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MATHEMATICS
V. D. GOLOVIN
ON A GENERALIZATION OF THE CONCEPT OF PERIODIC EXTENSION
(Presented by Academician S. N. Bernstein on 10 X 1962)
1. Let the complex numbers \(\lambda_k\) \((k=1,2,\ldots)\) be pairwise distinct. Associating with the sequence \(\Lambda\), in which each \(\lambda_k\) occurs a (finite) number \(\alpha_k\) of times, the sequence \(E_\Lambda\) of functions
\[ e_{kj}(t)=(it)^{j-1}e^{i\lambda_k t} \qquad (j=1,\ldots,\alpha_k;\ k=1,2,\ldots), \]
we denote by \(L^2_\Lambda(-\sigma,\sigma)\) the closed vector subspace generated by the sequence \(E_\Lambda\) in \(L^2(-\sigma,\sigma)\).
Definition 1. An extension of a function \(f(t)\in L^2_\Lambda(-\sigma,\sigma)\) with respect to the sequence \(E_\Lambda\) will mean a function \(F(t)\), defined on the whole real axis and coinciding with \(f(t)\) almost everywhere on \((-\sigma,\sigma)\), whose restriction to each finite interval \((-\tau,\tau)\) belongs to the subspace \(L^2_\Lambda(-\tau,\tau)\).
The periodic extension of a function \(f(t)\in L^2(-\pi,\pi)\) is, according to this definition, the extension of the function \(f(t)\) with respect to the sequence \(e^{ikt}\) \((k=0,\pm1,\pm2,\ldots)\).
We shall study the following problem:
What conditions must the sequence \(\Lambda\) and the number \(\sigma>0\) satisfy in order that every function \(f(t)\in L^2_\Lambda(-\sigma,\sigma)\) have a unique extension with respect to the sequence \(E_\Lambda\)?
The answer to this question is given by Theorems 1 and 2 of the present note. For what follows it is important that, by virtue of Banach’s theorem on open mappings, the posed problem may be given the following form:
What conditions must the sequence \(\Lambda\) and the number \(\sigma>0\) satisfy in order that for every \(\tau>\sigma\) there exist a constant \(M\), depending on \(\tau\), such that
\[ \int_{-\tau}^{\tau}\left|\sum_{k,j}c_{kj}e_{kj}(t)\right|^2\,dt \leq M\int_{-\sigma}^{\sigma}\left|\sum_{k,j}c_{kj}e_{kj}(t)\right|^2\,dt \tag{1} \]
for every finite sequence of complex numbers \(c_{kj}\).
Questions close in their formulation were studied earlier by A. F. Leont’ev \((^1)\) and Zh. P. Kagan \((^2)\).
2. Along with the functions \(e_{kj}(t)\) we shall have to consider functions of a somewhat more general form; their construction is as follows.
Divide the sequence \(\Lambda\) into nonempty pairwise disjoint finite sets \(\Lambda_k\) \((k=1,2,\ldots)\), denoting by \(\nu_k\) the number of elements of the set \(\Lambda_k\), and by \(\lambda_{k,0},\lambda_{k,1},\ldots,\lambda_{k,\nu_k-1}\) the elements themselves. The resulting partition will be denoted by \(\hat{\Lambda}\).
Put
\[ \hat e_{kj}(t)=\frac{1}{2\pi i}\int_{C_k}\frac{e^{izt}}{\omega_{kj}(z)}\,dz \qquad (j=1,\ldots,\nu_k;\ k=1,2,\ldots), \tag{2} \]
where
\[ \omega_{kj}(\zeta)=\frac{1}{(j-1)!}(\zeta-\lambda_{k,0})\ldots(\zeta-\lambda_{k,j-1}), \tag{3} \]
and \(C_k\) is a contour enclosing all points of the set \(\Lambda_k\).
It is verified directly that every linear combination of the functions \(e_{kj}(t)\) is at the same time a linear combination of the \(\hat e_{kj}(t)\), and conversely; thus, the sequence \(\hat E_\Lambda\) of functions \(\hat e_{kj}(t)\) \((j=1,\ldots,\nu_k;\ k=1,2,\ldots)\) and the sequence \(E_\Lambda\) generate one and the same vector subspace. For \(\lambda_{k,0}=\lambda_k,\ \nu_k=\alpha_k\) \((k=1,2,\ldots)\), the sequence \(\hat E_\Lambda\) coincides with \(E_\Lambda\).
