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MATHEMATICS
Academician of the Academy of Sciences of the Armenian SSR M. M. Dzhrbashyan
ON THE REPRESENTATION OF CERTAIN CLASSES OF ENTIRE AND QUASI-ENTIRE FUNCTIONS
In the present note we give formulations of several new results on the integral representation of certain general classes of analytic functions.
A. Parametric representation of entire functions
1°. In the author’s papers \((^{1,2})\) certain theorems of Paley–Wiener type \((^3)\) were established on the parametric representation of entire functions of finite order \(\rho \geq 1/2\) and normal type, square-integrable in modulus along certain systems of rays of the complex plane.
These representations were constructed with the aid of entire functions of Mittag-Leffler type
\[ E_\rho(z;\mu)=\sum_{k=0}^{\infty}\frac{z^k}{\Gamma(\mu+k\rho^{-1})}, \tag{1} \]
whose fine asymptotic properties formed the basis of the investigations mentioned.
A further development of the method by which these theorems were obtained has made it possible to establish a result of considerably more general nature, essentially of a final character.
Let us introduce the following notation. Let \(\{\vartheta_k\}_0^{\chi+1}\) be a set of numbers
\[ -\pi<\vartheta_0<\vartheta_1<\cdots<\vartheta_\chi\leq \pi<\vartheta_{\chi+1}=\vartheta_0+2\pi, \]
\[ \max_{0\leq k\leq \chi}\{\vartheta_{k+1}-\vartheta_k\}=\pi/\rho, \]
where \(\rho\geq 1/2,\ \chi=[\chi]\geq [2\rho]-1\).
Forming consecutive pairs \((\vartheta_k,\vartheta_{k+1})_0^\chi\), let us select from them (preserving their mutual order of succession) all those \((\vartheta_{r_k},\vartheta_{r_{k+1}})_0^p\) \((0\leq p\leq \chi)\) for which \(\vartheta_{r_{k+1}}-\vartheta_{r_k}=\pi/\rho\) \((k=0,1,\ldots,p)\), and put \(\theta_k=\frac12\{\vartheta_{r_k}+\vartheta_{r_{k+1}}\}\) \((k=0,1,\ldots,p)\). Finally, assuming that \(\omega\in(-1,+1)\) and \(\sigma_k\geq 0\) \((k=0,1,\ldots,p)\), consider the class \(W_\sigma^{(\rho)}(\omega;\{\vartheta_k\};\{\sigma_k\})\) of entire functions \(f(z)\) of order \(\rho\) and normal type \(\leq \sigma\), satisfying the conditions:
\[ 1)\quad \int_0^\infty \left|f\left(te^{-i\theta_k}\right)\right|^2 t^\omega\,dt<+\infty \quad (k=0,1,\ldots,\chi); \]
\[ 2)\quad h(-\theta_k;f)\leq \sigma_k\leq \sigma \quad (k=0,1,\ldots,p), \]
where \(h(\vartheta;f)\) is the indicator of the function \(f(z)\).
The following fundamental theorem on the parametric representation of the class \(W_\sigma^{(\rho)}(\omega;\{\vartheta_k\};\{\sigma_k\})\) has been established.
Theorem 1. a) The class \(W_\sigma^{(\rho)}(\omega;\{\vartheta_k\};\{\sigma_k\})\) coincides with the set of functions \(f(z)\) admitting a representation of the form
\[ f(z)=\sum_{k=0}^{p}\int_{0}^{\sigma_k} E_\rho\{e^{i\theta_k}zt^{1/\rho};\mu\}\varphi_k(t)t^{\mu-1}\,dt, \tag{2} \]
where \(\mu=(1+\omega+\rho)/2\rho\) and \(\varphi_k(t)\in L_2(0,\sigma_k)\) \((k=0,1,\ldots,p)\);
b) the functions \(\varphi_k(\tau)\) are unique, and almost everywhere
\[ \frac{i}{\sqrt{2\pi}\rho} \left\{ e^{-i\frac{\pi}{2}\mu}\Phi_{r_{k+1}}(-\tau) - e^{i\frac{\pi}{2}\mu}\Phi_{r_k}(\tau) \right\} = \begin{cases} \varphi_k(\tau), & \tau\in(0,\sigma_k),\\ 0, & \tau\in(\sigma_k,+\infty), \end{cases} \quad (k=0,1,\ldots,p) \tag{3} \]
where
\[ \Phi_k(\tau)= \frac{1}{\sqrt{2\pi}}\frac{d}{d\tau} \int_0^\infty f\left(e^{-i\vartheta_k}v^{1/\rho}\right) \frac{e^{-i\vartheta_k}-1}{-iv}\, v^{\mu-1}\,dv \quad (k=0,1,\ldots,\chi). \tag{3'} \]
\(2^\circ\). We shall give only one of the numerous consequences following from this theorem and having independent interest.
