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MATHEMATICS
V. D. GOLOVIN
ON A RIESZ BASIS OF EXPONENTIAL FUNCTIONS
(Presented by Academician S. N. Bernstein on 10 II 1962)
A basis \(\{x_k\}\) of a Hilbert space \(H\) will be called, following N. K. Bari \((^1)\), a Riesz basis if the series
\[ \sum_{k=-\infty}^{\infty} c_k x_k, \]
where \(c_k\) \((k=0,\ \pm 1,\ldots)\) are numbers, converges if and only if \(\{c_k\}\in l^2\).
As follows directly from the Riesz–Fischer theorem and Parseval’s equality, the trigonometric system \(\{e^{ikt}\}\) is, by virtue of the periodicity of its elements, a Riesz basis in its closed linear span relative to the space \(L^2(-\sigma,\sigma)\) for any \(\sigma \geq \pi\).
The purpose of the present work is to study conditions under which an analogous fact holds for an arbitrary system of exponential functions.
- A system \(\{x_k\}\) of elements of a Hilbert space \(H\) is usually called uniformly minimal \((^{1,2})\) if in \(H\) there exists a norm-bounded system \(\{h_k\}\) conjugate to \(\{x_k\}\), i.e. such that \((x_k,h_j)=\delta_{kj}\). A Riesz basis is, obviously, a uniformly minimal system.
Let \(\Lambda\) be an arbitrary sequence of complex numbers \(\lambda_k\) \((k=0,\ \pm 1,\ldots)\), and let \(E_\Lambda\) be the corresponding system of exponential functions \(\{e^{i\lambda_k t}\}\). By \(\tau=\tau(\Lambda)\) we shall denote the exact lower bound of the numbers \(\sigma>0\) such that the system \(E_\Lambda\) is uniformly minimal in \(L^2(-\sigma,\sigma)\).
If the system \(E_\Lambda\) is uniformly minimal in \(L^2(-\sigma,\sigma)\) for every \(\sigma>0\), then \(\tau=0\); if, however, it is not uniformly minimal in \(L^2(-\sigma,\sigma)\) for any \(\sigma>0\), then naturally one sets \(\tau=\infty\).
The system \(E_\Lambda\) may be in \(L^2(-\tau,\tau)\) both uniformly minimal and nonminimal even in the ordinary sense. Indeed, the trigonometric system \(\{e^{ikt}\}\) is uniformly minimal in \(L^2(-\pi,\pi)\) and is not minimal in \(L^2(-\sigma,\sigma)\) for any \(\sigma<\pi\). On the other hand, the system \(E_\Lambda\) with \(\lambda_k=k-\frac14\operatorname{sgn} k\) \((k=0,\ \pm1,\ldots)\) is uniformly minimal in \(L^2(-\sigma,\sigma)\) for every \(\sigma>\pi\) (this follows from Theorem 2), but is not minimal even in the ordinary sense in \(L^2(-\pi,\pi)\) \((^3)\).
- Definition. A sequence \(\Lambda\) of complex numbers \(\lambda_k\) \((k=0,\ \pm1,\ldots)\) belongs to the class \(K\) if
\[ \sup_{-\infty<k<\infty} |\operatorname{Im}\lambda_k|<\infty; \tag{1} \]
\[ \inf_{-\infty<m,n<\infty} |\lambda_m-\lambda_n|>0 \quad (m\ne n). \tag{2} \]
We shall also denote by \(L^2_\Lambda(-\sigma,\sigma)\) the closed linear span of the system \(E_\Lambda\) in \(L^2(-\sigma,\sigma)\).
Theorem 1. If the system \(E_\Lambda\) is a Riesz basis in \(L^2_\Lambda(-\sigma,\sigma)\) for some \(\sigma>0\), then the sequence \(\Lambda\) belongs to the class \(K\) and \(\sigma\geq \tau\).
