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MATHEMATICS
G. A. DZANASHIYA
ON CARLEMAN’S PROBLEM FOR THE CLASS OF GEVREY FUNCTIONS
(Presented by Academician A. N. Kolmogorov on 13 II 1962)
Let \(\{m_n\}\) be some sequence of positive numbers. Denote by \(C(m_n)\) the class of all infinitely differentiable functions \(f(x)\) on the segment \([-1,1]\) for each of which there exists an \(A>0\) such that
\[ |f^{(n)}(x)| \leq A^{n+1} m_n \]
for all \(x \in [-1,1]\) and all \(n=0,1,2,\ldots\).
In the present paper the following problem is solved. For any \(\alpha>1\) and any sequence of complex numbers \(\{v_n\}\) satisfying the condition \(|v_n|\leq B^{n+1} n^{n\alpha}\) for some \(B>0\), the existence is proved of a function \(f(x)\in C(n^{n\alpha})\) such that \(f^{(n)}(0)=v_n\). Moreover, the function \(f(x)\) is written explicitly.
Remark 1. This problem is analogous to the problem posed by Carleman \((^1)\) for the case when the class \(C(m_n)\) is quasianalytic. The class \(C(n^{n\alpha})\) is not quasianalytic for any \(\alpha>1\).
Remark 2. In the work of B. S. Mityagin \((^4)\) the fact of the existence of the above-mentioned function \(f(x)\) is proved. In the present paper the function \(f(x)\) is constructed explicitly in the form (1), which is an analogue of the Taylor series.
For the proof of the assertion formulated by us, consider the series
\[ \sum_{k=0}^{\infty}\frac{v_k}{k!}\,a_k(x)x^k, \tag{1} \]
where the numbers \(v_k\) are the same as above, and the functions \(a_k(x)\) are defined as follows.
Let \(b_k(x)\) be the functions given by the equalities
\[ b_k(x)= \begin{cases} 0, & \text{for } -1\leq x\leq -\sigma_k,\\[4pt] \exp\left(-\dfrac{k\sigma_k^{4r}}{x^{2r}(\sigma_k+x)^{2r}}\right), & \text{for } -\sigma_k\leq x\leq 0,\\[6pt] 0, & \text{for } 0\leq x\leq 1, \end{cases} \tag{2} \]
where \(r\) is any natural number such that \(\dfrac{1}{2r}<\alpha-1\), and \(\sigma_k=D^{-1}k^{-(\alpha-1)}\) for some \(D>0\). Put
\[ a_k(x)= \frac{\displaystyle\int_{-1}^{x} b_k(t)\,dt} {\displaystyle\int_{-1}^{1} b_k(t)\,dt} \tag{3} \]
for \(-1\leq x\leq 0\), and for \(0<x\leq 1\) put \(a_k(x)=a_k(-x)\).
We shall now prove that the series (1) converges, together with all its derivatives, uniformly on \([-1,1]\), and that its sum \(f(x)\) gives a solution of the problem, i.e. \(f(x)\in C(n^{n\alpha})\), and \(f^{(n)}(0)=v_n\). (The value of the number \(D>0\),
entering into \(b_k(x)\), will be specified later.) Let us estimate \(\left|a_k^{(n)}(x)\right|\).
First of all, let us estimate from below the integral \(\displaystyle \int_{-1}^{1} b_k(t)\,dt\). From (2) it is clear that \(b_k(x)\) increases for \(-\sigma_k < x < -\sigma_k/2\) and decreases for \(-\sigma_k/2 < x < 0\), and, moreover, \(b_k(-\sigma_k/2-t)=b_k(-\sigma_k/2+t)\) for \(0\le t\le \sigma_k/2\). Therefore,
\[ \int_{-1}^{1} b_k(t)\,dt = \int_{-\sigma_k}^{0} b_k(t)\,dt > \frac{\sigma_k}{2}\, b_k\!\left(-\frac{\sigma_k}{4}\right) \]
or, by virtue of (2),
\[ \int_{-\sigma_k}^{0} b_k(t)\,dt > \frac{\sigma_k}{2}\exp\!\left(-\frac{16^{2r}k}{3^{2r}}\right). \tag{4} \]
