Full Text
GENERAL SINGULAR EQUATION IN AN EXCEPTIONAL CASE
S. G. SAMKO
The general theory of singular equations arose on the basis of the integral equation with Cauchy kernel:
\[ a(t)\varphi(t)+b(t)\int_{\Gamma}\frac{\varphi(\tau)}{\tau-t}\,d\tau+\int_{\Gamma}k(t,\tau)\varphi(\tau)\,d\tau=f(t). \tag{1} \]
Singular equations may be characterized as those equations for which Fredholm’s theorems are not valid, but Noether’s theorems are. Among such equations are equation (1), equations of convolution type, and infinite algebraic systems with difference indices.
Questions of the general theory of such equations were considered in the works of Z. I. Khalilov [2], F. V. Atkinson [3], Yu. I. Cherskii [1], and other mathematicians. The abstract theory of singular equations proposed by Yu. I. Cherskii includes, as special cases, the equations listed above, including equations of convolution type. This theory applies only to equations of normal type.
In the present paper, the theory of Yu. I. Cherskii is generalized so as to include in the general scheme the so-called exceptional cases. We consider the equation
\[ A_1\varphi+A_2S\varphi+T\varphi=f, \tag{2} \]
where \(S\) is an operator satisfying the condition \(S^2=I\), and \(T\) is a completely continuous operator. If the operators \(A_1\pm A_2\) are invertible, equation (2) is of normal type. It was precisely in this case that equation (2) was studied by Yu. I. Cherskii. In the present paper we shall assume that the operators \(A_1\pm A_2\) are not invertible, but are representable in a certain special form. This case will be called exceptional.
The results obtained are valid for the exceptional case of equation (1) and for equations of convolution type. Here, as a realization, we shall consider an equation of convolution type.
§ 1. GENERAL THEORY OF THE EQUATION
Let \(X\) be a linear normed space containing a Banach subspace \(B\). We shall distinguish between invertibility of operators in the space \(X\) and invertibility in \(B\). If an operator \(A\) is invertible in \(B\), this means that there exists an inverse operator \(A^{-1}\), defined in \(B\) and acting in \(B\). If, however, the operator \(A\) is invertible in \(X\) (and not invertible in \(B\)), then \(A^{-1}\) does not map \(B\) into itself, acting from \(B\) into \(X\).
An operator \(S\) will be called singular if
1) \(S\) is a continuous linear operator;
\[ \begin{gathered} 2)\quad S^{2}=I,\quad S\ne \pm I . \end{gathered} \tag{3} \]
Let \(R\) be a ring of linear operators continuous in \(X\), such that
1) the operators \(A\in R\) have no roots in \(X\);
2) \(AS-SA\) is a completely continuous operator.
The operator \(S\) generates subspaces \(B_{\pm}\) of elements \(\varphi_{\pm}\) such that \(\varphi_{+}\in B_{+}\) if \((-I+S)\varphi_{+}=0\), and \(\varphi_{-}\in B_{-}\) if \((I+S)\varphi_{-}=0\). For every element \(\varphi\in X\) the representation
\[ \varphi=\varphi_{+}-\varphi_{-}, \tag{4} \]
is possible, and the elements \(\varphi_{\pm}\) are determined uniquely:
\[ \varphi_{+}=\frac{1}{2}(I+S)\varphi,\qquad \varphi_{-}=\frac{1}{2}(-I+S)\varphi . \]
Analogously, in the adjoint space \(\overline{B}\) the elements \(\overline{\varphi}_{+}\) and \(\overline{\varphi}_{-}\) are defined, such that \((I+S^{*})\overline{\varphi}_{+}=0,\ (I-S^{*})\overline{\varphi}_{-}=0\).
