Full Text
UDC 517.516
MATHEMATICS
S. P. GEISBERG
ON QUASIANALYTIC FUNCTIONS IN \(L(-\infty,\infty)\)
(Presented by Academician S. N. Bernstein on 22 III 1965)
1. Usually, in order to obtain necessary and sufficient conditions for the quasianalyticity of a function, one has to consider quasianalytic classes of functions \((^{1,2})\); this is caused by the difficulties that arise in proving that the corresponding sufficient conditions are also necessary. Thus, for example, according to Mandelbrojt \((^2)\), the class \(L(M_n)\) of functions \(f(x)\) infinitely differentiable in the space* and defined by the conditions \(\|f^{(n)}\|_L \leq M_n\), is called quasianalytic if, for every function \(f \in L(M_n)\), the closures of the linear spans of its derivatives \(f^{(n)}\) and of its translates \(f_x(t)=f(x+t)\) coincide. The question naturally arises of such a definition of the concept of a quasianalytic function which would make it possible to give necessary and sufficient conditions of quasianalyticity referring to a function, and not to a class of functions. In this note it is shown that for \(L\) one can propose the following definition.
Definition. A function \(f \in L\) is called \(A\)-quasianalytic if, for every functional \(m \in M\) and every \(\{x_k\}\), \(x_k \ne x_0\), \(x_k \to x_0\), from the conditions
\[ m(f_{x_k})=\int_{-\infty}^{\infty} f(x_k+t)m(t)\,dt=0,\qquad k=1,2,\ldots,\infty, \]
it follows that \(m(f_x)=0\).
Obviously, every function belonging to a quasianalytic class of Mandelbrojt is \(A\)-quasianalytic. The properties of \(A\)-quasianalytic functions are established in the following theorems.
Theorem 1. If \(f\) is an \(A\)-quasianalytic function, then \(f\) is infinitely differentiable, \(f^{(n)} \in L(-\infty,\infty)\), and* *
\[ \lim_{\tau\to0}\left\|\left(f_\tau-\sum_{j=0}^{n}\frac{\tau^j}{j!}f^{(j)}\right)\tau^{-(n+1)}-\frac{f^{(n+1)}}{(n+1)!}\right\|_L=0, \]
\[ f^{(0)}=f,\qquad n=0,1,\ldots,\infty . \tag{1} \]
Theorem 2. Let \(f(t)\) be an even function, \(f \in L\), and let its Fourier transform \(F(x)\) satisfy the conditions:
\(1^\circ.\) \(F(x)>0\), is differentiable, and for \(x>0\), \(-xF'(x)/F(x)\) strictly increases.
\(2^\circ.\) For some \(\beta>0\), \(|F'(x)|=O(|x|^\beta+1)\).
Then, in order that \(f(t)\) be an \(A\)-quasianalytic function, it is necessary and sufficient that the following conditions hold:
\(3^\circ.\) \(f(t)\) is infinitely differentiable, \(f^{(n)}\in L\), \(n=1,2,\ldots,\infty\), and (1) holds.
\(4^\circ.\)
\[ \int_{1}^{\infty}\frac{\log F(x)}{x^2}=-\infty . \]
* In what follows, \(L\) and \(M\) denote the real space \(L(-\infty,\infty)\) and its conjugate.
** Formula (1) should be regarded as a recurrence relation defining the successive derivatives of the function \(f\).
Conditions \(1^\circ, 2^\circ\) occur in \((^2)\). Theorem 2, together with \((^2)\), p. 52, Lemma 1; p. 38, Theorem 1, shows that for functions satisfying conditions \(1^\circ, 2^\circ\), \(A\)-quasianalyticity and quasianalyticity in the sense of Mandelbrojt coincide. Theorems 1 and 2 may also be regarded as a development of Wiener’s theorem \((^3)\), since they give conditions for closedness in \(L\) of any system of translates \(f_x\) having a limit point \(x_k \to x_0\).
