Abstract
Full Text
MATHEMATICS
Academician of the Academy of Sciences of the Armenian SSR M. M. Dzhrbashyan
A GENERALIZED RIEMANN–LIOUVILLE OPERATOR AND SOME OF ITS APPLICATIONS
In the author’s monograph ((^1)), the operator of fractional integro-differentiation of Riemann—Liouville found an essential application in the theory of the parametric representation of meromorphic functions.
In the present note we give the construction of a generalized operator of Riemann—Liouville type, which makes it possible, from a considerably more general point of view, to approach the solution of questions in the theory of classes of analytic functions and to establish a number of new results in this direction. Below we give the statements of these results, as well as a new sufficient condition for the solvability of the Hausdorff moment problem that is naturally connected with them.
(1^\circ). Denote by (\Omega) the set of functions (\omega(x)) satisfying the conditions:
1) (\omega(x)) is nonnegative and continuous on ([0,1)), and
[
\omega(0)=1,\qquad \int_0^1 \omega(x)\,dx<+\infty;
]
2) for every (r) ((0\le r<1))
[
\int_r^1 \omega(x)\,dx>0.
]
Further, for (\omega(x)\in\Omega) define the function
[
p(0)=1,\qquad p(\tau)=\tau\int_\tau^1 \frac{\omega(x)}{x^2}\,dx,\qquad \tau\in(0,1],
\tag{1}
]
continuous on ([0,1]), and we shall agree to write (p(\tau)\in P_\omega).
Finally, for (p(\tau)\in P_\omega) introduce into consideration the function
[
\Delta(r)=(1+r)\int_0^1 \tau^r\,dp(\tau),\qquad r\in[0,+\infty),
\tag{2}
]
continuous on the half-axis ([0,+\infty)) and representable also in the form
[
\Delta(0)=1,\qquad \Delta(r)=r\int_0^1 \omega(x)x^{r-1}\,dx,\quad r\in(0,+\infty).
\tag{2′}
]
Having then defined the sequence of positive numbers
[
\Delta_k=\Delta(k)\qquad (k=0,1,2,\ldots),
]
we introduce into consideration the power series
[
C(z;\omega)=\sum_{k=0}^{\infty}\Delta_k^{-1}z^k,\qquad
S(z;\omega)=1+2\sum_{k=1}^{\infty}\Delta_k^{-1}z^k,
\tag{3}
]
whose radius of convergence is equal to unity.
Concerning these functions in a neighborhood of their singular point (z=1), the following has been established.
Lemma 1. a) If
[
0<\underline{\lim}{x\to 1-0}\,\omega(x)\le
\overline{\lim}\,\omega(x)<+\infty,
]
then
[
0<\underline{\lim}_{r\to 1-0}(1-r)C(r;\omega)<+\infty.
]
b) If (\displaystyle \lim_{x\to 1-0}\omega(x)=0), then (\displaystyle \lim_{r\to 1-0}(1-r)C(r;\omega)=\infty).
c) If (\displaystyle \lim_{x\to 1-0}\omega(x)=+\infty), then (\displaystyle \lim_{r\to 1-0}(1-r)C(r;\omega)=0).
A more delicate property of the functions (C(z;\omega)) and (S(z;\omega)) is contained in the following theorem:
Theorem 1. If the function (\omega(x)\in\Omega) is nondecreasing on ([0,1)), then
[
\operatorname{Re} C(z;\omega)\geq 0,\qquad \operatorname{Re} S(z;\omega)\geq 0\qquad (|z|<1).
\tag{4}
]
Let us note that in the special case when (\omega(x)=(1-x)^\alpha) ((-1<\alpha<+\infty)), we immediately obtain
[
\Delta_k=\Gamma(1+\alpha)\Gamma(1+k)/\Gamma(1+\alpha+k)\qquad (k=0,1,\ldots),
\tag{5}
]
[
C(z;\omega)=\frac{1}{(1-z)^{1+\alpha}},\qquad
S(z;\omega)=\frac{2}{(1-z)^{1+\alpha}}-1.
