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V. B. DYBIN
AN EXCEPTIONAL CASE OF CONVOLUTION-TYPE INTEGRAL EQUATIONS IN THE CLASS OF GENERALIZED FUNCTIONS
(Presented by Academician N. I. Muskhelishvili, 31 VIII 1964)
The following convolution-type integral equations in the class of generalized functions (g.f.) are considered: the equation “with two kernels”
\[ \lambda f(x)+\frac{1}{\sqrt{2\pi}}\int_0^\infty k_1(x-t)f(t)\,dt +\frac{1}{\sqrt{2\pi}}\int_{-\infty}^0 k_2(x-t)f(t)\,dt =g(x), \]
\[ -\infty<x<\infty, \tag{1} \]
and the “paired” equations
\[ \lambda_1 f(x)+\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty k_1(x-t)f(t)\,dt =g(x),\qquad x>0, \]
\[ \lambda_2 f(x)+\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty k_2(x-t)f(t)\,dt =g(x),\qquad x<0. \tag{2} \]
Equations (1), (2) are investigated by reducing them to a problem of linear conjugation (^1), whose coefficients are allowed zeros of integral order at a finite number of points of the contour, including the infinitely distant point.
In the class \(L_2(-\infty,\infty)\) an analogous case was considered in (^2).
- We introduce the following notation. Let \(n\geq 0\), \(m\geq 0\) be integers. \(L_2(-n,-m)\) is the space of basic functions \(\varphi(x)\), defined on the real axis, differentiable \(m\) times, with norm
\[ \|\varphi\|=\left(\int_{-\infty}^{\infty} \left|\left(\frac{d}{dx}+1\right)^m\{(x+i)^n\varphi(x)\}\right|^2\,dx\right)^{1/2}. \tag{3} \]
Then \(L_2(n,m)\) is the space of g.f. \(f\) over the space \(L_2(-n,-m)\), i.e. of linear functionals continuous with respect to the norm (3).
The Cauchy operator \(S\), acting in \(L_2(0,m)\) by the formula
\[ (Sf,\varphi)=\left(f,-\frac{1}{\pi i}\int_{-\infty}^{\infty}\frac{\varphi(t)\,dt}{t-x}\right), \qquad f\in L_2(0,m),\quad \varphi\in L_2(0,-m), \]
is linear and maps \(L_2(0,m)\) onto itself. By \(L_2^{\pm}(0,m)\) we denote the subspaces of the space \(L_2(0,m)\) of g.f. \(f^{\pm}\) satisfying the equations \(f^{\pm}=\pm Sf^{\pm}\).
By \(L_{2\pm}(n,m)\) we denote the subspaces of the space \(L_2(n,m)\) of g.f. \(f_{\pm}\). The Fourier operator \(V\) establishes a homeomorphism between the spaces \(L_2(n,m)\) and \(L_2(m,n)\). In this case the following properties hold:
\[ (x\pm i)^{-m}F^{\pm}=(x\pm i)^{-m}Vf_{\pm}\in L_2^{\pm}(0,n), \]
where \(f_{\pm}\in L_{2\pm}(n,m)\), \(F^{\pm}\in L_2^{\pm}(m,n)=V[L_{2\pm}(n,m)]\).
Let \(r\) be an integer. By \(L_2(r,n,\rho(x))\) we denote the space of g.f. \(G\) over the basic space \(L_2(-r,-n,\rho(x))\) of functions \(\psi(x)\),
defined on the real axis, differentiable \(n\) times, with norm
\[ \|\psi\|=\left(\int_{-\infty}^{\infty}\rho(x)\left|\left(\frac{d}{dx}+1\right)^n\{(x+i)^r\psi(x)\}\right|^2 dx\right)^{1/2}, \]
where \(\rho(x)\) is a nonnegative function bounded on the real axis.