- A family of points \(x_i\) \((i\in I)\) of a topological vector space \(E\) over the field of real or complex numbers will be called completely free if there exists in \(E\) a neighborhood of zero \(U\) such that for every \(k\in I\) the neighborhood \(x_k+U\) does not intersect the closed vector subspace generated by those \(x_i\) for which \(i\ne k\). In order that a family of points \(x_i\) \((i\in I)\) of a locally convex space \(E\) be completely free, it is necessary and sufficient that there exist on \(E\) an equicontinuous family of linear forms \(f_i\) \((i\in I)\) forming with \((x_i)_{i\in I}\) a biorthogonal system.
By \(\tau(\hat E_\Lambda)\) we shall denote the exact lower bound of those \(\sigma>0\) for which the sequence \(\hat E_\Lambda\) is completely free in \(L^2(-\sigma,\sigma)\). If the sequence \(\hat E_\Lambda\) is not completely free in \(L^2(-\sigma,\sigma)\) for any \(\sigma\), we shall say that \(\tau(\hat E_\Lambda)=\infty\).
We shall call a partition \(\hat\Lambda\) regular if: 1) all numbers \(\lambda_k\) lie in some strip \(|\operatorname{Im}\lambda|\le h\); 2) the numbers \(\nu_k\) are bounded in the aggregate; 3) for \(\lambda\in\Lambda_k,\ \mu\in\Lambda_j\) \((k\ne j)\) the inequality \(\inf|\lambda-\mu|>0\) holds, where the lower bound is taken over all \(0<k,j<\infty\); 4) there exists a constant \(M\) such that \(|\lambda_{k,0}-\lambda_{k,j}|\le M\) \((j=0,\ldots,\nu_k-1;\ k=1,2,\ldots)\).
Proposition 1. If the partition \(\hat\Lambda\) is regular, then \(\tau(\hat E_\Lambda)<\infty\).
Proposition 2. For a given sequence \(\Lambda\), the number \(\tau(\hat E_\Lambda)\) has one and the same value \(\hat\tau_\Lambda\) for any regular partition \(\hat\Lambda\).
We omit the proofs of these propositions.
- Definition 2. The sequence \(\Lambda\) is called regular if: 1) all numbers \(\lambda_k\) \((k=1,2,\ldots)\) lie in some strip \(|\operatorname{Im}\lambda|\le h\); 2) there exists a constant \(N>0\) such that no rectangle of the form
\[ R_t=\{\lambda:\ t\le \operatorname{Re}\lambda\le t+1;\ |\operatorname{Im}\lambda|\le h\}\quad(-\infty<t<\infty) \tag{4} \]
contains more than \(N\) numbers \(\lambda_k\), counted with their multiplicities \(\alpha_k\).
It is easy to show (see (3)) that the sequence \(\Lambda\) is regular if and only if it possesses at least one regular partition.
Theorem 1. If the sequence \(\Lambda\) is regular, then for \(\sigma>\hat\tau_\Lambda\) every function \(f(t)\in L_\Lambda^2(-\sigma,\sigma)\) has a unique continuation with respect to the sequence \(E_\Lambda\).
The proof is based on the following propositions (cf. (4)).
Proposition 3. If the partition \(\hat\Lambda\) is regular, then for \(\sigma>0\) there exists a constant \(B\) such that
\[ \int_{-\sigma}^{\sigma}\left|\sum_{k,j} c_{kj}\hat e_{kj}(t)\right|^2\,dt \le B\sum_{k,j}|c_{kj}|^2 \tag{5} \]
for any finite sequence of complex numbers \(c_{kj}\).
