Reports of the Academy of Sciences of the USSR
1958. Volume 121, No. 2

MATHEMATICS

Academician of the Academy of Sciences of the Armenian SSR M. M. Dzhrbashyan and A. B. Nersesyan

ON THE APPLICATION OF CERTAIN INTEGRO-DIFFERENTIAL OPERATORS

In the present note we give formulations of a number of new results in the theory of Dirichlet series and in the theory of quasianalytic classes of functions. These results are obtained by introducing special integro-differential operators connected with the concept of fractional integration in the sense of Riemann—Liouville or H. Weyl.

\(1^\circ\). Let the function \(F(\sigma)\) be defined and continuous on the half-axis \((\sigma_0,+\infty)\). For any \(\alpha>0\) define the operator

\[ \frac{d_e^{-\alpha}F(\sigma)}{d_e\sigma^{-\alpha}} \equiv \frac{1}{\Gamma(\alpha)} \int_{\sigma}^{+\infty}(e^{-\sigma}-e^{-u})^{\alpha-1}e^{-u}F(u)\,du, \tag{1} \]

calling it the fractional integral of the function \(F(\sigma)\) of order \(\alpha\) with endpoint at \(+\infty\). It is easy to see that
\[ \lim_{\alpha\to+0}\frac{d_e^{-\alpha}F(\sigma)}{d_e\sigma^{-\alpha}}\equiv F(\sigma), \]
and therefore it is natural to regard the function \(F(\sigma)\) itself as the integral of order zero.

Let the sequence \(\{\mu_n\}\) \((n\geqslant 0)\) satisfy the condition

\[ \mu_0=0;\qquad 0<\mu_{k+1}-\mu_k\leqslant 1\quad (k\geqslant 0);\qquad \lim_{k\to\infty}\mu_k=+\infty. \tag{2} \]

Denote \(\alpha_k=1-(\mu_k-\mu_{k-1})\) \((k=1,2,\ldots)\), and introduce the operators

\[ L^{(\mu_0)}F(\sigma)\equiv F(\sigma),\qquad L^{(\mu_k)}F(\sigma)\equiv -\frac{d_e^{-\alpha_k}}{d_e\sigma^{-\alpha_k}} \left\{e^\sigma\frac{d}{d\sigma}L^{(\mu_{k-1})}F(\sigma)\right\} \quad (k\geqslant 1), \tag{3} \]

assuming that all of them exist and are continuous on the half-axis \((\sigma_0,+\infty)\).

We shall agree to say that the function \(L^{(\mu_k)}F(\sigma)\) is continuous on \((\sigma_0,+\infty]\) if: 1) it is continuous on the interval \((\sigma_0,+\infty)\); 2) there exists a finite limit
\[ L^{(\mu_k)}F(+\infty)=\lim_{\sigma\to+\infty}L^{(\mu_k)}F(\sigma). \]
We shall say that \(F(\sigma)\in\mathscr{L}(\mu_n;\sigma_0)\) if all the functions \(L^{(\mu_k)}F(\sigma)\) \((k\geqslant 0)\) exist and are continuous on \((\sigma_0,+\infty]\), while the functions
\[ e^\sigma\frac{d}{d\sigma}L^{(\mu_k)}F(\sigma)\quad (k=0,1,2,\ldots) \]
are continuous and absolutely integrable on \((\sigma_1,+\infty)\), where \(\sigma_1>\sigma_0\) is arbitrary.

Theorem 1. If \(F(\sigma)\in\mathscr{L}(\mu_n;\sigma_0)\), then for any \(n\geqslant 0\) and \(\sigma\in(\sigma_0,+\infty]\) the formula holds

\[ F(\sigma)= \sum_{k=0}^{n} \frac{L^{(\mu_k)}F(+\infty)}{\Gamma(1+\mu_k)}e^{-\mu_k\sigma} + \frac{1}{\Gamma(\mu_{n+1})} \int_{\sigma}^{+\infty} (e^{-\sigma}-e^{-u})^{\mu_{n+1}-1}e^{-u} L^{(\mu_{n+1})}F(u)\,du. \tag{4} \]

This formula is, in a certain sense, an analogue of Taylor’s formula and, as is not difficult to see, in the particular case when \(\mu_n=n\) \((n=0,1,2,\ldots)\), after the change of variable \(e^{-\sigma}=x\), coincides with it.

