Abstract Generated abstract
This note studies a generalized M. Riesz identity for trigonometric polynomials and uses it to investigate when a convolution operator defined on polynomials of order at most n admits a norm-preserving extension to periodic function spaces. It establishes general criteria and examples for such extensions in spaces of continuous functions and Lp functions, including positive kernels, arbitrary kernels in L2, and a discrete operator arising from the generalized Riesz formula in the spaces of continuous and integrable periodic functions. The paper also proves positivity of associated kernels for derivatives of the Dirichlet kernel, derives norm values for these cases, and obtains pointwise inequalities for Fourier sums and their derivatives, including a formulation connected with Bernstein’s classical inequality for algebraic polynomials.
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.