MATHEMATICS

G. V. BADALYAN

A CONDITION FOR THE EXPANDABILITY OF FUNCTIONS IN A QUASI-POWER SERIES IN THE BASIC INTERVAL

(Presented by Academician M. V. Keldysh on 10 VII 1950)

In the paper \((^1)\) a condition was established for the expandability of functions of certain quasi-analytic classes in the quasi-power series

\[ \varphi(t)=\sum_{n=0}^{\infty} a_n \omega_n(u,t), \tag{1} \]

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

\[ \omega_n(u,t)=\int_u^t t_1^{\gamma_1-1}\,dt_1 \int_u^{t_1} t_2^{\gamma_2-\gamma_1-1}\,dt_2\cdots \int_u^{t_{n-1}} t_n^{\gamma_n-\gamma_{n-1}-1}\,dt_n; \]

\[ 0=\gamma_0<\gamma_1\leqslant\gamma_2\leqslant\cdots\to\infty, \qquad \sum_{\nu=1}^{\infty}\frac{1}{\gamma_\nu}=\infty; \]

\[ a_0=\varphi(u),\qquad a_k=u^{-\gamma_k+\gamma_{k-1}+1}\varphi_k(u),\qquad k=1,2,\ldots; \]

\[ \varphi_1(u)=\varphi'(u),\qquad \varphi_{k+1}(u)= \left( \frac{\varphi_k(u)}{u^{\gamma_k-\gamma_{k-1}-1}} \right)', \qquad k=1,2,\ldots. \tag{2} \]

In the particular case when \(\gamma_k=k,\ k=0,1,2,\ldots\), series (1) coincides with the ordinary Taylor series.

In the present paper theorem (2,7) of \((^1)\) is refined, and an analogous theorem is also proved for slow growth of the sequence \(\{\gamma_\nu\}\)—a case which was not considered in \((^1)\).

Definition 1. We shall agree to say that a function \(\varphi(t)\) belongs to the class \(C_{\{\dot m_n\}}([0,1])\) of analytic or quasi-analytic functions if the following conditions are satisfied:

1) \(|\varphi^{(n)}(t)|\leqslant \dot m_n,\ n=0,1,2,\ldots,\ t\in[0,1]\), where the sequence \(\{\dot m_n\}\) is convexly regularized with respect to logarithms:

\[ 2)\quad \int^{\infty}\frac{\log \dot T(r)}{r^2}\,dr=\infty, \qquad \dot T(r)=\sup_{n\geqslant 1}\frac{r^n}{\dot m_n}; \]

3) the counting function \(n(r)=n(r,\beta)\) of the sequence

\[ \left\{\beta\,\frac{\dot m_n}{\dot m_{n-1}}\right\}, \]

\(0<\beta<\infty\), is such that

\[ \varlimsup_{r\to\infty} \frac{r^2}{n(r)} \int_r^\infty \frac{n(t)\,dt}{t(t^2+r^2)} <\Delta_1=\log\Delta. \tag{3} \]

Definition 2. We shall agree to say that

\[ \varphi(t)\in C_{\{\dot m_n\},\,\varkappa}([0,1]), \]

if

\[ t^{\varkappa}\varphi(t)\in C_{\{\dot m_n\}}([0,1]), \]

where \(\varkappa\) is an arbitrary number.

Theorem 1. Every function \(\varphi(t)\) belonging on \([0,1]\) to some class \(C_{\{\dot m_n\},\varkappa}\) of analytic or quasi-analytic functions is expanded into the quasipower series, convergent on \((0,u]\), \(0<u<1\),

\[ \varphi(t)=\sum_{n=0}^{\infty} a_n\omega_n(u,t) \]

for the sequence \(\{\gamma_\nu\}\), if the conditions

\[ \gamma_0=0,\qquad \gamma_\nu=\beta\,\frac{\dot m_\nu}{\dot m_{\nu-1}},\qquad \nu=1,2,\ldots, \]

are satisfied, where \(\beta>6ea\Delta\), \(a\geq 2\left(1+\sqrt{\frac{1}{1-u}}\right)\), and \(\Delta\) is defined in (3).

