Abstract
Full Text
Mathematics
L. Sh. Khodzhaev
On the Operator of Analytic Continuation of Generalized Functions
(Presented by Academician N. N. Bogolyubov on 10 V 1962)
The present paper has as its aim to extend the Cauchy integral operator
\[ K\rho=\frac{1}{2\pi i}\int_{-\infty}^{+\infty}\frac{\rho(x')}{x'-z}\,dx', \qquad \operatorname{Im} z\ne 0, \tag{1} \]
to the class of generalized functions. In connection with this we introduce for consideration certain spaces of generalized and analytic generalized functions that are of particular interest to us.
Let \(\mathscr D\) denote the space of real finite and infinitely differentiable functions \(\psi(x)\). One says that a sequence \(\{\psi_n\}\) of functions \(\psi_n\in\mathscr D\) tends to zero in \(\mathscr D\) if: 1) there exists a bounded interval \(\Delta\) of the \(x\)-axis outside which all the functions \(\psi_n\) vanish; 2) the functions \(\psi_n\) and their derivatives of arbitrary order tend to zero uniformly for each arbitrarily fixed order of derivatives. Convergence in \(\mathscr D\) will be denoted by \(\psi_n\Rightarrow\psi\) in \(\mathscr D\). Let \((\rho,\psi)\) be the number obtained by applying the functional \(\rho\) to the function \(\psi\in\mathscr D\). Such functionals are called generalized functions. (For details on generalized functions see \((^1)\).)
Let us define the notion of an analytic generalized function. Let \(\mathscr R\) be the space of holomorphic functions \(\Phi(z)\) defined in the strip
\[ t_a=\{z\mid |\operatorname{Im}z|<a\}, \qquad \text{where } 0<a\leqslant\infty, \tag{2} \]
and satisfying the inequality
\[ \left|\Phi^{(k)}(z)\right|\leqslant \frac{C^{(k)}}{|z|^{k+1}} \quad \text{for } |z|\to\infty, \tag{3} \]
where \(C^{(k)}\) is some constant number, \(k=0,1,2\ldots\).
We introduce the notion of convergence in \(\mathscr R\). One says that a sequence \(\{\Phi_n(z)\}\) of functions \(\Phi(z)\in\mathscr R\) converges to zero in \(\mathscr R\) if
\[ \left|z^{k+1}\Phi_n^{(k)}(z)\right|\leqslant C_n^{(k)}, \tag{4} \]
where \(C_n^{(k)}\to 0\) as \(n\to\infty\) and for arbitrarily fixed values \(k=0,1,2,\ldots\) in every bounded domain of the strip \(t_a\); here uniform convergence of \(z^{k+1}\Phi_n^{(k)}(z)\) to zero as \(n\to\infty\) with respect to \(z\) is required. Functionals in \(\mathscr R\), which we denote by \(\langle\rho(z),\Phi\rangle\), always have integral representations
\[ \langle\rho(z),\Phi\rangle=\int_C \rho(z)\Phi(z)\,dz \tag{5} \]
for all \(\Phi(z)\in\mathscr R\), where \(C\) is a certain contour in the \(z\)-plane. The set of linear continuous functionals in \(\mathscr R\) will be denoted by \(\mathscr R'\). Such func-
we shall call functionals analytic generalized functions.
Finally, we introduce for consideration the space \(\mathscr L\) of functions \(\Phi(x)\), defined on the whole \(x\)-axis, infinitely differentiable, satisfying the inequality
\[ \left|\Phi^{(k)}(x)\right| \leq \frac{C^{(k)}}{|x|^{k+1}} \quad \text{as } |x| \to \infty, \tag{6} \]
where \(C^{(k)}\) is some constant number, \(k=0,1,2,\ldots\), and representable in the form
\[ \Phi(x)=\overline K\Phi(z)=\frac{1}{2\pi i}\int_L \frac{\Phi(z)}{x-z}\,dz \tag{7} \]
for any \(\Phi(z)\in \mathscr R\), where \(L\) is a contour consisting of two straight lines parallel to the \(x\)-axis: one, situated in \(\operatorname{Im} z>0\), is directed in the positive direction of the \(x\)-axis, and the other, situated in \(\operatorname{Im} z<0\), is directed opposite to the positive direction of the \(x\)-axis.
The notion of convergence in \(\mathscr L\) is analogous to convergence in \(\mathscr R\).