Proof. Since the series
\[ \sum_{-\infty}^{\infty} c_k \int_{-\sigma}^{\sigma} f(t)e^{i\lambda_k t}\,dt \]
converges for every function \(f(t)\in L^2(-\sigma,\sigma)\) and every sequence \(\{c_k\}\in l^2\), it follows that
\[ \sum_{-\infty}^{\infty}\left|\int_{-\sigma}^{\sigma} f(t)e^{i\lambda_k t}\,dt\right|^2<\infty \]
and, consequently,
\[ \frac{\operatorname{sh}(2\sigma \operatorname{Im}\lambda_k)}{\operatorname{Im}\lambda_k} = \int_{-\sigma}^{\sigma}\left|e^{i\lambda_k t}\right|^2\,dt =O(1), \]
whence, as \(k\to\infty\), \(\operatorname{Im}\lambda_k=O(1)\).
Further, the system \(E_\Lambda\) is uniformly minimal in \(L^2(-\sigma,\sigma)\), and therefore it has an adjoint system \(\{h_k(t)\}\), bounded in norm. The equality
\[ \int_{-\sigma}^{\sigma}\left(e^{i\lambda_k t}-e^{i\lambda_j t}\right)\overline{h_k(t)}\,dt=1 \qquad (k\ne j) \]
shows that the limiting relation
\[ \lim_{k,j\to\infty}|\lambda_k-\lambda_j|=0 \qquad (k\ne j) \]
is impossible. Thus, the theorem is completely proved.
Theorem 2. If the sequences \(\Lambda=\{\lambda_k\}\) and \(M=\{\mu_k\}\) both belong to the class \(K\) and \(\lambda_k=\mu_k+O(1)\) as \(k\to\infty\), then \(\tau(\Lambda)=\tau(M)\).
We do not give the proof of this theorem here for lack of space.
Corollary. If the sequence \(\Lambda\) belongs to the class \(K\), then \(\tau(\Lambda)<\infty\).
Indeed, every sequence \(\Lambda\) of class \(K\) is contained in some sequence of the form \(\{\alpha k+O(1)\}\), where \(\alpha\) is a constant, and for such a sequence, by Theorem 2, \(\tau<\infty\).
3. Theorem 3. If the sequence \(\Lambda\) belongs to the class \(K\), then the system \(E_\Lambda\) is a Riesz basis in \(L^2_\Lambda(-\sigma,\sigma)\) for every \(\sigma>\tau(\Lambda)\).
For \(\sigma=\tau\) the assertion of the theorem, generally speaking, does not hold, since in \(L^2(-\tau,\tau)\) the system \(E_\Lambda\) may fail to be even minimal.
We precede the proof of Theorem 3 by several auxiliary propositions.
Lemma 1. If the sequence \(\Lambda\) belongs to the class \(K\), then for every \(\sigma>0\) there exists a constant \(B\) such that
\[ \int_{-\sigma}^{\sigma}\left|\sum c_k e^{i\lambda_k t}\right|^2\,dt \leq B\sum |c_k|^2 \]
for every finite system of complex numbers \(c_k\).
Proof. Let \(n_k\) be the integer nearest to \(\lambda_k\). We may assume from the outset that \(n_k\ne n_j\) for \(k\ne j\), since the whole sequence \(\Lambda\) can be split into a finite number of subsequences possessing this property.
First of all it is clear that
\[ \int_{-\sigma}^{\sigma}\left|\sum c_k e^{i\lambda_k t}\right|^2\,dt = \sum_{k\ne j}\sum c_k\overline{c_j} \int_{-\sigma}^{\sigma} e^{i(\lambda_k-\overline{\lambda_j})t}\,dt + \sum_k |c_k|^2 \int_{-\sigma}^{\sigma}\left|e^{i\lambda_k t}\right|^2\,dt, \]
where the last integral on the right-hand side has a finite upper bound with respect to \(k\). Integrating by parts, we obtain
\[ \int_{-\sigma}^{\sigma} e^{i(\lambda_k-\overline{\lambda_j})t}\,dt = \frac{1}{n_k-n_j} \left\{ -ie^{i(\lambda_k-\overline{\lambda_j})t}\bigg|_{-\sigma}^{\sigma} - \int_{-\sigma}^{\sigma}(\delta_k-\overline{\delta_j})e^{i(\lambda_k-\overline{\lambda_j})t}\,dt \right\}, \]
where \(\delta_k=\lambda_k-n_k=O(1)\) as \(k\to\infty\). To complete the proof, it remains to…
it remains now only to apply Hilbert’s theorem, by virtue of which
\[ \left|\sum_{k\ne j}\sum \frac{A_kB_j}{k-j}\right| <2\pi\left\{\sum |A_k|^2\sum |B_j|^2\right\}^{1/2} \]
for arbitrary complex \(A_k, B_j\) \((^5)\).