Let \(U,\ \Psi_{+},\ \Psi_{-}\) be invertible operators in \(B\) belonging to \(R\), determined by the following properties:
1) \(U\varphi_{+}\in B_{+}\), if \(\varphi_{+}\in B_{+}\);
2) \(U^{-1}\varphi_{-}\in B_{-}\), if \(\varphi_{-}\in B_{-}\);
3) there exists a unique (up to a constant multiplier) element \(h_{+}\in B_{+}\), such that \(U^{-1}h_{+}\in B_{-}\) and \(h_{+}\ne 0\);
4) there exists a unique (up to a constant multiplier) element \(\overline{h}_{-}\in \overline{B}_{-}\), such that \(\overline{h}_{-}\ne 0\) and \(U^{*}\overline{h}_{-}\in \overline{B}_{+}\);
5) \(\Psi_{+}\varphi_{+}\in B_{+},\ \Psi_{+}^{-1}\varphi_{+}\in B_{+}\), if \(\varphi_{+}\in B_{+}\);
6) \(\Psi_{-}\varphi_{-}\in B_{-},\ \Psi_{-}^{-1}\varphi_{-}\in B_{-}\), if \(\varphi_{-}\in B_{-}\);
7) the class \(E_{+}\,(E_{-})\) of operators \(\Psi_{+}\,(\Psi_{-})\) contains all isomorphic* operators obtained from \(\Psi_{+}\,(\Psi_{-})\) by the automorphism \(U\):
\[ U^{-1}\Psi_{+}U\in E_{+},\quad U\Psi_{+}U^{-1}\in E_{+} \quad \left(U^{-1}\Psi_{-}U\in E_{-},\quad U\Psi_{-}U^{-1}\in E_{-}\right); \]
8) \(E_{+}\,(E_{-})\) also contains the isomorphic operators obtained from \(\Psi_{+}\,(\Psi_{-})\) by any automorphism \(\Psi'_{-}\,(\Psi'_{+})\):
\[ (\Psi'_{-})^{-1}\Psi_{+}\Psi'_{-}\in E_{+} \quad \left((\Psi'_{+})^{-1}\Psi_{-}\Psi'_{+}\in E_{-}\right); \]
9) the operator \(U-\lambda I\) is invertible in \(X\) for every \(\lambda\), but is not invertible in \(B\) for every \(\lambda\);
10) if \(\lambda\) is such that \(U-\lambda I\) is not invertible in \(B\), then
\[ (U-\lambda I)^{-1}h_{+}\in B,\qquad (U-\lambda I)^{*-1}\overline{h}_{-}\in \overline{B}; \]
11) for the operator \(U-\lambda I\) a property of type 7) holds:
\[ (U-\lambda I)^{-1}\Psi_{+}(U-\lambda I)\in E_{+},\qquad (U-\lambda I)\Psi_{+}(U-\lambda I)^{-1}\in E_{+}, \]
\[ (U-\lambda I)^{-1}\Psi_{-}(U-\lambda I)\in E_{-},\qquad (U-\lambda I)\Psi_{-}(U-\lambda I)^{-1}\in E_{-}. \]
We note that from 7) it follows that for any integer \(K\)
\[ U^{-k}\Psi_{+}U^{k}\in E_{+},\qquad U^{-k}\Psi_{-}U^{k}\in E_{-}. \]
* Operators \(A\) and \(B\) are called isomorphic if there exists an automorphism \(C\) carrying \(A\) into \(B\), i.e. \(B=C^{-1}AC\).
We shall use the definition of the index of an operator given by Yu. I. Cherskii ([1], p. 283).
Definition. If an operator \(A\) invertible in \(B\) is representable in the form
\(A=\Psi_+ U^\varkappa \Psi_-\), then it is said to have an index equal to \(\varkappa\):
\[ \operatorname{ind} A=\varkappa . \]
Using properties 1)—11), one can prove the following assertion.
Theorem 1. If the operators \(A_1\) and \(A_2\) have indices \(\varkappa_1\) and \(\varkappa_2\), respectively, then their composition also has an index, and
\[ \operatorname{ind} A_1 A_2=\operatorname{ind} A_2 A_1 =\operatorname{ind} A_1+\operatorname{ind} A_2 =\varkappa_1+\varkappa_2 . \]
Hence we obtain
Corollary 1. If an operator \(A\) has an index, then the operator \(A^{-1}\) also has an index, and
\(\operatorname{ind} A^{-1}=-\operatorname{ind} A\).
Corollary 2. If an operator \(A\) has an index, then \(UA=A_1U\), where the operator \(A_1\) has the same index.
Let us also note that the elements \(h_+, Uh_+, \ldots, U^n h_+, \ldots\) and \(U^{-1}h_+\), \(U^{-2}h_+, \ldots, U^{-n}h_+, \ldots\) form systems of linearly independent elements belonging to \(B_+\) and \(B_-\), respectively (see [1]).
