MATHEMATICS

M. I. KADETS

THE EXACT VALUE OF THE PALEY–WIENER CONSTANT

(Presented by Academician S. N. Bernstein on 16 XII 1963)

A basis \(\{e_k\}\) of a Hilbert space is called a Riesz basis if, for every element \(x=\sum_{k=1}^{\infty} a_k e_k\), the inequality

\[ A\left(\sum |a_k|^2\right)^{1/2} \leqslant \|x\| \leqslant B\left(\sum |a_k|^2\right)^{1/2} \qquad (0<A\leqslant B<\infty) \]

holds. Let \(\lambda_k=k+\delta_k\) \((k=0,\pm1,\pm2,\ldots)\) be real numbers satisfying the condition

\[ \sup_k |\delta_k|=d<D. \tag{1} \]

Paley and Wiener \((^1)\) showed that if \(D=1/\pi^2\), then the sequence \(\{e^{i\lambda_k t}\}_{-\infty}^{\infty}\), where the \(\lambda_k\) are subject to condition (1), is a Riesz basis in the space \(L_2(-\pi,\pi)\). Duffin and Eachus \((^2)\) (see also \((^3)\)) established that this result is valid for \(D=\ln 2/\pi \approx 0.22\), and V. D. Golovin \((^4)\) raised the value of \(D\) to 0.24. According to a theorem of Levinson \((^5)\), for \(D>0.25\) the assertion ceases to be true. In the present note we shall show that the exact boundary of admissible \(D\)’s is \(D=0.25\). As in the papers \((^{1-4})\), the starting point of our considerations will be the

Paley–Wiener Lemma. If the system \(\{e^{i\lambda_k t}\}\) is close to the system \(\{e^{ikt}\}\) in the sense that

\[ \left\|\sum_k a_k e^{ikt}-\sum_k a_k e^{i\lambda_k t}\right\| \leqslant \theta \left\|\sum_k a_k e^{ikt}\right\| = \theta \left(\sum_k |a_k|^2\right)^{1/2} \]

for some \(\theta<1\) and all finite sets of numbers \(a_k\), then the system \(\{e^{i\lambda_k t}\}\) is a Riesz basis in \(L_2(-\pi,\pi)\).

Theorem 1. If the sequence \(\lambda_k=k+\delta_k\) is subject to the condition

\[ \sup_k |\delta_k|=d<0.25\quad (k=0,\pm1,\pm2,\ldots), \]

then the system \(\{e^{i\lambda_k t}\}\) is a Riesz basis in \(L_2(-\pi,\pi)\).

Proof. According to the Paley–Wiener lemma, the question reduces to the investigation of the upper bound of the expression

\[ U=\left\|\sum_k a_k(1-e^{i\delta_k t})e^{ikt}\right\| =\left\{\frac{1}{2\pi}\int_{-\pi}^{\pi} \left|\sum_k a_k(1-e^{i\delta_k t})e^{ikt}\right|^2 dt\right\}^{1/2}, \tag{2} \]

taken over all finite sets of numbers \(a_k\) such that \(\sum_k |a_k|^2 \leqslant 1\). If this upper bound is less than one, then the system \(\{e^{i\lambda_k t}\}\) is a Riesz basis.

Expand the function \(\psi(t)=1-e^{i\delta t}\) \((-\pi\leqslant t\leqslant \pi)\) in a Fourier series with respect to the orthogonal system \(\{1;\cos \nu t;\sin(\nu-\tfrac12)t\}\) \((\nu=1,2,\ldots)\):

\[ 1-e^{i\delta t} = \left(1-\frac{\sin \pi\delta}{\pi\delta}\right) + \sum_{\nu=1}^{\infty} \frac{(-1)^\nu 2\delta\sin \pi\delta}{\pi(\nu^2-\delta^2)}\cos \nu t + \]

\[ {}+ i\sum_{\nu=1}^{\infty} \frac{(-1)^\nu 2\delta\cos \pi\delta}{\pi[(\nu-\tfrac12)^2-\delta^2]} \sin(\nu-\tfrac12)t. \tag{3} \]

Substitute (3) into (2) and change the order of summation:

\[ \begin{aligned} U=\Bigg\|& \sum_k \left(1-\frac{\sin \pi \delta_k}{\pi \delta_k}\right)a_k e^{ikt} +\sum_{\nu=1}^{\infty}\cos \nu t \sum_k \frac{(-1)^\nu 2\delta_k \sin \pi\delta_k}{\pi(\nu^2-\delta_k^2)}a_k e^{ikt} \\ &\quad +i\sum_{\nu=1}^{\infty}\sin(\nu-\tfrac12)t \sum_k \frac{(-1)^\nu 2\delta_k \cos \pi\delta_k}{\pi[(\nu-\tfrac12)^2-\delta_k^2]}a_k e^{ikt} \Bigg\|. \end{aligned} \]