2. The linear conjugation problem. Find generalized functions \(F^+\in L_2^+(m,n)\) and \(F^-\in L_2^-(m,n)\) satisfying the relation
\[ \frac{\prod_{k=1}^{p}(x-a_k)^{\alpha_k}}{(x+i)^{\alpha+\alpha_0}}\Omega(x)F^+ = \frac{\prod_{j=1}^{q}(x-b_j)^{\beta_j}}{(x+i)^{\beta+\beta_0}}T(x)F^-+G, \tag{4} \]
where \(\operatorname{Im} a_k=\operatorname{Im} b_j=0,\ 1\le k\le p,\ 1\le j\le q;\ \alpha_k,\ 0\le k\le p,\) and \(\beta_j,\ 0\le j\le q,\) are positive integers, and moreover
\[ \sum_{k=1}^{p}\alpha_k=\alpha,\qquad \sum_{j=1}^{q}\beta_j=\beta,\qquad \max_{\substack{1\le k\le p\\ 1\le j\le q}}\{\alpha_k,\beta_j\}=n_0\le n; \]
\(\Omega(x)\) and \(T(x)\) are \(n\)-times continuously differentiable functions, nonzero on the real axis, \(\Omega^{(n)}(x)\in H_\mu,\ T^{(n)}(x)\in H_\mu,\ 0<\mu<1\). The free term of the problem is \(G\in L_2(r,n-n_0,\rho_1(x))\), where \(r\le m-m_1=s,\ m_1=\max\{\alpha_0,\beta_0\}\),
\[ \rho_1(x)= \left| \frac{\prod_{k=1}^{p_1}(x-\tilde a_k)} {(x+i)^{p_1}} \right|^{\varepsilon_1}; \qquad 0<\varepsilon_1<1; \]
\(\tilde a_k\) are those among \(a_k,b_j\) for which \(\alpha_k=n_0,\beta_j=n_0\).
We shall call the number \(\chi=\chi_1+\beta+\beta_0+s\) the index of problem (4), where
\[ \chi_1=\frac{1}{2\pi i}\,[\ln T(x)-\ln \Omega(x)]_{-\infty}^{+\infty}. \]
Let us formulate the final result.
Theorem. If \(\chi>0\), the homogeneous problem (4) is solvable and has \(\chi\) linearly independent solutions; if \(\chi\le 0\), it has no solutions distinct from the identically zero one. Its general solution has the form
\[ F^+=(x+i)^{\alpha_0+s}X^+(x)\frac{P_{\chi-1}(x)}{(x+i)^{\chi-1}} \sum_{k=1}^{p}\sum_{s=0}^{\alpha_k-1}A_{ks}\delta^{(s)+}(x-a_k), \]
\[ F^-=(x-i)^{\beta_0+s}X^-(x)\frac{P_{\chi-1}(x)}{(x-i)^{\chi-1}} \sum_{j=1}^{q}\sum_{l=0}^{\beta_j-1}B_{jl}\delta^{(l)-}(x-b_j). \tag{5} \]
Here \(P_{\chi-1}(x)\) is a polynomial of degree \(\chi-1\) with arbitrary coefficients; \(A_{ks}\) and \(B_{jl}\) are completely determined constants; \(X^\pm(x)\) are the boundary values of the canonical function of the homogeneous problem with coefficient \(((x+i)/(x-i))^{\chi_1}T(x)/\Omega(x)\).
The nonhomogeneous problem (4) is solvable for \(\chi\ge0\) and has \(\chi\) linearly independent solutions if \(\chi>0\); for \(\chi=0\) its solution is unique. The general solution of the nonhomogeneous problem is the sum of the general solution (5) of the homogeneous problem and a particular solution of the nonhomogeneous problem. The latter has the form
\[ (Y^+,\Phi)= \left( (x+i)^{\alpha_0+s}X^+(x)G_1^+,\, \frac{(x+i)^\alpha\Phi(x)-Q(x)} {\prod_{k=1}^{p}(x-a_k)^{\alpha_k}} \right), \]
\[ (Y^-,\Phi)= \left( (x-i)^{\beta_0+s}\left(\frac{x+i}{x-i}\right)^\chi X^-(x)G_1^-,\, \frac{(x+i)^\beta\Phi(x)-Q_1(x)} {\prod_{j=1}^{q}(x-b_j)^{\beta_j}} \right), \tag{6} \]
where \(Q(x)\) is the Hermite interpolation polynomial (see (4)) of degree \(a-1\) for the function \((x+i)^\alpha\Phi(x)\) with interpolation nodes \(a_k\) of multiplicity \(\alpha_k\); \(Q_1(x)\) is the analogous interpolation polynomial for the function \((x+i)^\beta\Phi(x)\); \(G_1=[(x+i)^sX^+(x)\Omega(x)]^{-1}G\), \(G_1^\pm=\pm \frac12 G_1+SG_1\).