Let us estimate from above \(\left|b_k^{(n)}(x)\right|\). We shall use the fact that the function \(b_k(x)\) is analytic on the interval \((-\sigma_k,0)\), and apply Cauchy’s formula to the derivative \(b_k^{(n)}(x)\), taking as the contour of integration the circle with radius equal to \(-hx\), and center at the point \(x\in[-\sigma_k/2,0)\). We obtain
\[ b_k^{(n)}(x) = \frac{n!}{2\pi i} \int_{0}^{2\pi} \frac{b_k\!\left(x-hxe^{i\varphi}\right)} {\left(-hxe^{i\varphi}\right)^n}\,d\varphi . \]
Hence
\[ \left|b_k^{(n)}(x)\right| \le \frac{n!}{2\pi}\, \frac{1}{h^n|x|^n} \int_{0}^{2\pi} \left|b_k\!\left(x-hxe^{i\varphi}\right)\right|\,d\varphi . \]
On the other hand, after simple transformations we obtain
\[ \left|b_k\!\left(x-hxe^{i\varphi}\right)\right| = \exp\!\left( \frac{-k\sigma_k^{4r}\cos(2r\beta+2r\gamma)} {x^{2r}(1-2h\cos\varphi+h^2)^r\left[(\sigma_k+x-hx\cos\varphi)^2+h^2x^2\sin^2\varphi\right]^r} \right), \]
where \(\beta\) and \(\gamma\) are defined by the relations
\[ \sin\beta= \frac{h\sin\varphi}{(1-2h\cos\varphi+h^2)^{1/2}}, \qquad \sin\gamma= \frac{hx\sin\varphi} {\left[(\sigma_k+x-hx\cos\varphi)^2+h^2x^2\sin^2\varphi\right]^{1/2}} . \]
We see that if \(h\) is sufficiently small, then
\[ \left|b_k\!\left(x-hxe^{i\varphi}\right)\right| \le \exp\!\left(-\frac{L\sigma_k^{2r}k}{x^{2r}}\right), \]
where \(L>0\) depends only on \(r\). Consequently,
\[ \left|b_k^{(n)}(x)\right| \le \frac{n!}{h^n|x|^n} \exp\!\left(-\frac{L\sigma_k^{2r}k}{x^{2r}}\right). \]
It is easy to show that
\[ \max_{\sigma_k/2\le x\le 0} \left\{ \frac{1}{|x|^n} \exp\!\left(-\frac{L\sigma_k^{2r}k}{x^{2r}}\right) \right\} \le \frac{1}{\sigma_k^n} \left(\frac{n}{k}\right)^{n/2r} \left(\frac{1}{2rL}\right)^{n/2r} e^{-n/2r}. \]
Thus, we have the following estimate for \(\left|b_k^{(n)}(x)\right|\):
\[ \left|b_k^{(n)}(x)\right| \le \frac{T^n n! n^{n/2r}}{\sigma_k^n k^{n/2r}} \le \frac{T^n D^n n^n n^{n/2r} k^{n(\alpha-1)}}{k^{n/2r}} . \tag{5} \]
Here \(T=\max (T_1,1)\), where
\[ T_1=\frac{1}{h}\left(\frac{1}{2rL}\right)^{1/2r} e^{-1/2r}. \]
It is clear that \(T\) does not depend on \(n\) and \(k\). Using inequalities (4) and (5), we arrive at the following estimate for \(\left|a_k^{(n)}(x)\right|\):
\[ \left|a_k^{(n)}(x)\right|\leq 2\exp\left(\frac{16^{2r}k}{3^{2r}}\right)T^nD^n \frac{n^n n^{(n-1)/2r}k^{n(\alpha-1)}}{k^{(n-1)/2r}} . \tag{6} \]
Differentiating the series (1) \(n\) times, we obtain a series with general term
\[ w_k(x)=\frac{v_k}{k!}\sum_{i=0}^n {n\choose i} a_k^{(n-i)}(x)(x^k)^{(i)} . \]
On the basis of (6)
\[ \begin{aligned} |w_k(x)| &\leq 2\frac{|v_k|}{k!}\exp\left(\frac{16^{2r}k}{3^{2r}}\right) \sum_{i=0}^n {n\choose i}T^{\,n-i}D^{\,n-i} \frac{(n-i)^{n-i}(n-i)^{(n-i-1)/2r}} {k^{-(n-i)(\alpha-1)}k^{(n-i-1)/2r}}\,k^i\delta_k^{\,k-i} \leq \\ &\leq 2\frac{|v_k|}{k!}\exp\left(\frac{16^{2r}k}{3^{2r}}\right) \sum_{i=0}^n {n\choose i}T^{\,n-i}D^{\,n-i} \frac{(n-i)^{n-i}(n-i)^{(n-i-1)/2r}} {k^{-(n-i)(\alpha-1)}k^{(n-i-1)/2r}}\,k^iD^{-k+i}k^{-(k-i)(\alpha-1)} \leq \\ &\leq 2\frac{|v_k|}{k!}\exp\left(\frac{16^{2r}}{3^{2r}}\right) \sum_{i=0}^n {n\choose i}T^nD^nD^{-k} \frac{(n-i)^{n-1}(n-i)^{(n-i-1)/2r}k^i k^{-k(\alpha-1)}} {k^{-n(\alpha-1)}k^{(n-i-1)/2r}} \leq \\ &\leq 2\frac{|v_k|}{k!}\exp\left(\frac{16^{2r}k}{3^{2r}}\right) D^{-k}k^{-k(\alpha-1)} \sum_{i=0}^n {n\choose i}T^nD^n \frac{n^{n-i}n^{(n-i-1)/2r}k^i} {k^{-n(\alpha-1)}k^{(n-i-1)/2r}} . \end{aligned} \]
For \(k\leq n\) we shall have