2. Proof of Theorem 1. We restrict ourselves to the case \(n=1\). For \(\tau>0\) put \(\varphi_\tau=(f_\tau-f)/\tau\), and first show that \(\|\varphi_\tau\|_L=O(1)\). Indeed, otherwise it is not difficult to show that \(\psi_\tau=\varphi_\tau/\|\varphi_\tau\|_L\) tends weakly to zero as \(\tau\to 0+\). Then, by induction, one can construct a sequence of functionals \(m_{\tau_k}\in M\) such that
\[ m_{\tau_0}(\psi_{\tau_0})=1;\quad m_{\tau_k}(f)=0,\ k=0,1,\ldots,\infty;\quad m_{\tau_k}(\psi_{\tau_j})=0,\quad 0\le j<k; \]
\[ m_{\tau_k}(\psi_{\tau_k})=-\sum_{j=0}^{k-1} m_{\tau_j}(\psi_{\tau_k});\quad \|m_{\tau_k}\|_M<2^{-k}, \]
where \(\{\psi_{\tau_k}\}\) is a suitably chosen subsequence of \(\{\psi_\tau\}\), \(\tau_k\downarrow 0\). Consequently, for the functional \(m=\sum m_j\) we have \(m\in M\) and
\(m(f_{\tau_0})=m(\psi_{\tau_0})=1\), \(m(f_{\tau_k})=m(\psi_{\tau_k})=0\), \(k=1,2,\ldots,\infty\), which contradicts the \(A\)-quasianalyticity of the function \(f\).
Next, without loss of generality, one may suppose that for some \(\sigma\), \(F(x)\ne 0\) for \(|x|\le \sigma\). Denote by \(\sigma_{\alpha,\beta}(z)\) the entire function of degree \(\sigma\) for which \(\sigma_{\alpha,\beta}(0)=\alpha\), \(\sigma'_{\alpha,\beta}(0)=\beta\), and \(\|\sigma_{\alpha,\beta}(x)\|_L<\infty\). From the Wiener–Paley theorems \((^4)\) and Wiener \((^3)\) it follows that there exists \(m_{\alpha,\beta}\in M\) such that
\(\sigma_{\alpha,\beta}(x)\equiv m_{\alpha,\beta}(f_x)\). We show that this implies the weak convergence of \(\{\varphi_\tau\}\) as \(\tau\downarrow 0\). In fact, since \(\|\varphi_\tau\|_L=o(1)\), otherwise there would exist a functional \(m\in M\), subsequences \(\{\tau'_j\}\), \(\{\tau''_j\}\), and numbers \(a<b\) such that \(\tau'_j\downarrow 0\),
\[ \tau''_j\downarrow 0,\quad m(\varphi_{\tau'_j})\to a,\quad m(\varphi_{\tau''_j})\to b. \]
Set \(m(f)=\alpha\), \((a+b)/2=\beta\), \(\overline m=m-m_{\alpha,\beta}\). Then \(\overline m\in M\) and
\[ \overline m(f_{\tau'_j})=m(f)+\tau'_j\overline m(\varphi_{\tau'_j}) =\tfrac12(a-b)\tau'_j+o(\tau'_j);\quad \overline m(f_{\tau''_j})=m(f)+\tau''_j\overline m(\varphi_{\tau''_j}) \]
\[ =\tfrac12(b-a)\tau''_j+o(\tau''_j). \]
This means that the continuous function \(\overline m(f_x)\) changes sign in every neighborhood to the right of zero. Consequently, the point \(x=0\) is a limit point of its zeros. Since, obviously, \(\overline m(f_x)\not\equiv 0\), this contradicts the \(A\)-quasianalyticity of the function \(f\).
By the weak completeness of \(L\), the sequence \(\{\varphi_\tau\}\) as \(\tau\downarrow 0\) has a weak limit \(\varphi_{0+}\in L\). Considering the functions \(\varphi_\tau-\varphi_{0+}\), which converge weakly to \(0\), and using arguments analogous to those given above, we obtain that, as \(\tau\downarrow 0\), \(\|\varphi_\tau-\varphi_{0+}\|_L\to 0\). In exactly the same way one establishes the existence of a function \(\varphi_{0-}\in L\) for which \(\|\varphi_\tau-\varphi_{0-}\|_L\to 0\) as \(\tau\uparrow 0\). Obviously, the Fourier transforms of the functions \(\varphi_{0+}\) and \(\varphi_{0-}\) coincide. Consequently, \(\varphi_{0+}\) and \(\varphi_{0-}\) coincide almost everywhere. Therefore
\[ \lim_{\tau\to 0}\|\varphi_\tau-\varphi_{0+}\|_L = \lim_{\tau\to 0}\|(f_\tau-f)/\tau-\varphi_{0+}\|_L =0,\quad \text{i.e. } f'=\varphi_{0+}. \]
3. Proof of Theorem 2. First of all, conditions \(1^\circ-4^\circ\) are sufficient, since from \(1^\circ-4^\circ\) and \((^2)\), p. 52, Lemma 1; p. 38, Theorem 1, it follows that for every \(m\in M\) the function
\(\mu(x)=m(f_x)\) belongs to the quasianalytic class of functions (1); consequently, its zeros have no limit point, i.e. \(f\) is an \(A\)-quasianalytic function.
