MATHEMATICS

G. V. BADALYAN

GENERALIZED REGULARLY MONOTONE FUNCTIONS AND A CRITERION FOR ABSOLUTE CONVERGENCE OF A QUASI-POWER SERIES

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

Definition 1. A function \(\varphi(t)\) on \((0,u]\), \(u>0\), belongs to the class of generalized regularly monotone functions with respect to the sequence of numbers

\[ 0=\gamma_0<\gamma_1\leqslant \gamma_2\leqslant\cdots, \tag{1} \]

briefly, to the class \(R_\gamma(0,u]\), if on \((0,u]\) there exists a sequence of functions

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

and the conditions

\[ (-1)^k\varphi_k(t)\geqslant 0. \tag{3} \]

are satisfied.

Definition \(1'\). A function \(\varphi(t)\) belongs to the class of functions \(R_{\gamma,\varkappa}(0,u]\) if it belongs to the class \(R_\gamma(0,u]\) and, moreover, the lower bound of all numbers \(\mu\) for which \(\lim\limits_{t\to+0} t^\mu\varphi(t)=0\) is equal to \(\varkappa\).

Definition 2. A function \(\varphi(t)\) belongs to the class of functions \(T_\gamma(0,u]\) if it can be expanded into a series converging on \((0,u]\) to the function \(\varphi(t)\),

\[ \varphi(t)=\sum_{k=0}^{\infty} a_k\omega_k\left(\frac{t}{u},\gamma\right) \tag{4} \]

(see \((^1)\)). If the order of uniform convergence of the series (4) on \([0,u]\) is then equal to \(\varkappa\), then the class \(T_\gamma(0,u]\) will be denoted by \(T_{\gamma,\varkappa}(0,u]\).

If, instead of ordinary convergence, absolute convergence of the series (4) on \((0,u]\) is allowed, then the corresponding classes of functions will be denoted by \(AT_\gamma(0,u]\) and \(AT_{\gamma,\varkappa}(0,u]\).

Definition 3. A function \(\varphi(t)\) belongs to the class of functions \(AC_{\gamma,\varkappa}(0,u]\), where \(\varkappa\geqslant 0\) is an arbitrary number, the sequence of numbers \(\{\gamma_\nu\}\) is defined in (1), if for \(t\in(0,u]\) there exists a sequence of functions (2) and, moreover, the conditions are satisfied

\[ \int_0^u |\varphi_{n+1}(t)|\, t^{\gamma_n+\varkappa_1}\,dt \leqslant C\prod_{\nu=1}^{n}(\varkappa' + \gamma_\nu), \quad n=1,2,\ldots, \tag{5} \]

where \(\varkappa_1\) and \(\varkappa'\) \((\varkappa_1>\varkappa'>\varkappa\geqslant 0)\) are arbitrary numbers; \(C\) is a constant independent of \(n\).

Theorem 1. Let there be given sequences of numbers \(0=\gamma_0<\gamma_1\leqslant\gamma_2\leqslant\cdots,\quad 0=\gamma'_0<\gamma'_1\leqslant\gamma'_2\leqslant\cdots\), where \(\gamma_\nu\leqslant\gamma'_\nu,\ \nu=1,2,\ldots,\)

and the corresponding classes of functions \(R_\gamma(0,u]\), \(R_{\gamma'}(0,u]\). Then
\[ R_\gamma(0,u]\subset R_{\gamma'}(0,u]. \]

The converse assertion, generally speaking, is false.

Theorem 2. Every function \(\varphi(t)\in R_\gamma(0,u]\) also belongs to the class \(T_\gamma(0,u]\), if and only if the sequence (1) satisfies the condition
\[ \sum_{\nu=1}^{\infty}\frac{1}{\gamma_\nu}=\infty . \tag{6} \]

Theorem 3. In order that a function \(\varphi(t)\in AT_\gamma(0,u]\), where the sequence \(\{\gamma_\nu\}\) defined in (1) satisfies condition (6), it is necessary and sufficient that it be representable as the difference of two functions of the class \(R_\gamma(0,u]\).

To clarify the interrelation of the classes of functions \(AT_\gamma(0,u]\) and \(R_\gamma(0,u]\), the following is proved:

Theorem 4. In order that \(\varphi(t)\in AT_{\gamma,\chi}(0,u]\), \(\chi\geq 0\), where the sequence \(\{\gamma_\nu\}\) is defined in (1) and (6), it is necessary and sufficient that \(\varphi(t)\) be representable as the difference of two functions \(\psi(t)\) and \(g(t)\), where \(\psi(t)\in R_{\gamma,\mu}(0,u]\), \(g(t)\in R_{\gamma,\mu'}(0,u]\), \(\mu\leq\chi\), \(\mu'\leq\chi\), and in at least one of the last inequalities the equality sign holds.

Theorem 5. In order that \(\varphi(t)\in AT_{\gamma,\chi}(0,u]\), \(\chi\geq 0\), where the sequence of numbers (1) satisfies condition (6), it is necessary and sufficient that \(\varphi(t)\in AC_{\gamma,\chi}(0,u]\).

Theorem 6. For the quasianalyticity of the class of functions \(AC_{\gamma\chi}(0,u]\) (see \((^1)\)) it is necessary and sufficient that the sequence of numbers (1) satisfy condition (6).

From Theorem 6 and Theorem 2 of note \((^2)\) it follows:

Theorem 7. The totality of all quasianalytic classes \(C_{\gamma,\chi}(0,u]\) (see \((^2)\)) coincides with the totality of all quasianalytic classes of functions \(AC_{\gamma,\chi}(0,u]\), whereas for the specific classes \(C_{\gamma,\chi}(0,u]\) and \(AC_{\gamma,\chi}(0,u]\) there is a strict inclusion
\[ AC_{\gamma,\chi}(0,u]\subset C_{\gamma,\chi}(0,u]. \]

Institute of Mathematics and Mechanics
Academy of Sciences of the Armenian SSR

Received
13 VII 1961

CITED LITERATURE

\(^6\) G. V. Badalyan, DAN, 136, No. 1 (1961). \(\quad\) \(^2\) G. V. Badalyan, DAN, 141, No. 5 (1961).