For a given integer \(p\geq 1\), denote by \(C_\sigma^{(\rho)}(\omega;\{\sigma\})\) the class of entire functions \(f(z)\) of order \(\rho\) \((p\leq \rho<2p)\) and of normal type \(\leq \sigma\), satisfying the following conditions:
1) the integrals
\[
\int_0^\infty |f(te^{-i\vartheta})|^2 t^\omega\,dt
\quad (-1<\omega<1)
\]
are finite in all intervals
\[
\frac{\pi}{p}k-\delta_p\leq \vartheta\leq \frac{\pi}{p}k+\delta_p
\quad (k=0,1,\ldots,2p-1),
\]
where
\[
\delta_p=\frac{\pi}{2}\left(\frac1p-\frac1\rho\right)<\frac{\pi}{p};
\]
2)
\[
h\left(-\frac{\pi}{p}\left(k+\frac12\right);f\right)\leq \sigma_k\leq \sigma.
\quad (k=0,1,\ldots,2p-1).
\]
Theorem 2. a) The class \(C_\sigma^{(\rho)}(\omega;\{\sigma_k\})\) coincides with the set of functions \(f(z)\) admitting the representation
\[ f(z)= \sum_{k=0}^{2p-1} \int_0^{\sigma_k} E_\rho \left\{ e^{i\frac{\pi}{p}\left(k+\frac12\right)} z\tau^{1/\rho}; \mu \right\} \varphi_k(\tau)\tau^{\mu-1}\,d\tau, \tag{4} \]
where \(\mu=(1+\omega+\rho)/2\rho\) and \(\varphi_k(\tau)\in L_2(0,\sigma_k)\) \((k=0,1,\ldots,2p-1)\);
b) the formulas
\[ \frac{i}{\sqrt{2\pi}\rho} \left\{ e^{-i\frac{\pi}{2}\mu}\psi_{k+1}^{(-)}(-\tau) - e^{i\frac{\pi}{2}\mu}\psi_k^{(+)}(\tau) \right\} = \begin{cases} \varphi_k(\tau), & \tau\in(0,\sigma_k),\\ 0, & \tau\in(\sigma_k,+\infty), \end{cases} \quad (k=0,1,\ldots,2p-1) \tag{5} \]
are valid, where
\[ \psi_k^{(\pm)}(\tau)= \frac{1}{\sqrt{2\pi}}\frac{d}{d\tau} \int_0^\infty \frac{e^{-i\tau v}-1}{-iv}\, f\left(e^{-i\left(\frac{\pi}{p}k+\delta_p\right)}v^{1/\rho}\right) v^{\mu-1}\,dv \quad (k=0,1,\ldots,2p-1). \tag{6} \]
Let us note that, in particular, for \(\rho=p=1\) and \(\omega=0\), this theorem yields the classical Paley–Wiener theorem on the representation of entire functions of exponential type belonging to the class \(L_2(-\infty,+\infty)\).
B. Representation of functions analytic on the Riemann surface of the logarithm.
\(1^\circ\). In a joint work of A. E. Avetisyan and the author [4], a representation was established for functions analytic in the angle
\[
\Delta(\alpha):\{|\Arg z|<\pi/2\alpha,\ 0<|z|<\infty\}
\quad (1/2<\alpha<\infty),
\]
of opening \(\pi/\alpha<2\pi\), and possessing
with a prescribed finite growth. This representation was constructed by means of a special contour integral transformation with a kernel of the form \(E_\rho(z\zeta;\mu)\).