\[ \begin{aligned} |w_k(x)| &\leq 2\frac{|v_k|}{k!}\exp\left(\frac{16^{2r}k}{3^{2r}}\right) D^{-k}k^{-k(\alpha-1)} \sum_{i=0}^n {n\choose i}T^nD^n \frac{n^{n-i}n^{(n-i-1)/2r}n^i} {n^{-n(\alpha-1)}n^{(n-i-1)/2r}} \leq \\ &\leq 2\frac{|v_k|}{k!}\exp\left(\frac{16^{2r}k}{3^{2r}}\right) D^{-k}k^{-k(\alpha-1)} \sum_{i=0}^n {n\choose i}T^nD^n n^{n\alpha}. \end{aligned} \]
By Stirling’s formula and the inequality \(|v_k|\leq B^{k+1}k^{k\alpha}\), we have
\[ |w_k(x)|\leq 2^{\,n-k+1}T^nD^n n^{n\alpha} \]
for sufficiently large \(D\). Therefore,
\[ \sum_{k=0}^n |w_k(x)|\leq 2^{\,n+2}T^nD^n n^{n\alpha}. \]
Now let \(k>n\). Then
\[ \begin{aligned} |w_k(x)| &\leq 2\frac{|v_k|}{k!}\exp\left(\frac{16^{2r}k}{3^{2r}}\right) D^{-k}k^{-k(\alpha-1)} \sum_{i=0}^n {n\choose i} \frac{k^{n-i}k^{(n-i-1)/2r}k^i} {k^{-n(\alpha-1)}k^{(n-i-1)/2r}} \leq \\ &\leq 2\frac{|v_k|}{k!}\exp\left(\frac{16^{2r}k}{3^{2r}}\right) D^{-k}k^{-k(\alpha-1)} \sum_{i=0}^n {n\choose i}T^nD^n k^{n\alpha}. \end{aligned} \]
Thus,
\[ \sum_{k=n+1}^{\infty}|w_k(x)| \leq 2^{\,n+1}T^nD^n \sum_{k=n+1}^{\infty} \frac{|v_k|}{k!} \exp\left(\frac{16^{2r}k}{3^r}\right) D^{-k}k^{-k(\alpha-1)}k^{n\alpha}. \]
Applying Stirling’s formula once more, we obtain
\[ \sum_{k=n+1}^{\infty} |w_k(x)| \leq 2^{2n+1}T^nD^nMn^{n\alpha}, \]
where for \(M\) one may take, for example, the number \((2\alpha)^\alpha\), and for \(D\) the number \(4B\exp\left(\frac{16^{2r}}{3^r}+2\right)\). As a result we have
\[ \sum_{k=0}^{\infty} |w_k(x)| \leq 2^{n+2}T^nD^n n^{n\alpha} +2^{2n+1}T^nM^nD^n n^{n\alpha} \leq 8^{n+1}T^nD^nMn^{n\alpha} \leq A^{n+1}n^{n\alpha}, \]
where \(A=8TDM\) and does not depend on \(n\).
Thus the uniform convergence of the series (1) and the membership of its sum \(f(x)\) in the class \(C(n^{n\alpha})\) have been proved. The fact that \(f^{(n)}(0)=v_n\) follows from the definition of \(a_k(x)\).
As an application, let us find, for the space \(S_\beta^\alpha\) (see (2)) with \(\alpha>1\), the general form of a linear continuous functional concentrated at a point. (This result was obtained by another method in the paper \({}^3\).)
Let \(H(\varphi)\) be a linear functional concentrated at zero, and \(\varphi\in S_\beta^\alpha\). From the preceding it is clear that the series
\[ \sum_{k=0}^{\infty} \frac{\varphi^{(k)}(0)}{k!}\,a_k(x)x^k \]
converges in \(S_\beta^\alpha\). Hence
\[ H(\varphi)=\sum_{k=0}^{\infty}\frac{\varphi^{(k)}(0)}{k!}\, H\bigl(a_k(x)x^k\bigr). \tag{7} \]
Since \(\varphi^{(k)}(0)\) may be any sequence of complex numbers satisfying the condition
\[
|\varphi^{(k)}(0)| \leq B^{k+1}k^{k\alpha},
\]
it follows that, for every \(\varepsilon>0\), starting from some \(k\),
\[ \left|H\bigl(a_k(x)x^k\bigr)\right| \leq \varepsilon^k k^{-k(\alpha-1)}. \]
Conversely, if the last condition is fulfilled, formula (7) defines a functional concentrated at the point \(x=0\).
Mathematical Institute named after A. M. Razmadze
Academy of Sciences of the Georgian SSR
Received
13 II 1962
REFERENCES
- T. Carleman, Leçons sur les fonctions quasi-analytiques, Paris, 1926.
- I. M. Gelfand, G. E. Shilov, Generalized Functions, vol. 2, Moscow, 1958.
- Ch. Roumieu, Ann. Sci. École Norm. Sup., 77, 1, 41 (1960).
- B. S. Mityagin, Dokl. Akad. Nauk SSSR, 138, No. 2 (1961).