r_{j_0}. \tag{14} \]
(13) and (14) imply Theorem 3.
* We assume that \(\|1\|=1\). The legitimacy of passing to the norm under the integral sign is easy to justify.
By virtue of Theorems 2 and 3 one might think that in any space \(E\) the convolution (3) is always its extension from \(\Pi_n\) to \(E\). In fact this is not so. In the case of the spaces \(\widetilde C\) and \(\widetilde L_1\) the situation changes substantially. This is seen from the following theorem.
Theorem 4. Let \(E=\widetilde C\) or \(E=\widetilde L_1\), and let the kernel \(\Phi(t)\) be such that
\[ \widetilde\Phi\left(\varphi_r-\frac{\alpha_n}{n}\right)\geqslant 0,\qquad r=1,2,\ldots,2n . \tag{15} \]
Then the extension of the convolution (3) from \(\Pi_n\) to \(\widetilde C\), or from \(\Pi_n\) to \(\widetilde L_1\)*, is the operator \(\overline U(f,x)\), which is given by formula (4). Moreover, \(\tau_n=\pi r_n\).
Proof. For definiteness we shall consider the case \(E=\widetilde C\). By identity (5) it is necessary only to show that \(\|U\|_{\widetilde C}^{\widetilde C}=\tau_n\). Let \(t_n^*(x)=\sin(nx+\alpha_n)\). One can verify that
\[ \overline U(t_n^*,0)=\frac{\pi}{2n}\sum_{r=1}^{2n}\widetilde\Phi\left(\varphi_r-\frac{\alpha_n}{n}\right)=\pi r_n . \]
Consequently,
\[ \|\overline U\|_{\widetilde C}^{\widetilde C}\geqslant \tau_n\geqslant \pi r_n . \tag{16} \]
On the other hand, from (4) it follows that for any \(f\in\widetilde C\)
\[ \|U(f)\|\leqslant \frac{\pi}{2n}\sum_{r=1}^{2n}\left|\widetilde\Phi\left(\varphi_r-\frac{\alpha_n}{n}\right)\right|. \]
Hence, by inequalities (15), we obtain that
\[ \|U\|_{\widetilde C}^{\widetilde C}\leqslant \pi r_n . \tag{17} \]
From (16) and (17) we conclude that \(\|\overline U\|_{\widetilde C}^{\widetilde C}=\tau_n=\pi r_n\).
Theorem 5. If the kernel \(\Phi(t)\) is the derivative of order \(k\) of the Dirichlet kernel
\[ \Phi(t)=D_n^{(k)}(t),\qquad k=1,2,\ldots, \tag{18} \]
then the associated kernel \(\widetilde\Phi(t)\) is nonnegative on the whole real axis.
We indicate the proof. Since
\[ D_n^{(k)}(t)=\sum_{\nu=1}^{n}\nu^k\sin\left(\nu t+\frac{k+1}{2}\pi\right),\qquad k=1,2,\ldots, \]
then, according to (2),
\[ \widetilde D_n^{(k)}(t)=n^k+2\sum_{\nu=1}^{n-1}\nu^k\cos(n-\nu)t,\qquad k=1,2,\ldots . \]
Applying L. Fejér’s theorem (5), we see that
\[ \widetilde D_n^{(k)}(t)\geqslant 0,\qquad -\infty<t<\infty,\qquad k=1,2,\ldots . \]
From Theorems 4 and 5 there follows
Corollary 1. If \(\Phi(t)\) is defined according to equality (18), then the extension of the convolution (3) from \(\Pi_n\) to \(\widetilde C\), or from \(\Pi_n\) to \(\widetilde L_1\), is the operator (4), where \(\tau_n=\pi n^k\).
3°. Equality (5) has so far been considered only for the case when \(f\in\Pi_n\). If it is considered for arbitrary \(f\in\widetilde C\), then the following holds.
Theorem 6. For any \(f\in\widetilde C\) there exists a set \(M\), consisting of at least \((2n+1)\) distinct points of \([0,2\pi)\), such that
\[ \overline U(f,x)=\sigma(f,x),\qquad x\in M . \]
If \(f\in\Pi_n\), then \(M=(-\infty,\infty)\).
* We assume that \(\widetilde L_1\) consists of everywhere finite functions.
