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).