The necessity of condition \(3^0\) follows from Theorem 1. Let us prove the necessity of \(4^0\). Suppose
\[ \int_1^\infty \frac{\log F(x)}{x^2}\,dx > -\infty . \]
Then, using \(1^0\), one can show that if \(\nu>0\), \(x(\nu)=-F'(\nu)\nu/F(\nu)>0\), then
\[ x(\nu)\uparrow\infty,\qquad \min_{\nu\leq t\leq \nu+1/2} F(t)t^{x(\nu)} \geq \frac12 F(\nu)\nu^{x(\nu)}. \tag{2} \]
Moreover, if \(x\geq 1\),
\[ v(x)=\max_{r\geq x} r^{x-[\ln x]}F(r),\qquad c>1,\qquad \psi_c(r)=\max_{r\geq x\geq 1}\frac{(cr)^x}{v(x)}, \]
then
\[ \int_1^\infty \frac{\ln \psi_c(r)}{r^2}\,dr<\infty . \tag{3} \]
Put \(v(x)=1\), \(0\leq x<1\), and choose \(\lambda_n\) so that \(\{\lambda_n\}\) is increasing and the \(\lambda_n\) run through all values from the set of numbers
\(2,4,6,\ldots,2n,\ldots;\ 1^2,3^2,5^2,\ldots,(2n+1)^2,\ldots\).
Then, for \(r\geq 1\) and some \(D>0\),
\[ \theta(r)=2\sum_{\lambda_n\leq r}\frac1{\lambda_n}\leq \ln Dr,\qquad U(r)=e^{\theta(r)}\leq Dr, \]
\[ V(r)=\sup_{0\leq x\leq r}\frac{[U(r)]^x}{v(x)}=O(\psi_D(r)), \]
and, by virtue of (3),
\[ \int_1^\infty \frac{\ln V(r)}{r^2}\,dr<\infty . \tag{4} \]
Next, let \(V(r)=V(-r)\) and \(V(r)=1\) for \(|r|<1\). Put
\[ g(z)=\frac{2x}{\pi}\int_{-\infty}^{\infty}\frac{\ln V(\tau)\,d\tau}{x^2+(y-\tau)^2},\qquad x>0,\qquad z=x+iy; \]
\((z)\) is the function conjugate to \(g(z)\); \(h(x)\equiv 0\) for \(x>0\);
\[ \omega(z)=e^{-g(z)-ih(z)}; \]
\[ G(z)=\prod_{n=1}^{\infty}\frac{\lambda_n-z}{\lambda_n+z}\,e^{2z/\lambda_n}; \qquad \Phi(z)=C^{-(1+z)}(1+z)^{-A}G(z)\omega(1+z), \]
where \(A\) and \(C\) are sufficiently large positive numbers. From (4) and (5), p. 19, it follows that, for \(x\geq 0\), \(\Phi(z)\) has the representation
\[ \Phi(z)=\int_0^\infty \varphi(u)u^z\,du, \]
where
\[ \int_0^\infty |\varphi(u)|u^x\,du=O(v(x)). \tag{5} \]
In particular, setting \(\varphi(-u)=\varphi(u)\), we have
\[ \operatorname{sgn}\int_{-\infty}^{\infty}|\varphi(u)|u^x\,du=\operatorname{sgn}G(x). \tag{6} \]
Let us show that \(u\varphi(u)/F(u)\in L\). Indeed, the function \(r^xF(r)\) attains its maximum at \(r\) satisfying the equality \(x=x(r)\)*. Therefore, setting \(x_n=x(n/4)\), \(n=1,2,\ldots,\infty\), we have
\[ v(x_n+3)=\max_{r\geq x_n+3} r^{x_n+3-[\ln(x_n+3)]}F(r) =O(1)\max_{r>0} r^{x_n}F(r) =O(1)(n/4)^{x_n}F(n/4). \]
\[ \text{* Recall that } x(\nu)=-F'(\nu)\nu/F(\nu)\text{ and }x'(\nu)\uparrow\infty. \]
Moreover, for \(n/4 \leqslant u \leqslant (n+1)/4\), by (2),
\(u^{x_n}F(u)\geqslant \frac12 (n/4)^{x_n}F(n/4)\). Hence, by (5),
\[ \int_{n/4}^{(n+1)/4} u\frac{|\varphi(u)|}{F(u)}\,du = \int_{n/4}^{(n+1)/4} u^{-2}\frac{|\varphi(u)|u^{3+x_n}}{F(u)u^{x_n}}\,du \leqslant \]