We outline the proof. By virtue of identity (5) and the linearity of the operators (3) and (4), we have
\[ \overline{U}(f)-\sigma(f)=\overline{U}(f-s_n(f))+\sigma(f-s_n(f)), \tag{19} \]
where \(s_n(f)\) is the partial sum of order \(n\) of the Fourier series of \(f\). Since \(\Phi\) is a polynomial of order \(n\), \(\sigma(f-s_n(f))=0\). Therefore, by (19),
\[ \overline{U}(f)-\sigma(f)=\overline{U}(f-s_n(f)). \tag{20} \]
Let us note that
\[ \int_0^{2\pi} \overline{U}(f-s_n(f),x)e^{ikx}\,dx=0,\qquad k=1,2,\ldots,n. \]
Consequently\(^6\), \(\overline{U}(f-s_n(f),x)\) has in the interval \([0,2\pi)\) no fewer than \((2n+1)\) distinct zeros. We now apply equality (20). It is obvious that identity (5) is a consequence of Theorem 6.
Theorem 7. If inequalities (15) are satisfied, then for every \(f\in \widetilde C\) there exists a set \(M\), consisting of at least \((2n+1)\) distinct points of \([0,2\pi)\), such that
\[ |\sigma(f,x)|\leq \pi r_n\sup_{\varphi_r}\left|f\left(x+\varphi_r-\frac{\alpha_n}{n}\right)\right|,\qquad x\in M. \]
If \(f\in \Pi_n\), then \(M=(-\infty,\infty)\).
We outline the proof. According to Theorem 6,
\[ |\sigma(f,x)|\leq \sup_{\varphi_r}\left|f\left(x+\varphi_r-\frac{\alpha_n}{n}\right)\right| \frac{\pi}{2n}\sum_{r=1}^{2n}\left|\widetilde{\Phi}\left(\varphi_r-\frac{\alpha_n}{n}\right)\right|,\qquad x\in M. \tag{21} \]
Since inequalities (15) are satisfied, the sum on the right-hand side of (21) is equal to \(2nr_n\).
Corollary 2. For any \(f\in \widetilde C\) there exists a set \(M\), consisting of at least \((2n+1)\) distinct points of \([0,2\pi)\), such that
\[ |s_n^{(k)}(f,x)|\leq n^k \sup_{r=1,2,\ldots,2n} \left|f\left(x+\frac{2r-n-2}{2n}\pi\right)\right|,\qquad x\in M,\quad k=1,2,\ldots . \]
If \(f\in \Pi_n\), then \(M=(-\infty,\infty)\).
This assertion follows from Theorems 7 and 5, when \(\Phi(t)=D_n^{(k)}(t)\).
Theorem 8. Let \(f(x)\) be continuous on \([-1,1]\). Then there exists a set \(M\), consisting of at least \(n\) points of \((-1,1)\), such that
\[ \left|\frac{d}{dx}s_n[f(\cos\theta),\arccos x]\right| \leq \frac{n}{\sqrt{1-x^2}}\sup_{\varphi_r}|f[\cos(\arccos x+\varphi_r)]|,\qquad x\in M. \tag{22} \]
If \(f\) is a polynomial of degree \(n\), then \(M=(-1,1)\).
It is clear that the most interesting case of (22) is the known inequality of S. N. Bernstein\(^7\), according to which
\[ |P'(x)|\leq \frac{n}{\sqrt{1-x^2}}\max_{-1\leq x\leq 1}|P(x)|,\qquad -1<x<1, \]
if \(P(x)\) is an algebraic polynomial of degree \(n\).
Leningrad Institute of Soviet Trade
named after F. Engels
Received
15 II 1966
References
- D. L. Berman, DAN, 161, No. 5 (1965).
- D. L. Berman, DAN, 163, No. 3 (1965).
- M. Riesz, C. R., 158, 1152 (1914).
- V. L. Goncharov, Theory of Interpolation and Approximation of Functions, Moscow, 1954.
- L. L. Fejér, Acta Lit. Sci. R. Univ. Hung., Francisco-Josephina, 2, 75 (1925).
- G. Pólya, G. Szegő, Problems and Theorems in Analysis, 1937.
- S. N. Bernstein, Extremal Properties of Polynomials, 1937, p. 168.