Let \(\mathscr L'\) be the set of linear continuous functionals in \(\mathscr L\). We note that the space \(\mathscr D \subset \mathscr L\), and the space \(\mathscr D' \subset \mathscr L'\). Therefore not every generalized function from \(\mathscr D'\) can belong to \(\mathscr L'\).
Theorem 1. In order that any generalized function \(\rho\in \mathscr D'\) belong simultaneously to the space \(\mathscr L'\), it is sufficient that it satisfy the inequality
\[ \left|(\rho,\psi(x+x_0))\right|\leq \frac{C(\psi)}{|x_0|^\alpha} \quad \text{as } |x_0|\to\infty \tag{8} \]
for any \(\psi\in \mathscr D\), where \(C(\psi)\) is some constant number depending on \(\psi\in \mathscr D\); \(x_0\) is an arbitrary fixed point of the \(x\)-axis, \(0<\alpha<1\).
Theorem 2. The operator \(K\), conjugate to the operator \(\overline K\), is defined on the linear manifold of any generalized functions \(\rho\in \mathscr D'\) (and also \(\rho\in \mathscr L'\)) satisfying inequality (8) for any \(\psi\in \mathscr D\) (and also \(\psi\in \mathscr L\)).
We indicate the scheme of proof of this theorem. Construct an averaging kernel \(\omega(x-y;h)\), which is a finite and infinitely differentiable function, i.e. \(\omega(x-y;h)\in \mathscr D\), and which has the property:
\[ \frac{1}{\varkappa h}\int_{-h}^{h}\omega(t;h)\,dt=1, \quad \text{where } \varkappa=\int_{-1}^{+1}\omega(t;1)\,dt. \]
Then the function
\[ \rho_h(x)=\frac{1}{\varkappa h}\,(\rho,\omega(x-y;h)) \]
will be infinitely differentiable and will satisfy the inequality
\[ |\rho_h(x)|\leq \frac{C_h}{|x|^\alpha} \quad \text{as } |x|\to\infty, \]
where \(C_h\) is some constant number; \(0<\alpha<1\).
Construct a sequence \(\{\rho_h\}\) of generalized functions \(\rho_h\) in \(\mathscr D\) by the formula
\[ (\rho_h,\psi)=\int_{-\infty}^{+\infty}\rho_h(x)\psi(x)\,dx \tag{9} \]
for any \(\psi\in \mathscr D\). On the basis of Theorem 1, the functional \(\rho_h\), satisfying condition (8), can be extended to the whole space \(\mathscr L\). Consequently, we obtain:
\[ \langle \breve{\rho}(z),\Phi\rangle=(\rho,\overline K\Phi) \quad \text{for any } \Phi\in \mathscr R. \tag{10} \]
The expression \(\langle \check{\rho}(z), \Phi\rangle\) depends linearly and continuously on \(\Phi\in\mathcal R\). Now, introducing norms in the spaces \(\mathcal L\) and \(\mathcal R\), respectively, by the formulas
\[ \|\Phi\|_{(-\infty,\infty)} = \max_{\substack{(-\infty,\infty)\\ k=0,1,2,\ldots}} \left|x^{k+1}\Phi^{(k)}(x)\right| \quad \text{for any } \Phi\in\mathcal L; \]
\[ \|\Phi\|_{G} = \max_{\substack{G\\ k=0,1,2,\ldots}} \left|z^{k+1}\Phi^{(k)}(z)\right| \quad \text{for any } \Phi\in\mathcal R, \]
we shall prove the linear and continuous dependence of \(\langle \check{\rho}(z),\Phi\rangle\) on the given generalized function \(\rho\). Thus, equality (10) defines, on the given manifold of our generalized functions, a linear operator of the form:
\[ \check{\rho}(z)=K\rho=\frac{1}{2\pi i}\left(\rho,\frac{1}{x-z}\right),\quad \operatorname{Im} z\ne 0, \tag{11} \]
where \(\dfrac{1}{x-z}\in\mathcal L\) for \(\operatorname{Im} z\ne 0\).
The operator \(K\) transforms manifolds of arbitrary generalized functions from \(\mathcal D'\) and \(\mathcal L'\), satisfying inequality (8), into analytic generalized functions defined by equality (11) and tending to zero at infinity.
We shall call the operator \(K\) the operator of analytic continuation of generalized functions. The uniqueness of this operator is evident.