Analogous representations can be established for analytic functions of finite growth, defined in an angle of type \(\Delta(\alpha)\), but of arbitrary opening \(\pi/\alpha\) \((0<\alpha<\infty)\), lying on the Riemann surface of the logarithmic function, i.e. in the domain \(G_\infty:\{-\infty<\operatorname{Arg} z<+\infty,\ 0<|z|<\infty\}\). However, in this case the representations obtained by us are integral transformations along special contours lying on \(G_\infty\), and with a kernel of the form \(v_\rho(z\zeta;\mu)\), where, by definition,
\[ v_\rho(z;\mu)=\int_0^\infty \frac{z^t}{\Gamma(\mu+t/\rho)}\,dt. \tag{7} \]
Let us note that the function \(v_\rho(z;\mu)\), being, obviously, a continual analogue of the function \(E_\rho(z;\mu)\), is connected with the well-known Volterra function
\[ v(z;\mu)=\int_0^\infty \frac{z^{t+\mu}}{\Gamma(1+\mu+t)}\,dt \tag{7'} \]
by the formula
\[ v_\rho(z;\mu)=\rho z^{\rho(1-\mu)}v\left(z^\rho;\mu-1\right). \tag{8} \]
For this reason, in establishing the two main theorems given below, we relied substantially on the important asymptotic properties of the function \(v(z;\mu)\), investigated in the author’s paper (5).
We introduce several preliminary notations. Let \(D_\rho(\vartheta;\nu)\) \((0<\rho<\infty,\ -\infty<\vartheta<\infty,\ 0\leqslant\nu<\infty)\) be the unbounded domain \(\operatorname{Re}(e^{-i\vartheta}\zeta)^\rho>\nu,\ |\operatorname{Arg}\zeta-\vartheta|<\pi/2\rho\) with boundary \(L_\rho(\vartheta;\nu):\{(e^{-i\vartheta}\zeta)^\rho=\nu,\ -\infty<\tau<+\infty\}\), lying on the surface \(G_\infty\).
The union of the domains \(\{D_\rho(\vartheta;\nu)\}\) over all values of the parameter \(\vartheta\in[-\pi/2\alpha,\pi/2\alpha]\) will be denoted by \(D^{(\alpha)}(\nu)\). The contour \(L_\rho^{(\alpha)}(\nu)\) of the unbounded domain \(D^{(\alpha)}(\nu)\in G_\infty\) consists of the arc \(-\pi/2\alpha\leqslant\operatorname{Arg}\zeta\leqslant\pi/2\alpha\), \(|\zeta|=\nu^{1/\rho}\), and of the unbounded curves beginning at its endpoints
\[ L_\rho^{(\pm)}\left(\pm\frac{\pi}{2\alpha};\nu\right):(e^{\pm i\pi/2\alpha}\zeta)^\rho=\nu\pm i\tau,\quad 0\leqslant\tau<+\infty. \]
Finally, denote by \(A^{(\alpha)}[\rho_1,\sigma_1]\) \((0<\alpha<\infty;\ 0\leqslant\sigma_1<\infty;\ 0<\rho_1<\infty)\) the class of functions \(F(z)\), analytic in the domain \(\Delta(\alpha)\in G_\infty\), for which the estimate \(|F(z)|\leqslant M_F e^{\sigma_1|z|^{\rho_1}},\ z\in\Delta(\alpha)\), holds.
The integral representation of the class \(A^{(\alpha)}[\rho_1,\sigma_1]\) is given by the theorem:
Theorem 3. If \(F(z)\in A^{(\alpha)}[\rho_1,\sigma_1]\), then for every \(\rho\geqslant\rho_1\) the following two assertions hold:
a) for each \(\vartheta\in[-\pi/2\alpha,\pi/2\alpha]\) the formula
\[ g_\rho(\zeta;F)=\rho\,(e^{-i\vartheta}\zeta)^{\mu\rho}\zeta^{-1} \int_0^\infty F\left(te^{-i\vartheta}\right)e^{-t^\rho(e^{-i\vartheta}\zeta)^\rho}t^{\mu\rho-1}\,dt,\quad \zeta\in D_\rho(\vartheta;\nu)\ (\mu>0) \tag{9} \]
defines a function analytic in the domain \(D_\rho^{(\alpha)}(\nu_0)\), where \(\nu_0=\sigma_1\) for \(\rho=\rho_1\) and \(\nu_0=0\) for \(\rho>\rho_1\);
b) the integral formula holds
\[ F(z)=\frac{1}{2\pi i}\int_{L_\rho^{(\alpha)}(\chi)} v_\rho(z\zeta;\mu)\,g_\rho(\zeta;F)\,d\zeta,\quad z\in\Delta(\alpha), \tag{10} \]
for any \(\chi>\nu_0\) and \(\mu\in(0,1/2]\).