By a general singular equation we shall mean the equation
\[ M\varphi\equiv(A_1+A_2S+T)\varphi=f, \]
where \(f\in B,\ \varphi\in X,\ A_1\in R,\ A_2\in R\), and the equation
\[ M_0\varphi\equiv(A_1+A_2S)\varphi=f \tag{5} \]
is called the characteristic singular equation.
Here we consider the characteristic equation (5) and the equation adjoint to it,
\[ M_0^*\overline{\varphi}\equiv(A_1^*+S^*A_2^*)\overline{\varphi}=\overline{g} \tag{6} \]
in the adjoint space, without dwelling on the full equation (2). We note only that, as soon as the characteristic equation is solved, the full equation is reduced to a Fredholm equation, i.e., to an equation \((I+T)\times\varphi=f\).
Equation (5), with the help of representation (4), reduces to the relation
\[ (A_1+A_2)\varphi_+=(A_1-A_2)\varphi_-+f, \tag{7} \]
which is an analogue of the Riemann problem.
In our exceptional case we assume that the operators \(A_1+A_2\) and \(A_1-A_2\) have the form
\[ A_1+A_2=\prod_{j=1}^{r}(U-\lambda_j I)^{\alpha_j}A', \quad A_1-A_2=\prod_{j=1}^{s}(U-\lambda_{r+j}I)^{\beta_j}A'', \tag{8} \]
where \(A'\) and \(A''\) are operators already invertible in \(B\), and \(\alpha_j,\ \beta_j\) are positive integers.
For the operator \(U-\lambda I\) the following has been obtained.
Theorem 2. If the operator \(U-\lambda I\) is invertible in \(B\), then it is either an operator of type \(U\) or an operator of type \(\Psi_+\) [5].
From this theorem follows the following conclusion. The representation (8) goes beyond the theory of the normal case if and only if \(\lambda_j\in C,\ j=1,2,\ldots,r+s,\ C\) is the spectrum of the operator \(U\). Therefore, in equal-
... properties (8), we assume that \(\lambda_j\notin C,\ j=1,\ldots,r+s\). Let
\[ \sum_{j=1}^{r}\alpha_j=l,\qquad \sum_{j=1}^{s}\beta_j=m. \]
Denoting, for brevity,
\[ U_l=\prod_{j=1}^{r}(U-\lambda_j I)^{\alpha_j},\qquad U_m=\prod_{j=1}^{s}(U-\lambda_{r+j}I)^{\beta_j}, \]
we can write (8) in the form
\[ A_1+A_2=U_l A',\qquad A_1-A_2=U_m A''. \tag{9} \]
Using properties 1)—11) of the operators \(U,\ \Psi_{\pm}\), we reduce problem (7) to the form
\[ U_l\varphi_+=U_m A\varphi_-+g, \]
where \(g=(A')^{-1}f\). We represent the element \(g\) in the form (4). After this, the solutions of the problem belonging to \(X\) are determined simply. For this it is enough to apply an analogue of Liouville’s theorem proved by Yu. I. Cherskii [1]. If, however, one seeks solutions from \(B\), then it becomes necessary to construct a particular solution of the inhomogeneous problem belonging to \(B\). For this, in addition to the indicated analogue of Liouville’s theorem, one needs the existence of a “polynomial” interpolating, in a certain sense, the element \(\varphi\in B\). By a “polynomial” in our abstract space we mean a construction
\[ P_n h_+=\sum_{k=0}^{n}C_k U^k h_+. \]
The following holds.
Theorem 3. If, for the element \(\varphi\in B\), one of the following conditions is fulfilled:
1) the representation
\[ \begin{aligned} \varphi_+&=(I+S)\varphi=a_0^{+}h_+ + a_1^{+}Uh_+ + \cdots + a_{l-1}^{+}U^{l-1}h_+ + U_l\vartheta_+,\\ \varphi_-&=(I-S)\varphi=a_1^{-}U^{-1}h_+ + \cdots + a_{m-1}^{-}U^{-m+1}h_+ + U^{-m}U_m\vartheta_-, \end{aligned} \tag{10} \]
2) the elements \(\varphi_{\pm}\) are expanded into series
\[ \varphi_+=\sum_{k=0}^{\infty}a_k^{+}U^k h_+,\qquad \varphi_-=\sum_{k=1}^{\infty}a_k^{-}U^{-k}h_+, \tag{11} \]
convergent in the norm of the space \(B\), and the series
\[ \sum_{k=0}^{\infty}a_k^{+}\lambda_j^k,\qquad \sum_{k=1}^{\infty}a_k^{-}\lambda_j^{-k} \tag{12} \]
converge for \(\lambda_j\notin C\), then there exists an “interpolation polynomial” \(P_{l+m-1}h_+\) such that
\[ \varphi_+-P_{l+m-1}h_+=U_l\Psi_+,\qquad \Psi_+\in B_+, \]
\[ \varphi_--P_{l+m-1}h_+=U_m\Psi,\qquad \Psi\in B. \tag{13} \]
To explain the meaning of requirements 1)—2), let us note that, for example, in the space \(B=L_2(-\infty,\infty)\), for \(S\varphi\equiv \operatorname{sign}x\,\varphi(x)\) and
\[ U\varphi=\varphi(x)-2\int_{-\infty}^{x} e^{-x+t}\varphi(t)\,dt, \]
for the fulfillment of 1) it is sufficient to require the existence of derivatives up to some order of the Fourier transform of ...