Apply the triangle inequality:

\[ \begin{aligned} U \leqslant& \left\|\sum_k \left(1-\frac{\sin \pi \delta_k}{\pi \delta_k}\right)a_k e^{ikt}\right\| +\sum_{\nu=1}^{\infty}\left\|\cos \nu t\sum_k \frac{(-1)^\nu 2\delta_k \sin \pi\delta_k}{\pi(\nu^2-\delta_k^2)}a_k e^{ikt}\right\| \\ &\quad +\sum_{\nu=1}^{\infty}\left\|\sin(\nu-\tfrac12)t\sum_k \frac{(-1)^\nu 2\delta_k \cos \pi\delta_k}{\pi[(\nu-\tfrac12)^2-\delta_k^2]}a_k e^{ikt}\right\|. \end{aligned} \]

Estimate each term:

\[ \left\|\sum_k \left(1-\frac{\sin \pi \delta_k}{\pi \delta_k}\right)a_k e^{ikt}\right\| \leqslant \left(1-\frac{\sin \pi d}{\pi d}\right) \left\|\sum_k a_k e^{ikt}\right\|, \]

\[ \left\|\cos \nu t\sum_k \frac{(-1)^\nu 2\delta_k \sin \pi\delta_k}{\pi(\nu^2-\delta_k^2)}a_k e^{ikt}\right\| \leqslant \frac{2d\sin \pi d}{\pi(\nu^2-d^2)} \left\|\sum_k a_k e^{ikt}\right\|, \]

\[ \left\|\sin(\nu-\tfrac12)t\sum_k \frac{(-1)^\nu 2\delta_k \cos \pi\delta_k}{\pi[(\nu-\tfrac12)^2-\delta_k^2]}a_k e^{ikt}\right\| \leqslant \frac{2d\cos \pi d}{\pi[(\nu-\tfrac12)^2-d^2]} \left\|\sum_k a_k e^{ikt}\right\|. \]

Thus:

\[ \begin{aligned} U \leqslant& \left\{1-\frac{\sin \pi d}{\pi d} +\sin \pi d\sum_{\nu=1}^{\infty}\frac{2d}{\pi(\nu^2-d^2)} +\cos \pi d\sum_{\nu=1}^{\infty}\frac{2d}{\pi[(\nu-\tfrac12)^2-d^2]}\right\} \left\|\sum_k a_k e^{ikt}\right\| \\ =&\left\{1-\frac{\sin \pi d}{\pi d} +\sin \pi d\left(\frac{1}{\pi d}-\operatorname{ctg}\pi d\right) +\cos \pi d\cdot \operatorname{tg}\pi d\right\} \left(\sum_k |a_k|^2\right)^{1/2} \\ =&\,(1-\cos \pi d+\sin \pi d)\left(\sum_k |a_k|^2\right)^{1/2}. \end{aligned} \]

Thus, the required upper bound of expression (2), for any \(d<0.25\), is strictly less than unity, which proves the theorem.

Duffin and Schaeffer \({}^{6}\) proved the following proposition:

Duffin–Schaeffer Theorem. If the system \(\{e^{i\lambda_k t}\}_{-\infty}^{\infty}\) is a Riesz basis in \(L_2(-\pi,\pi)\), and the real numbers \(\mu_k\) satisfy the condition \(\sup_k |\mu_k|<\infty\), then the system \(\{e^{(\mu_k+i\lambda_k)t}\}\) is also a Riesz basis.

From this proposition and Theorem 1 there follows directly

Theorem 2. If the numbers \(z_k\) are such that

\[ \sup_k |\operatorname{Im}(z_k-ik)|<0.25;\qquad \sup_k |\operatorname{Re} z_k|<\infty, \]

then the system \(\{e^{z_k t}\}_{-\infty}^{\infty}\) is a Riesz basis in \(L_2(-\pi,\pi)\).

Received
3 XII 1963

REFERENCES

\({}^{1}\) R. E. A. C. Paley, N. Wiener, Fourier Transforms in the Complex Domain, N. Y., 1934.
\({}^{2}\) R. J. Duffin, J. J. Eachus, Bull. Am. Math. Soc., 48, 850 (1942).
\({}^{3}\) F. Riesz, B. Sz.-Nagy, Lectures on Functional Analysis, IL, 1954.
\({}^{4}\) V. D. Golovin, Dokl. AN ArmSSR, 36, No. 2, 65 (1963).
\({}^{5}\) N. Levinson, Ann. Math., 37, 919 (1936).
\({}^{6}\) R. J. Duffin, A. C. Schaeffer, Trans. Am. Math. Soc., 72, 341 (1952).