From the class $\mathcal L(\mu_n;\sigma_0)$ we single out the subclass $\mathcal L^*(\mu_n;\sigma_0)$ of those functions $F(\sigma)$ for which, as $n\to\infty$, the integral remainder term in formula (4) tends uniformly to zero in any interval $[\sigma_1,+\infty]\subset(\sigma_0,+\infty]$. Consequently, if $F(\sigma)\in\mathcal L^*(\mu_n;\sigma_0)$, then the expansion into a Dirichlet series is valid:

\[ F(\sigma)=\sum_{k=0}^{\infty} \frac{L^{(\mu_k)}F(+\infty)}{\Gamma(1+\mu_k)}e^{-\mu_k\sigma}, \tag{5} \]

uniformly convergent on any half-line $[\sigma_1,+\infty]\subset(\sigma_0,+\infty]$. It is clear, moreover, that if $F(\sigma)\in\mathcal L^*(\mu_n;\sigma_0)$, then it admits an analytic continuation to the whole half-plane $\sigma=\operatorname{Re}s>\sigma_0$ $(s=\sigma+it)$, and the expansion (5) remains valid in the same half-plane.

The following criterion of necessary-and-sufficient type holds for the expansibility of functions in a Dirichlet series with respect to the given system $\{e^{-\mu_k s}\}$.

Theorem 2. Let the sequence $\{\mu_n\}$ $(n\geq 0)$ satisfy condition (1).

a) If $F(\sigma)\in\mathcal L(\mu_n;\sigma_0)$ and, in addition,

\[ \sup_{(\sigma_0,+\infty]} \left|L^{(\mu_k)}F(\sigma)\right| \leq Me^{-\sigma_0\mu_k}\Gamma(1+\mu_k) \quad (k\geq 0), \]

then $F(\sigma)\in\mathcal L^*(\mu_n;\sigma_0)$, i.e. the expansion

\[ F(\sigma)=\sum_{k=0}^{\infty} \frac{L^{(\mu_k)}F(+\infty)}{\Gamma(1+\mu_k)}e^{-\mu_k\sigma}, \qquad \sigma\in(\sigma_0,+\infty], \]

holds.

b) If, in addition to (1), the sequence $\{\mu_n\}$ also satisfies the condition

\[ \limsup_{k\to\infty}\frac{\log k}{\mu_k}<l<+\infty \tag{6} \]

and the expansion

\[ F(\sigma)=\sum_{k=0}^{\infty}a_k e^{-\mu_k\sigma}, \qquad \sigma\in(\sigma_0,+\infty], \]

holds, then for $\sigma_1=\sigma_0+l$ one has $F(\sigma)\in\mathcal L(\mu_n;\sigma_1)$; moreover,

\[ \sup_{(\sigma_1,+\infty]} \left|L^{(\mu_k)}F(\sigma)\right| \leq AB^{\mu_k}\Gamma(1+\mu_k) \quad (k\geq 0), \]

\[ a_k= \frac{L^{(\mu_k)}F(+\infty)}{\Gamma(1+\mu_k)} \quad (k\geq 0). \]

In the case when the condition $0<\mu_{k+1}-\mu_k\leq 1$ $(k\geq 0)$ is not fulfilled, the following result holds.

Theorem 3. Let the sequence $\{\mu_n\}$ $(n\geq 0)$ satisfy the conditions

\[ \mu_0\geq 0;\qquad 0<\mu_{k+1}-\mu_k\quad (k\geq 0);\qquad \lim_{k\to\infty}\mu_k=+\infty. \tag{7} \]

a) If $F(\sigma)$ is defined on $(\sigma_0,+\infty)$ and there exists an extension of $\{\mu_n\}$ to a sequence $\{\mu_n^*\}$ satisfying conditions (1), such that:

1) $F(\sigma)\in\mathcal L(\mu_n^*;\sigma_0)$;

2)

\[ \sup_{(\sigma_0,+\infty]} \left|L^{(\mu_k)}F(\sigma)\right| \leq Me^{-\sigma_0\mu_k}\Gamma(1+\mu_k) \quad (k\geq 0); \]