functions \(\varphi_+(x), \varphi_-(x)\) at the points
\[
x_j=-i\,\frac{\lambda_j+1}{\lambda_j-1},\qquad j=1,2,\ldots,r+s.
\]
As for condition 2), the expansion into the series (11) is always feasible, since it is an expansion in Laguerre polynomials with weight \(e^{-x}\), and it remains to verify the convergence of the series (12).
The proof of the theorem is almost trivial under the first assumption. In case 2), the proof is obtained as follows. Expanding the elements \(\varphi_\pm\) into the series (11) and using the continuity of the operator \(U\), we determine the coefficients \(a_k\) in the expansion so as to obtain the representation (13). This can be done by virtue of the arbitrariness of the coefficients of the “polynomial” \(P_{l+m-1}h_+\).
Let us return to the problem. Performing simple transformations, we obtain the formula
\[
\varphi=\varphi_+-\varphi_-=(A_1-A_2)^{-1}A_2\Psi_+U_mP_{\chi-l-1}h_+
\]
\[
\quad +(A_1-A_2)^{-1}\bigl[A_1-A_2\Psi_+U_l^{-1}(S-2N)U_l\Psi_+^{-1}\bigr](A_1+A_2)^{-1}f,
\tag{14}
\]
where \(N\) is the operator constructing the “interpolation polynomial,” and \(\Psi_+\) is the operator arising from the representation \(A=\Psi_+U^\chi\Psi_-\).
Let us formulate one more theorem on the number of solutions of the characteristic equation, depending on the index of the equation, which we define as
\[
\operatorname{ind}A=\operatorname{ind}A''(A')^{-1}=\operatorname{ind}A''-\operatorname{ind}A'=\chi.
\]
Theorem 4. Suppose that, for the element \(\Psi_+^{-1}(A')^{-1}f\), one of the conditions of Theorem 3 is satisfied. In the case \(\chi-l-1\ge 0\), the homogeneous equation has \(\chi-l\) linearly independent solutions from \(B\), and the nonhomogeneous equation is unconditionally solvable. In the case \(\chi-l\le 0\), the homogeneous equation has no nontrivial solutions in \(B\), while the nonhomogeneous equation has a unique solution in \(B\) when \(\chi=l\), and is solvable only under the fulfillment of certain solvability conditions when \(\chi<l\).
Considering the Riemann problem in the conjugate space \(\overline{B}\), we obtain analogous results for the adjoint equation (6); namely, Theorem 4 will be valid if \(\chi\) is replaced by \(-\chi\) and \(l\) by \(m\), and the formula will be:
\[
\overline{\varphi}=(A_1^*-A_2^*)^{-1}Z^*U_l^*U^{*\chi+m-l+1}P_{-\chi-m-1}\overline{h}_-
\]
\[
\quad +(A_1^*-A_2^*)^{-1}\bigl[A_1^*-Z^*U^{*m}U_m^{*-1}(S^*-2N^*)U_m^*U^{*-m}Z^{*-1}A_2^*\bigr](A_1^*+A_2^*)^{-1}\overline{g},
\tag{15}
\]
where
\[
Z^*=U^{*l-m}U_m^*U_l^{*-1}\Psi_-^{*-1};
\]
\(N^*\) is the operator of finding the “interpolation polynomial” in the conjugate space.
If solutions are sought in the space \(X\), then in the formulas obtained and in the theorem one should replace \(l\) by \(-m\) and set \(N=0\).
