MATHEMATICS

G. V. BADALYAN

A CRITERION FOR THE EXPANDABILITY OF FUNCTIONS IN A QUASI-POWER SERIES AND QUASI-ANALYTIC CLASSES OF FUNCTIONS

(Presented by Academician I. N. Vekua, 14 VII 1961)

We consider the quasi-power series

\[ \varphi(t)=\sum_{k=0}^{\infty} a_k \omega_k\left(\frac{t}{u},\gamma\right), \tag{1} \]

where \(t \in (0,u]\), \(u>0\),

\[ a_0=\varphi(u), \qquad a_k=\frac{(-1)^k u^{\gamma_k-1}\varphi_k(u)}{\prod_{\nu=1}^{k}\gamma_\nu}, \]

\[ \varphi_0(t)=\varphi(t), \qquad \varphi_1(t)=\varphi'(t), \qquad \varphi_{k+1}(t)=\left(\frac{\varphi_k(t)}{t^{\gamma_k-\gamma_{k-1}-1}}\right)', \qquad k=1,2,\ldots, \tag{2} \]

\[ 0=\gamma_0<\gamma_1\leqslant\gamma_2\leqslant\cdots, \tag{3} \]

\[ \omega_k\left(\frac{t}{u},\gamma\right) =(-1)^k u^{-\gamma_k}\prod_{\nu=1}^{k}\gamma_\nu \int_u^t t_1^{\gamma_1-1}\,dt_1 \int_u^{t_1} t_2^{\gamma_2-\gamma_1-1}\,dt_2 \cdots \int_u^{t_{n-1}} t_n^{\gamma_n-\gamma_{n-1}-1}\,dt_n = \]

\[ =\frac{\prod_{\nu=1}^{k}\gamma_\nu}{2\pi i} \int_C \frac{(t/u)^{-\xi}\,d\xi}{\prod_{\nu=0}^{k}(\xi+\gamma_\nu)}, \qquad k=0,1,2,\ldots, \tag{4} \]

the simple contour \(C\) surrounding neighborhoods of all poles of the integrand in (1).

Definition 1. The lower bound of all numbers \(\mu\) for which the series

\[ t^\mu \sum_{k=0}^{\infty} a_k\omega_k\left(\frac{t}{u},\gamma\right) \]

converges uniformly on \([0,u]\) will be called the order of uniform convergence of the series (1) on \([0,u]\).

Definition 2. A function \(\varphi(t)\) on \((0,u]\), \(u>0\), belongs to the class of functions \(T_{\gamma,\varkappa}\) \(\bigl(T_{\gamma,\varkappa}(0,u]\bigr)\), if it is expandable into a quasi-power series (1), convergent on \((0,u]\) to the function \(\varphi(t)\), with finite order of uniform convergence \(\varkappa\) on \([0,u]\).

Definition 3. An infinitely differentiable function \(\varphi(t)\) on \((0,u]\), \(u>0\), belongs to the class of functions \(C_{\gamma,\varkappa}\) \(\bigl(C_{\gamma,\varkappa}(0,u]\bigr)\), \(\varkappa\geqslant0\), if, for the given sequence of numbers (3), its

there exists a sequence of functions (2) satisfying the condition

\[ \left|\int_{x}^{u}\varphi_{n+1}(t)t^{\gamma_n+x_1}\,dt\right| \leq C\prod_{\nu=1}^{n}(x' + \gamma_\nu),\quad n=0,1,2,\ldots, \tag{5} \]

where \(x\in(0,u]\), \(x_1\) and \(x'\) (\(x_1>x'>x\geq 0\)) are arbitrary numbers, and \(C\) is a constant independent of \(n\) and \(x\).

Definition 4. The class of functions \(C_{\gamma,x}(0,u]\) is called a quasianalytic class if, from the vanishing of any function of the class and of all its successive derivatives at any point of the interval \((0,u]\), it follows that \(\varphi(t)\equiv 0\) on all of \((0,u]\).

Theorem 1. In order that

\[ \varphi(t)\in T_{\gamma,x}(0,u],\quad x\geq 0, \]

it is necessary and sufficient that:

1) \(\varphi(t)\in C_{\gamma,x}(0,u]\);

2) the sequence of numbers (3) satisfy the condition

\[ \sum_{\nu=1}^{\infty}\frac{1}{\gamma_\nu}=\infty. \tag{3'} \]

Theorem 2. For the quasianalyticity of the class of functions \(C_{\gamma,x}(0,u]\), it is necessary and sufficient that the sequence of numbers (3) satisfy condition (3').

From Theorems 1 and 2 it follows:

Theorem 3. In order that

\[ \varphi(t)\in T_{\gamma,x}(0,u],\quad u>0,\ x\geq 0, \]

it is necessary and sufficient that the function \(\varphi(t)\) belong to the quasianalytic class of functions \(C_{\gamma,x}(0,u]\).

Thus it turns out that the function \(\varphi(t)\) is expanded into a series of the form (1), convergent on \((0,u]\) to the function \(\varphi(t)\), if and only if it is uniquely determined by the totality of all its successive derivatives at one point of the domain of definition.

We note that Theorem 3 definitively solves Carleman’s problem on the representation of functions of quasianalytic classes by an analogue of the Taylor series \((^2)\).

In the case when \(\gamma_\nu=\nu\), \(\nu=0,1,2,\ldots\), the series (1) becomes the classical Taylor series of the function \(\varphi(t)\) in a neighborhood of the point \(u\).

Yerevan State
University

Received
13 VII 1961

REFERENCES

\(^1\) G. V. Badalyan, DAN, 136, No. 1 (1961). \(^2\) T. Carleman, Les fonctions quasianalytiques, Paris, 1926.