3) $L^{(\mu_k^*)}F(+\infty)=0$, if $\mu_k^*\notin\{\mu_n\}$,

then the expansion into a Dirichlet series is valid:

\[ F(\sigma)=\sum_{k=0}^{\infty} \frac{L^{(\mu_k)}F(+\infty)}{\Gamma(1+\mu_k)}e^{-\mu_k\sigma}, \qquad \sigma\in(\sigma_0,+\infty]. \]

In this case \(F(\sigma)\) can be analytically continued to the half-plane \(\sigma=\operatorname{Re}s>\sigma_0\) \((s=\sigma+it)\), and the expansion remains valid in the whole half-plane \(\sigma=\operatorname{Re}s>\sigma_0\).

b) Let \(\{\mu_n\}\) \((n\ge 0)\) satisfy conditions (7) and (6), and suppose that there is an expansion

\[ F(\sigma)=\sum_{k=0}^{\infty} a_k e^{-\mu_k\sigma}, \qquad \sigma\in(\sigma_0,+\infty). \]

For any completion of \(\{\mu_n\}\) to a sequence \(\{\mu_n^*\}\) satisfying conditions (1) and

\[ \limsup_{k\to+\infty}\frac{\log k}{\mu_k^*}<l^*<+\infty, \]

the following assertions hold for \(\sigma^*=\sigma_0+l^*\):

1) \(F(\sigma)\in \mathcal{L}(\mu_n^*;\sigma^*)\);

2)

\[ \sup_{(\sigma^*,+\infty)} \left|L^{(\mu_k)}(F(\sigma))\right| \le AB^{\mu_k}\Gamma(1+\mu_k)\qquad (k\ge 0); \]

3)

\[ a_k=\frac{L^{(\mu_k)}F(+\infty)}{\Gamma(1+\mu_k)}\qquad (k\ge 0), \]

where \(L^{(\mu_k)}F(\sigma)=L^{(\mu_{n_k}^*)}F(\sigma)\), if \(\mu_k=\mu_{n_k}^*\).

\(2^\circ\). For a function \(f(x)\), defined and continuous on the half-line \([0,+\infty)\), and for a given sequence \(\{\alpha_k\}\) \((k\ge 0)\), where \(0\le \alpha_k<1\) \((k\ge 0)\), we introduce the operators

\[ D^0 f(x)\equiv \frac{d^{-\alpha_0}}{dx^{-\alpha_0}} f(x);\qquad D^{(k)}f(x)\equiv \frac{d^{-\alpha_k}}{dx^{-\alpha_k}}\frac{d}{dx}D^{(k-1)}f(x)\qquad (k\ge 0), \]

where, for \(\alpha>0\),

\[ \frac{d^{-\alpha}}{dx^{-\alpha}}f(x)\equiv \frac{1}{\Gamma(\alpha)}\int_0^x (x-t)^{\alpha-1} f(t)\,dt \]

is the fractional integral of order \(\alpha\) of the function \(f(x)\) in the Riemann–Liouville sense.*

We shall call the function \(D^{(k)}f(x)\) (if it exists) the \(k\)-th generalized derivative of \(f(x)\) with respect to the given sequence \(\{\alpha_k\}\). We shall say that \(f(x)\in C\{\alpha_k\}\) if the functions \(D^{(k)}f(x)\) \((k\ge 0)\) are continuous on the half-line \([0,+\infty)\), and the functions

\[ \frac{d}{dx}D^{(k)}f(x)\qquad (k\ge 0) \]

are continuous on \((0,+\infty)\) and absolutely integrable on every interval \([0,\delta]\) \((\delta\ge 0)\).

Let \(\{m_n\}\) \((n\ge 0)\) be some sequence of positive numbers. We assign to the class \(C_{m_n}\{\alpha_k\}\) all those functions \(f(x)\) from the class \(C\{\alpha_k\}\) for which:

a)

\[ \left|D^{(k)}f(x)\right|\le AB^{\sum_1^k(1-\alpha_i)}\,m_k e^{Cx} \qquad (k\ge 0),\quad 0\le x<+\infty, \]

where \(A\), \(B\), and \(C\) are constants depending on the given function \(f(x)\);

b)

\[ e^{-Cx}\left|\frac{d}{dx}D^{(k)}f(x)\right|\in L_1(0,+\infty)\qquad (k\ge 0). \]

We shall say that the set of functions \(C_{m_n}\{\alpha_k\}\) constitutes a quasi-analytic class in the generalized sense if, for any functions \(f_1(x)\) and \(f_2(x)\in C_{m_n}\{\alpha_k\}\), the equalities \(D^{(k)}f_1(0)=D^{(k)}f_2(0)\) \((k\ge 0)\) imply the identity

\[ f_1(x)\equiv f_2(x),\qquad 0\le x<+\infty. \]

The following assertion holds—an analogue of the well-known Carleman–Ostrovsky theorem.