Lemma 2. If the sequence \(\Lambda\) belongs to the class \(K\), then for any sequence \(\{c_k\}\in l^2\) there exists an entire function \(F(\lambda)\) of exponential type not exceeding a given number \(\sigma>\tau(\Lambda)\), such that \(F(\lambda_k)=c_k\) \((k=0,\pm1,\ldots)\) and
\[ \int_{-\sigma}^{\sigma}|F(x)|^2\,dx \le M\sum_{-\infty}^{\infty}|c_k|^2, \]
where the constant \(M\) does not depend on the sequence \(\{c_k\}\).
Proof. Put, for \(\tau<\sigma_0<\sigma,\ 0<\omega<\sigma-\sigma_0\),
\[ F(\lambda)= \sum_{-\infty}^{\infty} c_k \frac{\sin\omega(\lambda-\lambda_k)}{\omega(\lambda-\lambda_k)} \int_{-\sigma_0}^{\sigma_0}\overline{h_k(t)}e^{i\lambda t}\,dt, \]
where \(\{h_k(t)\}\) is the system from \(L_\Lambda^2(-\sigma_0,\sigma_0)\) biorthogonal to \(E_\Lambda\). It is clear that \(F(\lambda)\) is an entire function of exponential type not greater than \(\sigma\), and \(F(\lambda_k)=c_k\) \((k=0,\pm1,\ldots)\). By Cauchy’s inequality,
\[ |F(\lambda)|^2\le \sum_{n=-\infty}^{\infty} \left|\frac{\sin\omega(\lambda-\lambda_n)} {\omega(\lambda-\lambda_n)}\right|^2 \sum_{k=-\infty}^{\infty} \left|c_k\int_{-\sigma_0}^{\sigma_0}\overline{h_k(t)}e^{i\lambda t}dt\right|^2. \]
Since the first sum on the right-hand side has a finite upper bound for all real \(\lambda\), it follows that
\[ \int_{-\infty}^{\infty}|F(x)|^2\,dx \le M_1\sum_{-\infty}^{\infty}|c_k|^2 \int_{-\sigma_0}^{\sigma_0}|h_k(t)|^2\,dt \le M\sum_{-\infty}^{\infty}|c_k|^2, \]
where \(M\), obviously, does not depend on the sequence \(\{c_k\}\).
Lemma 3. If the sequence \(\Lambda\) belongs to the class \(K\), then for every \(\sigma>\tau(\Lambda)\) there exists a constant \(A\) such that
\[ A\sum |c_k|^2 \le \int_{-\sigma}^{\sigma} \left|\sum c_ke^{i\lambda_k t}\right|^2\,dt \]
for every finite system of complex numbers \(c_k\).
Proof. Let the numbers \(c_k\) be included in some sequence \(\{c_k\}\in l^2\). By Lemma 2 there exists a function \(f(t)\in L^2(-\sigma,\sigma)\) such that
\[ \int_{-\sigma}^{\sigma} f(t)e^{i\lambda_k t}\,dt =\overline{c_k}\qquad (k=0,\pm1,\ldots), \]
therefore,
\[ \sum_{-\infty}^{\infty}|c_k|^2 = \int_{-\sigma}^{\sigma} f(t)\sum_{-\infty}^{\infty}c_ke^{i\lambda_k t}\,dt, \]
and, on the basis of Bunyakovsky’s inequality, the right-hand side does not exceed the magnitude of the expression
\[ \left\{ \int_{-\sigma}^{\sigma}|f(t)|^2\,dt \int_{-\sigma}^{\sigma}\left|\sum_{-\infty}^{\infty}c_ke^{i\lambda_k t}\right|^2\,dt \right\}^{1/2} \]
and, moreover,
\[ \int_{-\sigma}^{\sigma}|f(t)|^2\,dt \le M\sum_{-\infty}^{\infty}|c_k|^2. \]
Thus the lemma is completely proved.