Proposition 4. If the partition \(\hat{\Lambda}\) is regular, then for \(\sigma>\hat{\tau}_\Lambda\) there exists a constant \(A\) such that
\[ A \sum_{k,j} |c_{kj}|^2 \leq \int_{-\sigma}^{\sigma} \left| \sum_{k,j} c_{kj}\hat e_{kj}(t) \right|^2 dt , \tag{6} \]
for any finite sequence of complex numbers \(c_{kj}\).
5. Theorem 2. If, for some \(\sigma>0\), every function \(f(t)\in L^2_\Lambda(-\sigma,\sigma)\) has a unique extension with respect to the sequence \(E_\Lambda\), then the sequence \(\Lambda\) is regular and \(\sigma\geq \hat{\tau}_\Lambda\).
We shall briefly indicate the proof. First of all, for any \(\tau>\sigma\) and some \(M=M(\tau)\), inequality (1) must hold, whatever the finite sequence of complex numbers \(c_{kj}\). In particular, taking \(c_{nj}=\delta_{kn}\delta_{j0}\), we obtain:
\[ |\operatorname{sh}(2\tau \operatorname{Im}\lambda_k)| \leq M|\operatorname{sh}(2\delta \operatorname{Im}\lambda_k)| \]
for all \(k=1,2,\ldots\). Consequently, all the numbers \(\lambda_k\) lie in some strip \(|\operatorname{Im}\lambda|\leq h\).
Suppose that condition 2 of Definition 2 is not satisfied. Then there will be sequences \((t_k)\) and \((\varepsilon_k)\) of positive numbers such that in each rectangle \(R_{t_k}\) there are at least \(k\) numbers
\(\lambda^{(k)}_1,\lambda^{(k)}_2,\ldots,\lambda^{(k)}_k\) of the sequence \(\Lambda\), for which
\(|\lambda^{(k)}_1-\lambda^{(k)}_i|\leq \varepsilon_k\) \((i=1,2,\ldots,k)\), and moreover \(k!\varepsilon_k\to 0\) as \(k\to\infty\). Put
\[ \varphi_k(t)=\frac{(k-1)!}{2\pi i} \int_{L_k}\frac{e^{i\zeta t}\,d\zeta} {(\zeta-\lambda^{(k)}_1)\cdots(\zeta-\lambda^{(k)}_k)}, \]
where \(L_k\) is a contour enclosing the rectangle \(R_{t_k}\) and separated from its sides by a distance greater than one. Then
\[ \varphi_k(t)=(it)^{k-1}e^{i\lambda^{(k)}_1t}+o(1) \]
as \(k\to\infty\), which contradicts the inequality
\[ \int_{-\tau}^{\tau}|\varphi_k(t)|^2dt \leq M\int_{-\sigma}^{\sigma}|\varphi_k(t)|^2dt, \]
valid by virtue of (1). Thus it has been proved that the sequence \(\Lambda\) is regular.
Let \(\hat{\Lambda}\) be one of its regular partitions and let \(\hat E_\Lambda\) be the sequence of functions \(\hat e_{kj}(t)\) corresponding to this partition. Then for \(\tau>\hat{\tau}_\Lambda\) it follows from inequality (1) that \(\hat E_\Lambda\) is a completely free sequence in \(L^2_\Lambda(-\sigma,\sigma)\), i.e. \(\sigma\geq\hat{\tau}_\Lambda\).
Remark. Denote by \(T_\Lambda\) the exact lower bound of those \(\sigma>0\) for which every function \(f(t)\in L^2_\Lambda(-\sigma,\sigma)\) has a unique extension with respect to the sequence \(E_\Lambda\). The assertions of Theorems 1 and 2 may be combined by saying that \(T_\Lambda=\hat{\tau}_\Lambda<\infty\) if the sequence \(\Lambda\) is regular, and \(T_\Lambda=\infty\) if the sequence \(\Lambda\) is not regular.
Kharkov State University
named after A. M. Gorky
Received
9 X 1962
REFERENCES
- A. F. Leont’ev, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 39 (1951).
- J. P. Kahane, Ann. Inst. Fourier, 5, 39 (1953—1954).
- B. Ya. Levin, Zap. Kharkovsk. gos. univ. i Matem. obshch., 27, 4 (1961).
- V. D. Golovin, DAN, 145, 1, 27 (1962).