Abstract
Full Text
MATHEMATICS
G. V. BADALYAN
A CONDITION FOR THE EXPANDABILITY OF FUNCTIONS IN A QUASI-POWER SERIES OUTSIDE THE FUNDAMENTAL INTERVAL
(Presented by Academician M. V. Keldysh on 19 VII 1960)
In the paper \((^1)\) a condition was established for the expandability of functions in the quasi-power series
\[ \varphi(t)=\sum_{n=0}^{\infty} a_n \omega_n\left(\frac{t}{u},\gamma\right), \tag{1} \]
where \(t \in (0,u]\), \(\omega_0\left(\frac{t}{u},\gamma\right)=1\),
\[ \omega_n\left(\frac{t}{u},\gamma\right) = \prod_{\nu=1}^{n}\gamma_\nu \cdot u^{-\gamma_n} \int_u^t t_1^{\gamma_1-1}\,dt_1 \int_u^{t_1} t_2^{\gamma_2-\gamma_1-1}\,dt_2 \cdots \int_u^{t_{n-1}} t_n^{\gamma_n-\gamma_{n-1}-1}\,dt_n; \tag{2} \]
\[ a_0=\varphi(u), \qquad a_n=\frac{u^{\gamma_{n-1}+1}\varphi_n(u)}{\prod_{\nu=1}^{n}\gamma_\nu}, \qquad \nu=1,2,\ldots; \]
\[ \varphi_1(u)=\varphi'(u), \qquad \varphi_{k+1}(u)= \left( \frac{\varphi_k(u)}{u^{\gamma_k-\gamma_{k-1}-1}} \right)', \qquad k=1,2,\ldots; \tag{3} \]
\[ 0=\gamma_0<\gamma_1\leqslant \gamma_2\leqslant \cdots \leqslant \gamma_n\leqslant \cdots, \qquad \sum_{\nu=1}^{\infty}\frac{1}{\gamma_\nu}=\infty. \]
In the present paper we consider the question of expanding functions in the convergent series (1), but only for values \(t>u\); we call this interval the nonfundamental interval.
Definition. We shall agree to say that
\[ \varphi(t)\in C_{\{\prod_1 \gamma_\nu\}}^{*}([u,A]), \qquad 0<u<A<\infty, \]
if \(\varphi(t)\) is infinitely differentiable and
\[ |\varphi_n(t)|\leqslant C\prod_{\nu=1}^{n}\gamma_\nu t^{-\gamma_n-1}, \qquad t\in [u,A]. \tag{4} \]
Here \(\varphi_n(t)\) is defined in (3), and \(C>0\) is an absolute constant.
It turns out that the conditions for the expandability of functions in the series (1) on the fundamental interval \((0,u]\) differ essentially from the conditions for expandability outside the fundamental interval, as follows from the following theorems.
Theorem 1. For every function
\[ \varphi(t)\in C_{\{\prod_1 \gamma_\nu\}}^{*}([u,A]), \qquad t\in [u,A], \qquad 0<u<A, \]
the inequality
\[ |R_n(u,t)|\leqslant c\omega_n\left(\frac{t}{u},\gamma\right), \tag{5} \]
holds.
where
\[ R_n(u,t)=\varphi(t)-\sum_{k=0}^{n-1} a_k \omega_k\left(\frac{t}{u},\gamma\right). \tag{6} \]
Consider sequences of numbers
\[ \begin{gathered} 0=\alpha_0<\alpha_1<\alpha_2<\cdots,\\ 0=\beta_0<\beta_1<\beta_2<\cdots,\qquad \beta_\nu\leqslant\alpha_\nu,\quad \nu=1,2,\ldots \end{gathered} \tag{7} \]
and construct the functions \(\omega_n\left(\frac{t}{u},\alpha\right)\), \(\omega_n\left(\frac{t}{u},\beta\right)\).
Theorem 2. For \(t>u\) the inequalities
\[ \prod_{\nu=1}^{n}\frac{\alpha_\nu}{\beta_\nu}G_n(1)\omega_n\left(\frac{t}{u},\beta\right) \leqslant \omega_n\left(\frac{t}{u},\alpha\right) \leqslant \prod_{\nu=1}^{n}\frac{\alpha_\nu}{\beta_\nu}G_n\left(\frac{t}{u}\right) \left(\frac{t}{u}\right)^{-\beta_n} \omega_n\left(\frac{t}{u},\beta\right), \tag{8} \]
hold, where \(\{\alpha_\nu\}, \{\beta_\nu\}\) are defined in (7),
\[ G_n(x)=\frac{1}{2\pi i}\int_C \prod_{\nu=1}^{n-1} \frac{z+\beta_\nu}{z+\alpha_\nu}\, \frac{x^{-\zeta}\,d\zeta}{z+\alpha_n}, \]
and the simple contour \(C\) encloses neighborhoods of the zeros of the denominator of the integrand.