§ 2. EQUATION OF CONVOLUTION TYPE IN THE EXCEPTIONAL CASE
As \(B\) we take the space \(L_2(-\infty,\infty)\). The ring \(R\) will be interpreted as the ring of operators \(A\) of the form
\[
A\varphi\equiv \mu\varphi(x)+\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}a(x-t)\varphi(t)\,dt,
\]
where \(\mu\) is any number, \(a(x)\in L_2(-\infty,\infty)\), and the function \(A(x)=Va^*\) satisfies the Hölder condition. The identity holds
\[ A\varphi \equiv V^{-1}[\mu + A(x)]V\varphi, \tag{16} \]
so that \(R\) is a commutative ring [4]. Let, furthermore,
\[ S\varphi \equiv \operatorname{sign} x\,\varphi(x). \]
The operators \(A, S\) satisfy all the requirements of the abstract theory. Equation (5) takes the form
\[ M_0\varphi \equiv (\mu_1+\mu_2\operatorname{sign}x)\varphi(x) +\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} a_1(x-t)\varphi(t)\,dt + \]
\[ +\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} a_2(x-t)\operatorname{sign}t\,\varphi(t)\,dt =f(x), \tag{17} \]
and its adjoint is
\[ M_0^*\varphi=(\mu_1+\mu_2\operatorname{sign}x)\varphi(x) +\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} a_1(x-t)\varphi(t)\,dt + \]
\[ +\frac{\operatorname{sign}x}{\sqrt{2\pi}}\int_{-\infty}^{\infty} a_2(x-t)\varphi(t)\,dt =g(x). \tag{18} \]
Elements of type \(\varphi_\pm\) are the functions
\[ \varphi_+(x)= \begin{cases} \varphi(x), & x>0,\\ 0, & x<0, \end{cases} \qquad \varphi_-(x)= \begin{cases} 0, & x>0,\\ -\varphi(x), & x<0, \end{cases} \quad \varphi(x)\in L_2(-\infty,\infty). \]
The operators \(\Psi_\pm\) are the transformations
\[ \Psi_\pm\varphi=\varphi(x)+\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\psi_\pm(x-t)\varphi(t)\,dt, \tag{19} \]
where \(\psi_+(x)\in B_+\), \(\psi_-(x)\in B_-\), and the functions \(1+\Psi^\pm(x)\), where \(\Psi^\pm(x)=V\psi_\pm\), are boundary values of functions analytic respectively in the upper and lower half-planes and having no zeros in these domains [1].
As \(U\) take
\[ U\varphi \equiv \varphi(x)-2\int_{-\infty}^{x} e^{-(x-t)}\varphi(t)\,dt \]
and then
\[ h_+= \begin{cases} e^{-x}, & x>0,\\ 0, & x<0 \end{cases} \quad [1]. \]
Obviously, \(U\in R\), and, according to (16),
\[ U^n\varphi \equiv V^{-1}\left(\frac{x-i}{x+i}\right)^n V\varphi,\qquad n=0,\ \pm1,\ \ldots \tag{20} \]
\[ {}^*\ \text{Everywhere below the symbol } V \text{ denotes the Fourier transform:} \]
\[ V\varphi \equiv \lim_{N\to\infty}\frac{1}{\sqrt{2\pi}}\int_{-N}^{N} e^{ixt}\varphi(t)\,dt. \]
The representation of the operator \(A \in R\) in the form \(A=\Psi_{+}U^{\varkappa}\Psi_{-}^{-1}\) leads to the relation
\[ \mu + A(x)=\frac{\mu+\Psi^{+}(x)}{1+\Psi^{-}(x)} \left(\frac{x-i}{x+i}\right)^{\varkappa}, \tag{21} \]
where \(\varkappa\) should be taken equal to \(\operatorname{ind}[\mu+A(x)]\). Equality (21) is the condition of the Riemann problem. It can be solved in the usual way.
The operator \(A \in R\) is not invertible in \(L_{2}(-\infty,\infty)\) if and only if the function \(\mu+A(x)\) has zeros on the real axis (including the point at infinity). Here we shall present the case when \(\mu+A(x)\) has zeros only at finite points. A zero at the point at infinity corresponds to an equation of the first kind.