Corollary. Theorem 1 is valid, in particular, when condition (3) of Definition 1 is replaced by the condition

\[ \sup_{t>r}\frac{n(t)}{t}<\frac{n(r)}{r}\log\Delta . \tag{3′} \]

Theorem (2,7) of work \({}^{1}\) is a special case of the corollary of Theorem 1, since from the condition imposed there on \(\left\{\frac{\dot m_n}{\dot m_{n-1}}\right\}\) it follows that \(\frac{n(t)}{t}\downarrow\) monotonically, and moreover the number \(\beta\) obtained here is smaller than the corresponding coefficient of \(\frac{\dot m_n}{\dot m_{n-1}}\) in (1).

Let us note that Theorem 1 gives a solution of Carleman’s problem, posed in 1926, on representing a quasi-analytic function by its element (the set of successive derivatives at one point) under an additional restriction, namely under condition (3).

We now consider the case when the counting function \(n(r)\) of the sequence \(\left\{\frac{m_n}{m_{n-1}}\right\}\) is such that

\[ n(r)\Delta\geq r, \tag{4} \]

where \(\Delta>0\) is some number. In the latter case we shall say that the sequence \(\left\{\frac{m_n}{m_{n-1}}\right\}\) grows slowly.

Definition 1′. A function \(\varphi(t)\) on \([0,1]\) belongs to the class \(C_{\{m_n\}}([0,1])\) of analytic or quasi-analytic functions if the following conditions are satisfied:

1) \(|\varphi^{(n)}(t)|\leq m_n,\quad n=0,1,2,\ldots,\ t\in[0,1]\), where the sequence \(\{m_n\}\) is logarithmically convex regularized;

2)

\[ \int^{\infty}\frac{\log T(r)}{r^2}\,dr=\infty,\qquad T(r)=\sup_{n\geq1}\frac{r^n}{m_n}. \]

The class of functions \(C_{\{\dot m_n\}}([0,1])\) differs from the class \(C_{\{m_n\}}([0,1])\) in that in the latter case the third condition (3) is omitted.

Analogously, the class

\[ C_{\{m_n,\varkappa\}}([0,1]) \]

is defined.

Theorem 2. Every function

\[ \varphi(t)\in C_{\{m_n,\varkappa\}}([0,1]) \]

can be expanded into a quasi-power series convergent on \((0,u]\), \(0<u<1\),

\[ \varphi(t)=\sum_{n=0}^{\infty} a_n\omega_n(u,t) \]

for the sequence \(\{\gamma_\nu\}\), if the conditions

\[ \gamma_0=0,\qquad \gamma_\nu=\beta\,\frac{m_\nu}{m_{\nu-1}},\qquad \nu=1,2,\ldots, \]

are satisfied, where \(\beta>12\cdot 3^\Delta e\), and \(\Delta>0\) is defined in (4).

In contrast to the proof of theorem (2,7) of paper \({}^{1}\), here the function \(\varphi(t)\) is first expanded in a Fourier series in Legendre polynomials, then generalizedly differentiated in the sense of (2), after which \(|\varphi_{n+1}(t)|\) is estimated from above. The proof is completed by the estimate

\[ R_n(u,t)=\varphi(t)-\sum_{k=0}^{n} a_k\omega_k(u,t)= \]

\[ =\int_u^t \varphi_{n+1}(t_0)\,dt_0 \int_{t_0}^t t_1^{\gamma_1-1}\,dt_1 \int_{t_0}^{t_1} t_2^{\gamma_2-\gamma_1-1}\,dt_2\cdots \int_{t_0}^{t_{n-1}} t_n^{\gamma_n-\gamma_{n-1}-1}\,dt_n . \]

Let us note that the estimate of \(|\varphi_{n+1}(t)|\) under the conditions of the second theorem differs in many respects from the same estimate in theorem 1.

Institute of Mathematics and Mechanics
Academy of Sciences of the ArmSSR

Received
21 V 1960

REFERENCES

\({}^{1}\) G. V. Badalyan, Izv. AN ArmSSR, 6, No. 5—6 (1953).