2°. We shall call a function \(f(z)\) quasi-entire if it is regular on the whole Riemann surface \(G_\infty\), except for its branch points \(z=0\) and \(z=\infty\), and if, for any way of tending with \(z\in G_\infty\) to the point \(z=0\), there exists a finite limit
\[
f(0)=\lim_{z\to 0} f(z).
\]
We assign to the class \(C_{(\rho,\sigma)}\) quasi-entire functions \(f(z)\) satisfying the conditions:
1) \[ M_f(r)=\sup_{-\infty<\vartheta<\infty}|f(re^{i\vartheta})|<+\infty,\qquad 0<r<\infty; \]
2) the order of the function \(M_f(r)\) is not greater than \(\rho\), i.e.
\[
\overline{\lim}_{r\to\infty}(\log r)^{-1}\log_2 M_f(r)\le \rho,
\]
and, in the case of equality, we also have
\[
\overline{\lim}_{r\to\infty} r^{-\rho}\log M_f(r)\le \sigma.
\]
As follows easily from the asymptotics of the function \(\nu(z;\mu)\), the simplest and most important example of a function of the class \(C_{(\rho,\sigma)}\) is the function \(\nu_\rho(z\xi;\mu)\) for any \(\rho>0\), \(\mu\in(0,+\infty)\), and \(|\xi|=\sigma^{1/\rho}\) \((^5)\).
For each function \(f(z)\in C_{(\rho,\sigma)}\) and for any \(\vartheta\in(-\infty,+\infty)\), one can define its Borel-type transform \(g_\rho(\zeta;f)\) according to formula (9). It is analytic on the entire Riemann surface \(G_\infty(\chi):\{-\infty<\operatorname{Arg}\zeta<+\infty,\ \chi<|\zeta|<\infty\}\), where \(\chi=\sigma^{1/\rho}\) if the function \(M_f(r)\) has order \(\rho\) and type \(\sigma\), and \(\chi=0\) if the order or the type of this function is lower than \(\rho\) or \(\sigma\).
We note that in the work of A. Pfluger \((^6)\) a formula was proposed which makes it possible to reconstruct a function \(f(z)\in C_{(1,\sigma)}\) by means of its Borel transform \(g_1(\zeta;f)\), analogously to Pólya’s well-known theorem for entire functions of exponential type.
But the indicated formula of A. Pfluger has the essential shortcoming that, being an integral transform of the function \(g_1(\zeta;f)\) with kernel \(e^{z\zeta}\) along a special contour depending on a parameter \(\Phi\in(-\infty,\infty)\), it represents the function \(f(z)\) not on the whole surface \(G_\infty\), but only in the corresponding half-plane
\[
\operatorname{Re}(e^{-i\Phi}z)>0.
\]
A natural and complete solution of the problem of representing quasi-entire functions of the class \(C_{(\rho,\sigma)}\) is given by the following theorem.
Theorem 4. If \(f(z)\in C_{(\rho,\sigma)}\) and \(g_\rho(\zeta;f)\) is its Borel-type transform, then for any \(\tau>\chi\) and \(\mu\in(0,\tfrac12]\) the integral formula is valid
\[
f(z)=\frac{1}{2\pi\rho i}\int_{-\infty}^{\infty}
\nu_\rho'(\tau z e^{i\vartheta};\mu)\,
g_\rho(\tau e^{i\vartheta};f)\,d(\tau e^{i\vartheta}),
\qquad z\in G_\infty .
\tag{11}
\]
In conclusion, we note that Theorems 3 and 4 may be regarded as peculiar approximation theorems for functions of the classes \(A^{(\alpha)}[\rho_1,\sigma_1]\) and \(C_{(\rho,\sigma)}\) by quasi-entire functions of the simplest nature, namely the function \(\nu_\rho(z\xi;\mu)\).
Institute of Mathematics and Mechanics
Academy of Sciences of the Armenian SSR
Received
17 VII 1964
REFERENCES
- M. M. Dzhrbashyan, Matem. sborn., 33 (75), 3, 485 (1953).
- M. M. Dzhrbashyan, Izv. AN SSSR, ser. matem., 19, 133 (1955).
- R. Paley, N. Wiener, Fourier Transforms in the Complex Domain, N. Y., 1934.
- M. M. Dzhrbashyan, A. E. Avetisyan, Sibirsk. matem. zhurn., 1, 3, 383 (1960).
- M. M. Dzhrbashyan, Izv. AN SSSR, ser. matem., 24, 387 (1960).
- A. Pfluger, Comm. Math. Helv., 8, 89 (1935/36).