Let
\[ \mu_{1}+\mu_{2}+A_{1}(x)+A_{2}(x)= \frac{\prod_{j=1}^{r}(x-x_{j})^{\alpha_{j}}}{(x+i)^{l}} [\nu_{1}+D_{1}(x)], \]
\[ \mu_{1}-\mu_{2}+A_{1}(x)-A_{2}(x)= \frac{\prod_{j=1}^{s}(x-x_{r+j})^{\beta_{j}}}{(x+i)^{m}} [\nu_{2}+D_{2}(x)], \]
where \(x_{1},\ldots,x_{r+s}\) are points of the real axis; \(\nu_{1}=\mu_{1}+\mu_{2}\), \(\nu_{2}=\mu_{1}-\mu_{2}\); \(D_i(x)\in L_{2}(-\infty,\infty)\) and satisfy the Hölder condition, and the functions \(\nu_i+D_i(x)\) nowhere vanish on the axis, \(i=1,2\). Denoting
\[ \lambda_j=\frac{x_j-1}{x_j+1}, \qquad j=1,\ldots,r+s, \]
we write:
\[ (A_{1}+A_{2})\varphi = V^{-1} \frac{\prod\left(\dfrac{x-i}{x+i}-\lambda_j\right)^{\alpha_j}} {\prod(1-\lambda_j)^{\alpha_j}} [\nu_{1}+D_{1}(x)]V\varphi, \]
\[ (A_{1}-A_{2})\varphi = V^{-1} \frac{\prod\left(\dfrac{x-i}{x+i}-\lambda_{r+j}\right)^{\beta_j}} {\prod(1-\lambda_{r+j})^{\beta_j}} [\nu_{2}+D_{2}(x)]V\varphi. \tag{22} \]
The spectrum \(C\) of the operator \(U\) is the circle \(|\lambda|=1\). We do not discuss here the case \(\lambda=1\), corresponding to the case of a zero at the point at infinity, which also justifies the representation (22), where we have assumed that \(1-\lambda_j\ne0\).
In the general theory the operator \(U-\lambda I\) was invertible in the space \(X\) containing \(B\). We shall now introduce this space as follows. Functions with polar singularities are transformed by Fourier transform, regarding them as generalized functions. We obtain a certain class of functions, which we shall take as \(X\). Let \(n\) be any integer such that
\[ n>\max_j\{\alpha_j;\beta_j\}. \]
By the space \(L_{2}(x_j;n)\) we shall mean the space of functions \(H(x)\) belonging to \(L_{2}(-\infty,\infty)\) everywhere except, possibly, in neighborhoods of the points \(x_j\), \(j=1,\ldots,r+s\), in which singularities are allowed for \(H(x)\) such that
\[ H(x)\prod_{j=1}^{r+s}\left(\frac{x-x_j}{x+i}\right)^n \in L_{2}(-\infty,\infty). \]
Further, \(\omega(x) \in L_2(x_j; -n)\), if
1) \(\omega(x)\) has derivatives up to order \(n\) inclusive;
2) \(\omega^{(k)}(x) \in L_2(-\infty,\infty)\), \(k=0,1,\ldots,n\);
3) \(\omega(x)\) has, at the points \(x_j\), \(j=1,\ldots,r+s\), zeros of such order that
\[ \omega(x)\prod_{j=1}^{r+s}\left(\frac{x+i}{x-x_j}\right)^n \in L_2(-\infty,\infty). \]
By the space \(\overline{L_2(x_j;-n)}\) we shall mean the space of functions \(\Omega(x)\) that are Fourier transforms of functions \(\omega(x)\): \(\Omega(x)=V\omega\).
Considering \(\overline{L_2(x_j;-n)}\) as a class of generalized functions defined over the space \(L_2(x_j;-n)\) of basic functions \(\omega(x)\), we introduce in \(\overline{L_2(x_j;-n)}\) the inverse Fourier transform \(h(x)=V^{-1}H\) by the formula
\[ (h,\Omega)=(H,\omega)=\int_{-\infty}^{\infty} H(x)\omega(x)\,dx. \tag{23} \]
As \(X\) we shall also take the space of obtained Fourier transforms. By analogy with the preceding notation, \(X=L_2(x_j;n)\).