- Proof of Theorem 3*. Let \(f(t) \in L^2_{\Lambda}(-\sigma,\sigma)\), i.e.
\[ \lim_{n\to\infty}\int_{-\sigma}^{\sigma}\left|f(t)-\sum_{k=-n}^{n}c_{kn}e^{i\lambda_k t}\right|^2\,dt=0 \]
for some complex numbers \(c_{kn}\) \((k=0,\ \pm 1,\ldots,\pm n)\). Put
\[ P_n(t)=\sum_{k=-n}^{n}c_{kn}e^{i\lambda_k t};\qquad S_n(t)=\sum_{k=-n}^{n}e^{i\lambda_k t}\int_{-\sigma}^{\sigma}f(t)\overline{h_k(t)}\,dt, \]
where \(\{h_k(t)\}\) is the system in \(L^2_{\Lambda}(-\sigma,\sigma)\) conjugate to \(E_{\Lambda}\). Then
\[ \int_{-\sigma}^{\sigma}|f(t)-S_n(t)|^2\,dt \leq 2\int_{-\sigma}^{\sigma}|f(t)-P_n(t)|^2\,dt + 2\int_{-\sigma}^{\sigma}|S_n(t)-P_n(t)|^2\,dt, \]
and the second integral on the right-hand side is equal to the limit, as \(N\to\infty\), of the expression
\[ \int_{-\sigma}^{\sigma} \left| \sum_{k=-n}^{n}(c_{kN}-c_{kn})e^{i\lambda_k t} \right|^2\,dt \leq B\sum_{k=-N}^{N}|c_{kN}-c_{kn}|^2. \]
The latter sum, in turn, does not exceed, up to a constant factor, the value of the integral
\[ \int_{-\sigma}^{\sigma} \left| \sum_{k=-N}^{N}(c_{kN}-c_{kn})e^{i\lambda_k t} \right|^2\,dt = \int_{-\sigma}^{\sigma}|P_N(t)-P_n(t)|^2\,dt. \]
Thus,
\[ \int_{-\sigma}^{\sigma}|f(t)-S_n(t)|^2\,dt \leq M\int_{-\sigma}^{\sigma}|f(t)-P_n(t)|^2\,dt, \]
i.e. \(E_{\Lambda}\) is a basis in \(L^2_{\Lambda}(-\sigma,\sigma)\). That this basis is a Riesz basis follows from Lemmas 1 and 3.
We shall indicate some consequences of the theorem proved.
Corollary 1. If \(|\lambda_n-n|\leq D<\infty\) \((n=0,\ \pm1,\ldots)\) and the sequence \(\Lambda\) belongs to the class \(K\), then \(E_{\Lambda}\) is a Riesz basis in \(L^2_{\Lambda}(-\sigma,\sigma)\) for every \(\sigma>\pi\).**
Corollary 2. If the sequence \(\Lambda\) belongs to the class \(K\), then for every function \(f(t)\in L^2_{\Lambda}(-\sigma_0,\sigma_0)\), \(\sigma_0>\tau\), there exists, for every \(\sigma>\sigma_0\), a unique function \(g(t)\in L^2_{\Lambda}(-\sigma,\sigma)\) such that \(g(t)=f(t)\) almost everywhere on the interval \((-\sigma_0,\sigma_0)\).***
Kharkov State University
named after A. M. Gorky
Received
30 I 1962
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* On the basis of results of N. K. Bari (1), the assertion of Theorem 3 follows directly from Lemmas 1 and 3. We give the proof only because this fact is elementary.
** In a number of works, conditions were studied under which the system \(E_{\Lambda}\), where \(|\lambda_k-k|<D\), is a Riesz basis in \(L^2(-\pi,\pi)\) (see (6–8)).
*** For similar facts see (6, 9, 10).