]
In this case, for the values (-1<\alpha\leq 0), property (4) of these functions is easily verified.
(2^\circ). Assuming that (\omega(x)\in\Omega) and (p(\tau)\in P_\omega), we introduce into consideration the operator
[
L^{(\omega)}[\varphi(x)]\equiv
-\frac{d}{dx}\left{x\int_0^1 \varphi(x\tau)\,dp(\tau)\right},
\qquad x\in(0,1),
\tag{6}
]
assuming that, on the appropriate classes of admissible functions, its right-hand side exists at least almost everywhere on ((0,1)).
Let us note that in the simplest case, when (\omega(x)\equiv 1), as is easy to see, almost everywhere
[
L^{(\omega)}[\varphi(x)]=\varphi(x)
]
for (\varphi(x)\in L(0,1)). In the case (\omega(x)=(1-x)^\alpha) ((-1<\alpha<+\infty)), one can establish the formula
[
L^{(\omega)}[\varphi(x)]
=\Gamma(1+\alpha)x^{-\alpha}D^{-\alpha}\varphi(x),
\qquad x\in(0,1),
\tag{7}
]
where (D^{-\alpha}) is the Riemann–Liouville operator, i.e.
[
D^0\varphi(x)=\varphi(x),\qquad
D^{-\alpha}\varphi(x)=\frac{1}{\Gamma(\alpha)}
\int_0^x (x-t)^{\alpha-1}\varphi(t)\,dt
\qquad (0<\alpha<+\infty),
]
[
D^{-\alpha}\varphi(x)=\frac{d}{dx}D^{-(1+\alpha)}\varphi(x)
\qquad (-1<\alpha<0).
\tag{8}
]
In the general case, under certain additional conditions imposed on the admissible functions (\omega(x)\in\Omega) or on the functions (\varphi(x)), the operator (L^{(\omega)}) can be represented in other forms.
Lemma 2. a) Let (\omega(x)\in\Omega) be continuous on ([0,1]), (\omega(1)=0), and, in addition, (\omega'(x)\in L(0,1)). Then, in the class of bounded summable functions (\varphi(x)) on ((0,1)),
[
L^{(\omega)}[\varphi(x)]=-\int_0^1 \varphi(x\tau)\omega'(\tau)\,d\tau.
\tag{9}
]
b) Let (\omega(x)\in\Omega). Then in the class (C_1[0,1]) of functions (\varphi(x)) continuously differentiable on ([0,1]),
[
L^{(\omega)}[\varphi(x)]
=\varphi(0)+x\int_0^1 \varphi'(x\tau)\omega(\tau)\,d\tau.
\tag{10}
]
Finally, let us note the formulas
[
L^{(\omega)}[1]=1,\qquad
L^{(\omega)}[x^r]=\Delta(r)x^r,\qquad r\in(0,+\infty),\quad x\in[0,1],
\tag{11}
]
which follow from (10) and ((2')).
By a direct application of formulas (10) and (11) one establishes
Theorem 2. a) Let the function
[
f(re^{i\varphi})=\sum_{k=0}^{\infty} a_k (re^{i\varphi})^k
\tag{12}
]
holomorphic in the disk (|z|<R). Then the function
[
L^{(\omega)}[f(re^{i\varphi})]\equiv f_{(\omega)}(re^{i\varphi})
=\sum_{k=0}^{\infty}\Delta_k a_k (re^{i\varphi})^k
\tag{13}
]
is holomorphic in the same disk (|z|<R).
b) For any (\rho) ((0<\rho<R)) the integral formulas
[
f(z)=\frac{1}{2\pi}\int_0^{2\pi} C\left(e^{-i\theta}\frac{z}{\rho};\omega\right)
f_{(\omega)}(\rho e^{i\theta})\,d\theta
\qquad (|z|<\rho),
\tag{14}
]
[
f(z)=i\,\operatorname{Im} f(0)+\frac{1}{2\pi}\int_0^{2\pi}
S\left(e^{-i\theta}\frac{z}{\rho};\omega\right)
\operatorname{Re} f_{(\omega)}(\rho e^{i\theta})\,d\theta
\qquad (|z|<\rho).