Corollary. Under the conditions of Theorem 2 the inequalities
\[ \prod_{\nu=1}^{n}\frac{\alpha_\nu}{\beta_\nu}\omega_n\left(\frac{t}{u},\beta\right) \leqslant \omega_n\left(\frac{t}{u},\alpha\right) \leqslant \prod_{\nu=1}^{n}\frac{\alpha_\nu}{\beta_\nu} \prod_{\nu=1}^{n-1}\frac{\alpha_n-\beta_\nu}{\alpha_n-\alpha_\nu} \left(\frac{t}{u}\right)^{\alpha_n-\beta_n} \omega_n\left(\frac{t}{u},\beta\right). \tag{9} \]
With the aid of inequalities (5) and (9) one proves:
Theorem A. Let
\[ \varphi(t)\in C^{*}_{\left\{\prod_{1}^{n}\gamma_\nu\right\}}([u,v]), \]
\[ 0=\gamma_0<\gamma_1<\gamma_2<\cdots,\qquad \gamma_\nu\leqslant \omega\nu,\quad \omega>0, \]
\[ \prod_{\nu=1}^{n}\frac{\gamma_\nu}{\omega\nu} \left[\left(\frac{v}{u}\right)^{\omega}-1\right]^n\to 0, \qquad \text{as } n\to\infty. \tag{10} \]
Then the series
\[ \varphi(t)=\sum_{n=0}^{\infty} a_n\omega_n\left(\frac{t}{u},\gamma\right) \]
converges uniformly for all \(t\in [u,v]\).
Theorem B. Let
\[ \varphi(t)\in C^{*}_{\left\{\prod_{1}^{n}\gamma_\nu\right\}}([u,v]), \]
\[ \gamma_0=0,\qquad \gamma_\nu\geqslant \omega\nu,\quad \omega>0, \]
\[ \lim_{n\to\infty}\prod_{\nu=1}^{n}\frac{\gamma_\nu}{\omega\nu} \prod_{\nu=1}^{n-1}\frac{\gamma_n-\omega\nu}{\gamma_n-\gamma_\nu} \left(\frac{v}{u}\right)^{\gamma_n-\omega n} \left[\left(\frac{v}{u}\right)^{\omega}-1\right]^n=0. \tag{11} \]
Then the series
\[ \varphi(t)=\sum_{n=0}^{\infty} a_n\omega_n\left(\frac{t}{u},\gamma\right) \]
converges uniformly on the whole interval \([u,v]\).
Remark. Theorems A and B show that, as the rate of growth of the sequence \(\{\gamma_\nu\}\) increases, the interval of convergence \([u, v]\) of the series (1) generally narrows, and there is reason to suppose that, for sufficiently rapid growth of \(\{\gamma_\nu\}\), the series (1) can converge only at the point \(t=u\).
That this is indeed so, and that the sufficient conditions given in Theorems A and B cannot be substantially sharpened (in the sense of describing the phenomenon), is proved by the following propositions.
Theorem \(A_1\). Under the conditions of Theorem A there exists a function \(\varphi(t)\) for which, when
\[ \frac{1}{n}\prod_{\nu=1}^{n-1} \frac{|\mu-\gamma_\nu|}{\omega n-\gamma_\nu} \left(\frac{v_1}{u}\right)^{\omega\nu} \left[\left(\frac{v_1}{u}\right)^\omega-1\right]^n > c > 0, \tag{12} \]
where \(u<v_1<v\), \(\mu\) is a certain number, the series
\[ \varphi(t)=\sum_{n=0}^{\infty} a_n \omega_n\left(\frac{t}{u}, \gamma\right) \]
diverges for all \(t\in [v_1,v]\).
Theorem \(B_1\). Under the conditions of Theorem B there exists a function \(\varphi(t)\) such that, when
\[ \gamma_0=0,\qquad \gamma_\nu>\omega(\nu+1),\qquad \omega>0,\qquad \nu=1,2,\ldots; \]
\[ \frac{1}{n}\prod_{\nu=1}^{n-1} \frac{|\mu-\gamma_\nu|}{\omega\nu} \left[\left(\frac{v_1}{u}\right)^\omega-1\right]^n > c > 0 \]
the series
\[ \varphi(t)=\sum_{n=0}^{\infty} a_n \omega_n\left(\frac{t}{u}, \gamma\right) \]
diverges for all \(t\in [v_1,v]\).
Theorem \(B'_1\). Under the conditions of Theorem B, if
\[ \lim_{\nu\to\infty}\frac{\gamma_\nu}{\nu}=\infty, \]
there exists a function \(\varphi(t)\) such that the series
\[ \varphi(t)=\sum_{n=0}^{\infty} a_n \omega_n\left(\frac{t}{u}, \gamma\right) \]
diverges for all \(t>u\).
Institute of Mathematics and Mechanics
Academy of Sciences of the Armenian SSR
Received
21 V 1960
References
- G. V. Badalyan, Izv. AN ArmSSR, 6, No. 5—6 (1953).