* It is easy to see the connection between the operators \(L^{(\mu_k)}F(\sigma)\) and \(D^{(k)}f(x)\), if one makes the change of variable \(x=e^{-\sigma}\).

Theorem 4. For the quasianalyticity of the class \(C_{m_n}\{\alpha_k\}\) it is necessary and sufficient that

\[ \int_{1}^{+\infty} \frac{\log T_{\alpha}(r)}{r^{2}}\,dr=+\infty, \]

where

\[ T_{\alpha}(r)=\sup_{n\geq 1}\frac{r^{\sum_{1}^{n}(1-\alpha_k)}}{m_n}. \]

3°. Let \(f(x)\) be given on \((-\infty,+\infty)\).

The Weyl integral of order \(\alpha>0\) of the function \(f(x)\) is the function

\[ W^{\alpha}f(x)=\frac{1}{\Gamma(\alpha)}\int_{-\infty}^{x}(x-t)^{\alpha-1}f(t)\,dt. \]

It is natural to set \(W^{0}f(x)\equiv f(x)\).

For a given sequence \(\{\alpha_k\}\) \((k\geq 0;\ 0\leq \alpha_k<1)\) we introduce the operations

\[ R^{(0)}f(x)\equiv W^{\alpha_0}f(x);\qquad R^{(k)}f(x)\equiv W^{\alpha_k}\frac{d}{dx}R^{(k-1)}f(x)\quad (k\geq 1). \]

It is easy to see that if \(\alpha_k=0\) \((k\geq 0)\), then \(R^{(k)}f(x)\equiv f^{(k)}(x)\).

Let the function \(p(x)\) be defined and continuously differentiable on the half-axis \([0,+\infty)\), and let \(\lim_{t\to+\infty}p'(t)=+\infty\), while the function \(q(x)\), as in \((^1)\), is conjugate to \(p(x)\) in the sense of Young. Let, further, \(\{m_n\}\) be some sequence of positive numbers.

We assign to the class \(C_{m_n}\{p(x);\alpha_k\}\) all functions satisfying the conditions:

a) The functions \(R^{(k)}f(x)\) and \(\dfrac{d}{dx}R^{(k)}f(x)\) exist and are continuous on the whole axis \((-\infty,+\infty)\);

b)

\[ |R^{(k)}f(x)|\leq m_k\omega_f(x)e^{-p_1(x)} \quad (k=0,1,2,\ldots),\qquad -\infty<x<+\infty, \]

where \(\omega_f(x)\geq 0\) is summable on \((-\infty,+\infty)\), and \(p_1(x)\equiv p(|x|)\) for \(x\leq 0\), while \(p_1(x)\geq C_0\) for \(x>0\) (\(C_0\) is a real constant);

c)

\[ \left|(1+x^2)\frac{d}{dx}R^{(k)}f(x)\right|\leq C_k \quad (k=0,1,2,\ldots),\qquad -\infty<x<+\infty, \]

where \(C_k>0\) are some constants.

Theorem 5. The class \(C_{m_n}\{p(x);\alpha_k\}\) is empty, in other words, contains only the function \(f(x)\equiv 0\), \(-\infty<x<+\infty\), if

\[ \lim_{R\to+\infty}\inf\left\{\frac{q(R)}{R}-\frac{2}{\pi}\int_{1}^{R}\frac{\log T_{\alpha}(r)}{r^{2}}\,dr\right\}=-\infty, \]

where

\[ T_{\alpha}(r)=\sup_{n\geq 1}\frac{r^{\sum_{1}^{n}(1-\alpha_k)}}{m_n}. \]

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

Received
21 IV 1958

CITED LITERATURE

1. M. M. Dzhrbashyan, Izv. AN ArmSSR, ser. phys.-mat. sciences, 10, No. 6, 7 (1957).