\tag{15}
]
3°. We shall now give a representation of some general classes of harmonic and analytic functions in the disk (|z|<1) associated with the given function (\omega(x)\in\Omega). We note that in the case (\omega(x)\equiv 1) these classes and their representations are well known (see, for example, (2)). The more general case (\omega(x)=(1-x)^\alpha) ((-1<\alpha<+\infty)) was recently considered by us (1).
Denote by (U_\omega) the set of functions (u(z)) harmonic in the disk (|z|<1) for which
[
U[u;\omega]=\sup_{0\le r<1}\left{\int_0^{2\pi}
|u_{(\omega)}(re^{i\varphi})|\,d\varphi\right}<+\infty,
\tag{16}
]
where, as usual, (\omega(x)\in\Omega) and (u_{(\omega)}(re^{i\varphi})=L^{(\omega)}[u(re^{i\varphi})]). The function
[
P(\theta,r;\omega)=\operatorname{Re} S(re^{i\theta};\omega)
=1+2\sum_{k=1}^{\infty}\Delta_k^{-1} r^k\cos k\varphi
\tag{17}
]
belongs to the class (U_\omega), since, by virtue of (10) and (11),
[
L^{(\omega)}[P(\theta,r;\omega)]
=1+2\sum_{k=1}^{\infty} r^k\cos k\varphi
=\frac{1-r^2}{1-2r\cos\theta+r^2}\equiv P(\theta,r)\ge 0,
\tag{18}
]
and thus (P_{(\omega)}(\theta,r;\omega)) is the Poisson kernel. We also note that if (\omega(x)\in\Omega) does not decrease on ([0,1)), then, according to Theorem 1, we shall have
[
P(\theta,r;\omega)\ge 0
\qquad (0\le r<1,\ 0\le \theta\le 2\pi).
\tag{19}
]
Theorem 3. a) The class (U_\omega) coincides with the set of functions of the form
[
u(re^{i\varphi})=\frac{1}{2\pi}\int_0^{2\pi}
P(\varphi-\theta,r;\omega)\,d\psi(\theta),
\tag{20}
]
where (\psi(\theta)) is an arbitrary function of bounded variation on ([0,2\pi]), and, for some sequence of numbers (\rho_n\uparrow 1),
[
\psi(\theta)=\lim_{n\to\infty}\int_0^\theta
u_{(\omega)}(\rho_n e^{i\varphi})\,d\varphi,
\qquad \theta\in[0,2\pi].
\tag{21}
]
b) The class (U_\omega^*\subset U_\omega) of functions (u(z)) for which (u_{(\omega)}(z)\ge 0) ((|z|<1)) coincides with the set of functions of the form (20), where (\psi(\theta)) is an arbitrary nondecreasing function.
In the case (\omega(x)\equiv 1), this theorem contains the known theorem (2) on the class of functions (U) representable by the Poisson–Stieltjes integral, since then (u_{(\omega)}(z)\equiv u(z)) and (P(\theta,r,1)\equiv P(\theta,r)).
Relying on Lemma 1 and Theorem 3, one can prove:
a) (U_\omega\subset U), if (\omega(x)\uparrow\infty) as (x\uparrow 1);
b) (U\subset U_\omega), if (\omega(x)\downarrow 0) as (x\uparrow 1),
and both inclusions are proper.
From Theorem 3 there follows the following generalization of the Herglotz theorem (3).
Theorem 4. The class (C_\omega) of functions (f(z)) analytic in the disk (|z|<1) for which
[
\operatorname{Re} f_{(\omega)}(z)\ge 0
\qquad (|z|<1),
]
coincides with the set of functions of the form
[
f(z)=iC+\frac{1}{2\pi}\int_{0}^{2\pi} S(e^{-i\theta}z;\omega)\,d\psi(\theta)
\quad (|z|<1),
\tag{23}
]
where (\psi(\theta)) is an arbitrary function of bounded variation on ([0,2\pi]).