On the basis of (10) we have
\[ \langle \check{\rho}^{(k)}(z),\Phi\rangle = (-1)^k\langle \check{\rho}(z),\Phi^{(k)}\rangle = \]
\[ = \left(\rho,\frac{k!}{2\pi i}\int_L \frac{\Phi(z)}{(x-z)^{k+1}}\,dz\right) \quad \text{for any } \Phi\in\mathcal R. \tag{12} \]
This equality defines the derivative of order \(k\) of the analytic generalized function \(\check{\rho}(z)\) defined by formula (11). Thus,
\[ \check{\rho}^{(k)}(z) = \frac{k!}{2\pi i} \left(\rho,\frac{1}{(x-z)^{k+1}}\right), \quad \operatorname{Im} z\ne 0,\quad k=0,1,2,\ldots \tag{13} \]
Theorem 3. If the conditions of Theorem 2 are satisfied, the formula
\[ \rho_+ - \rho_- = \rho, \tag{14} \]
holds, where
\[ \check{\rho}_{\pm} = -\frac{1}{2\pi i}\left(\rho * P\frac{1}{x}\right) \pm \frac{1}{2}\rho. \tag{15} \]
Here the convolution \(\left(\rho * P\dfrac{1}{x}\right)\) exists and is defined by the formula
\[ \left(\rho * P\frac{1}{x},\varphi\right) = \left(\rho,\int_0^\infty \frac{1}{y}\,[\varphi(x+y)-\varphi(x-y)]\,dy\right) \tag{16} \]
for any \(\varphi\in\mathcal L\). The generalized functions \(\check{\rho}_{\pm}\) mean:
\[ (\check{\rho}_{\pm},\varphi) = \lim_{\varepsilon\to 0} \int_{-\infty}^{+\infty} \check{\rho}(x+i\varepsilon)\varphi(x)\,dx \tag{17} \]
for any \(\varphi\in\mathcal L\).
Let us give several simplest examples.
- \(\rho=\delta^{(k)}(x)\), \(k=0,1,2,\ldots\). By definition \((\delta^{(k)}(x),\varphi)=(-1)^k\varphi^{(k)}(0)\) for any \(\varphi\in\mathcal L\). Then, according to formula (11), we obtain
which gives the corresponding analytic generalized function
\[ \check{\delta}^{\,k}(z)=\frac{(-1)^{k+1}k!}{2\pi i}\,\frac{1}{z^{k+1}},\qquad k=0,1,2,\ldots; \tag{18} \]
\[ \check{\delta}^{(k)}_{\pm}=(-1)^{k+1}\frac{k!}{2\pi i}\,P_f\frac{1}{x^{k+1}}\pm\frac{1}{2}\delta^{(k)}(x); \tag{19} \]
\[ \check{\delta}^{(k)}_{+}-\check{\delta}^{(k)}_{-}=\delta^{(k)}(x),\qquad k=0,1,2,\ldots. \tag{20} \]
- \(\rho=P_f\dfrac{1}{x^m}\), where \(m\) is a positive integer. By definition
\[ \left(P_f\frac{1}{x^m},\varphi\right)= \int_{0}^{\infty}\frac{1}{x^m} \left[\varphi(x)+(-1)^m\varphi(-x)-2\sum_{k=0}^{m-1}\frac{x^k}{k!}\varphi^{(k)}(0)\right]\,dx \tag{21} \]
for every \(\varphi\in\mathcal L\).
Using this formula, from (11) we obtain the corresponding analytic generalized function, namely:
\[ \check{\rho}(z)= \begin{cases} \dfrac{1}{2z^m}, & \operatorname{Im} z>0,\\[6pt] -\dfrac{1}{2z^m}, & \operatorname{Im} z<0; \end{cases} \tag{22} \]
\[ \check{\rho}_{\pm}=\pm\frac{1}{2}P_f\frac{1}{x^m} +\frac{i\pi}{2}\frac{(-1)^{m-1}}{(m-1)!}\delta^{(m-1)}(x), \tag{23} \]
whence
\[ \check{\rho}_{+}-\check{\rho}_{-}=P_f\frac{1}{x^m} \tag{24} \]
and so on.
It should be noted that in their recent work Bremermann and Durand \({}^{(2)}\) made an attempt to define the notions of the Cauchy integral for generalized functions, with the aim of defining a generalized function as the weak jump of two holomorphic functions on the real axis.
Taking this opportunity, I express my deep gratitude to Academician N. N. Bogolyubov for valuable comments during the preparation of this work.
United Institute
for Nuclear Research
Received
4 V 1962
REFERENCES
- I. M. Gelfand, G. E. Shilov, Generalized Functions, 1, 2nd ed., Moscow, 1959.
- H. J. Bremermann, L. Durand, J. Math. Phys., 2, No. 2 (1961).