\[ \leqslant \left(\frac n4\right)^{-2} \frac{2}{(n/4)^{x_n}F(n/4)} \int_0^\infty |\varphi(u)|u^{x_n+3}\,du = O(1)\frac{\nu(x_n+3)}{(n/4)^{x_n}F(n/4)} = O(n^{-2}). \]
Thus,
\[ \int_{-\infty}^{\infty} |u|\frac{|\varphi(u)|}{F(u)}\,du = O(1)\left[\sum_1^\infty n^{-2}+\int_0^1 |u\varphi(u)|\,du\right]<\infty. \]
Next, put
\[ m(x)=\int_{-\infty}^{\infty}\frac{u\varphi(u)}{F(u)}\sin ux\,du. \]
Then \(m(x)\in M\) and
\[ m(fx)=\int_{-\infty}^{\infty}u\varphi(u)\sin ux\,du=\Psi(x). \]
We shall show that \(x=0\) is a limit point of the zeros of \(\Psi(x)\), and thereby prove the theorem\(^*\). For \(2n<x<2(n+1)\) we have
\[ \int_0^\infty \left(\int_{-\infty}^{\infty}\varphi(u)(1-\cos u\tau)\,du\right) \tau^{-(x+1)}\,d\tau = \]
\[ = \int_0^\infty \left(\int_{-\infty}^{\infty} \varphi(u)\left(\sum_{k=0}^{n}(-1)^k\frac{\tau^{2k}u^{2k}}{(2k)!}-\cos u\tau\right)\,du\right) \tau^{-(x+1)}\,d\tau = \]
\[ = \int_{-\infty}^{\infty}\varphi(u)\,du \int_0^\infty \left(\sum_{k=0}^{n}(-1)^k\frac{\tau^{2k}u^{2k}}{(2k)!}-\cos u\tau\right) \tau^{-(x+1)}\,d\tau = \]
\[ = (-1)^n\frac{\Gamma(x+1)}{\Gamma(x+1-2n)} \int_0^\infty\frac{1-\cos u\tau}{\tau^{x+1-2n}}\,d\tau \int_{-\infty}^{\infty}\varphi(u)|u|^x\,du, \]
since, by (6),
\[ \int_{-\infty}^{\infty}\varphi(u)u^{2k}=0,\qquad k=1,2,\ldots,\infty. \]
Consequently, if
\[ \omega(\tau)=\int_{-\infty}^{\infty}\varphi(u)\cos u\tau\,du, \]
then, by (6), for \(2n<x<2n+1\),
\[ \operatorname{sgn}\int_0^\infty \frac{\omega(0)-\omega(\tau)}{\tau^{x+1}}\,d\tau = (-1)^n\operatorname{sgn}G(x). \]
It follows from this and from the definition of \(G(x)\) that on each interval
\[ (2n+1)^2-1<x<(2n+1)^2+1 \]
the function
\[ R(x)=\int_0^\infty \frac{\omega(0)-\omega(\tau)}{\tau^{x+1}}\,d\tau \]
changes sign. Since \(\omega(\tau)=O(1)\), this is possible only if \(\omega(\tau)\equiv 0\) in some right-hand neighborhood of \(0\), or if \(\omega(\tau)\) changes sign in every right-hand neighborhood of \(0\). In both cases \(\tau=0\) must be a limit point of the zeros of \(\omega'(\tau)\). But \(\omega'(\tau)=\Psi(\tau)\); therefore \(\tau=0\) is a limit point of the zeros of \(\Psi(\tau)\).
Leningrad Civil Engineering Institute
Received
12 III 1965
CITED LITERATURE
- S. Mandelbrojt, Adjacent Series. Regularization of Sequences. Applications, Moscow, 1955.
- S. Mandelbrojt, Closure Theorems and Composition Theorems, Moscow, 1960.
- N. Wiener, The Fourier Integral and Some of Its Applications, Moscow, 1963.
- N. Wiener, R. Paley, Fourier Transforms in the Complex Domain, Moscow, 1964.
- W. H. J. Fuchs, J. Lond. Math. Soc., 22, 19 (1947).
\[ {}^*\ \text{Obviously, } \Psi(x)\not\equiv 0. \]