4°. We now give an application of the functions of the class (\Omega) to the Hausdorff moment problem ((^{4,5})), which also allows us to construct the inverse of the operator (L^{(\omega)}) in the case when (\omega(x)) is a nondecreasing function.
Theorem 5. Let the function (\omega(x)\in\Omega) on ([0,1]) be nondecreasing. Then:
a) There exists a nondecreasing function (a_\omega(x)), bounded on ([0,1]), continuous at the point (x=0), and such that the function
[
U(r)=\Delta^{-1}(r)=\left{r\int_{0}^{1}\omega(x)x^{r-1}\,dx\right}^{-1}
\tag{24}
]
admits the representation
[
U(r)=\int_{0}^{1}x^r\,da_\omega(x),\qquad r\in[0,+\infty).
\tag{25}
]
b) The Hausdorff moment problem
[
\mu_n=\Delta_n^{-1}=\int_{0}^{1}x^n\,da(x)\qquad (n=0,1,2,\ldots)
]
has the solution (a(x)=a_\omega(x)) in the class of nondecreasing functions bounded on ([0,1]).
We note that for (\omega(x)\equiv 1) we also have (U(r)=\Delta^{-1}(r)\equiv 1), (r\in[0,+\infty)), and therefore in this case, in the representation (25), the function (a_\omega(x)) must have the form
[
a_\omega(x)\equiv c\quad (0\le x<1),\qquad a_\omega(1)=c+1.
\tag{26}
]
Finally, under the assumption that the function (\omega(x)\in\Omega) on ([0,1]) is nondecreasing, along with the operator (L^{(\omega)}[\varphi(x)]) we consider the integral operator
[
M^{(\omega)}[\varphi(x)]\equiv\int_{0}^{1}\varphi(x\tau)\,da_\omega(\tau),\qquad x\in(0,1),
\tag{27}
]
assuming that (\varphi(x)) is at least continuous on ([0,1]), while (a_\omega(\tau)) is the function whose existence was asserted in Theorem 5.
We note that in the case (\omega(x)\equiv 1) we shall have (M^{(\omega)}[\varphi(x)]\equiv\varphi(x)), since then (a_\omega(x)) is determined by (26). In the more general case, when (\omega(x)=(1-x)^\alpha) ((-1<\alpha<0)), one can, along with (7), establish the formula
[
M^{(\omega)}[\varphi(x)]=\Gamma^{-1}(1+\alpha)D^\alpha{x^\alpha\varphi(x)}.
\tag{28}
]
In this case the operators (L^{(\omega)}) and (M^{(\omega)}) will be inverses of each other, since for (-1<\alpha<0) the operators (D^\alpha) and (D^{-\alpha}) are known ((^{1})) to possess this property. In the general case the following is true.
Theorem 6. In the class (C_1[0,1]) of functions (\varphi(x)) continuously differentiable on ([0,1]), the operators (L^{(\omega)}) and (M^{(\omega)}) are inverses of each other; that is, for any function (\varphi(x)\in C_1[0,1]),
[
L^{(\omega)}M^{(\omega)}[\varphi(x)]\equiv M^{(\omega)}L^{(\omega)}[\varphi(x)]\equiv\varphi(x),\qquad x\in[0,1].
\tag{29}
]
In conclusion, we note that the operators (L^{(\omega)}) and (M^{(\omega)}), as well as their special cases when (\omega(x)=(1-x)^\alpha) ((-1<\alpha<+\infty)), can also be applied in the theory of the representation of meromorphic functions.
Institute of Mathematics and Mechanics
Academy of Sciences of the Armenian SSR
Received
12 IX 1967
REFERENCES
- M. M. Dzhrbashyan, Integral Transforms and Representations of Functions in the Complex Domain, vol. IX, “Nauka,” 1966.
- I. I. Privalov, Boundary Properties of Analytic Functions, Moscow–Leningrad, 1950.
- G. Herglotz, Leipzig Ber., 63, 501 (1911).
- F. Hausdorff, Math. Zs., 9, 280 (1921).
- J. Shohat, J. Tomarkin, The Problem of Moments, N. Y., 1943, pp. 86–89.