The space \(X\) contains certain functions having polynomial growth at infinity. This can be verified by transforming, according to (23), the function \(H(x)=\dfrac{1}{(x-x_j)^k}\). In this way one obtains the formula
\[ V^{-1}\frac{1}{(x-x_j)^k} = -\frac{(-i)^k}{(k-1)!}\sqrt{\frac{\pi}{2}}\,e^{-ixx_j}x^{k-1}\operatorname{sign}x. \]
In the space constructed we have the possibility of taking Fourier transforms of functions with power singularities, and, consequently, in this space the operators \(A_1+A_2\) are invertible.
The “interpolation polynomial” is constructed in the following way. In our case the role of the “polynomial” is played by the function
\[ P_nh_+=\sum_{k=0}^{n}c_k V^{-1}\left(\frac{x-i}{x+i}\right)^k Vh_+. \]
Using the easily derived formula
\[ V^{-1}\frac{1}{(x+i)^k} = \frac{(-i)^{k+1}}{k!}\sqrt{2\pi} \begin{cases} x^k e^{-x}, & x>0,\\ 0, & x<0, \end{cases} \]
we find that
\[ U^kh_+ = \begin{cases} e^{-x}L_k(2x), & x>0,\\ 0, & x<0, \end{cases} \]
where \(L_k(x)\) are the Laguerre polynomials. As is known, the functions \(\psi_k(x)=e^{-x}L_k(2x)\) form a basis in \(L_2(0,\infty)\).
Turning to the conditions of Theorem 3, we note that (10) holds if one requires that the function \(\Phi^+(x)=V\varphi_+\) \((\Phi^-(x)=V\varphi_-)\) have, at the points \(x_j\) \((x_{r+j})\), continuous derivatives up to order \(\alpha_j-1\) \((\beta_j-1)\) inclusive.
As for (11)—(12), (11) requires expansion in the functions \(\psi_k(x)\) in \(L_2(0,\infty)\) and in \(\psi_k(-x)\) in \(L_2(-\infty,0)\), which is always possible; (12) forces one to require that the series
\[ \sum_{k=0}^{\infty} a_k^{\pm} e^{\pm ik\frac{x_j-i}{x_j+i}}, \]
converge, where \(a_k^{+}\), \(a_k^{-}\) are the coefficients of the expansions of the functions \(\varphi_+(x)\), \(\varphi_-(x)\).
We formulate the final results. For equation (17), Theorem 4 of § 1 holds. Formula (14) gives, for equation (17),
\[ V\varphi = \frac{1+\Psi^{+}(x)}{\nu_2+D_2(x)} \sum_{k=0}^{\varkappa-l-1}\frac{c_k}{(x+i)^{k+1}} + \]
\[ +\frac{1}{Q(x)[\nu_2+D_2(x)]} \left[ \frac{\mu_1+A_1(x)}{\nu_1+D_1(x)}F(x) -\frac{R_2(x)}{\pi i} \int_{-\infty}^{\infty} \frac{F(t)}{R_1(t)} \times \right. \]
\[ \left. \times \frac{dt}{t-x} +2R_1(x)N\left(\frac{F(x)}{R_2(x)}\right) \right], \]
where \(\varphi(x)\) is a solution of equation (17) in \(L_2(-\infty,\infty)\),
\[ R_1(x)=[\nu_1+D_1(x)][1+\Psi^{+}(x)],\qquad R_2(x)=[\mu_2+A_2(x)][1+\Psi^{+}(x)], \]
\[ Q(x)= \frac{ \displaystyle \prod_{j=1}^{r}(x-x_j)^{\alpha_j} \prod_{j=1}^{s}(x-x_{r+j})^{\beta_j} } {(x+i)^{l+m}}, \qquad F(x)\equiv Vf . \]
An analogous formula is obtained for the adjoint equation (18). It is not difficult to obtain a realization of the theory also for equation (1). We shall not dwell on this.
References
-
Cherskii Yu. I. General singular equation and equations of convolution type. Mat. sb., new series, vol. 41, no. 3, 1957.
-
Khalilov Z. I. Linear Equations in Linear Normed Spaces. Publ. House of the Academy of Sciences of the Azerbaijan SSR, 1949.
-
Atkinson F. V. Normal solvability of linear equations in normed spaces. Mat. sb., vol. 28 (70), 1951.
-
Titchmarsh E. Introduction to the Theory of Fourier Integrals. Gostekhizdat, Moscow–Leningrad, 1948.
-
Samko S. G. Scientific Reports of Rostov University. Publ. House of Rostov University, 1964.
Received by the editors
January 13, 1965
Rostov State University