If \(\chi<0\), then the nonhomogeneous problem is, generally speaking, unsolvable; for its solvability the following \(|\chi|\) additional conditions imposed on the right-hand side \(G\) are necessary and sufficient:
\[ \left(\,[X^+(x)\Omega(x)]^{-1}G,\ \frac{1}{(x+i)^{k+s}}\,\right)=0,\qquad k=1,2,\ldots,|\chi|. \]
When these conditions are satisfied, the solution of the problem is unique and is given by formulas (6).
3. Integral equations of convolution type
The solutions of equations (1) and (2) are sought in the space \(L_2(n,m)\). The following restrictions are imposed on the convolution kernels \(k_j(x)\): \(K_j(x)=V[k_j(x)]\in C^{(n)}\), and \(K_j^{(n)}(x)\in H_\mu\) \((0<\mu<1)\), \(j=1,2\).
The conventional notation (1) for \(\lambda=0\) is understood in the following sense:
\[ \left(f_+,\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} k_1(t-x)\varphi(t)\,dt\right) - \left(f_-,\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} k_2(t-x)\varphi(t)\,dt\right) =(g,\varphi) \tag{7} \]
for any test function \(\varphi\in L_2(-n,-m)\), \(f_\pm\) being unknown generalized functions from the space \(L_{2\pm}(n,m)\).
Taking the Fourier transform of equation (7), we arrive at the following boundary-value problem in the space \(L_2(m,n)\):
\[ K_1(x)F^+=K_2(x)F^-+G. \tag{8} \]
Let the functions \(K_j(x)\) admit the representations
\[ K_1(x)= \frac{\displaystyle\prod_{k=1}^{p}(x-a_k)^{\alpha_k}} {(x+i)^{\alpha+\alpha_0}}\Omega(x), \qquad K_2(x)= \frac{\displaystyle\prod_{j=1}^{q}(x-b_j)^{\beta_j}} {(x+i)^{\beta+\beta_0}}T(x), \]
where \(a_k,b_j,\alpha_k,\beta_j,\Omega(x),T(x)\) satisfy the conditions of problem (4). Then, for \(G\in L_2(r,n-n_0,\rho(x))\), \(r\le s\), the results of the theorem formulated above are valid for problem (8), and its solution can be found from formulas (5), (6). Once the general solution \(F^\pm\) of problem (8) has been found, the general solution of equation (7) is given by the formula \(f_\pm=V^{-1}F^\pm\).
Equations (2), in exact notation for \(\lambda_1=\lambda_2=0\), have the form
\[ \left(f,\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} k_1(t-x)\varphi_+(t)\,dt\right) =(g,\varphi_+), \]
\[ \left(f,\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} k_2(t-x)\varphi_-(t)\,dt\right) =(g,\varphi_-), \tag{9} \]
where \(\varphi_\pm\in L_{2\pm}(-n,-m)\), i.e., they are test functions from the space \(L_2(-n,-m)\) equal to zero for \(x<0\) and \(x>0\), respectively. Extending equalities (9) to the whole \(x\)-axis and applying the Fourier transform to them, we obtain the equivalent system of equations
\[ K_1(x)F=G^++\Psi^-, \qquad K_2(x)F=G^-+\Psi^+, \tag{10} \]
where \(\Psi^+\in L_2^+(m-\beta_0,n)\), \(\Psi^-\in L_2^-(m,\alpha_0,n)\) are new unknown generalized functions.
Let the coefficients \(K_j(x)\) have the form
\[ K_1(x)=\frac{A(x)C(x)\Omega(x)}{(x+i)^{\alpha+\gamma+\alpha_0}}, \qquad K_2(x)=\frac{B(x)C(x)T(x)}{(x+i)^{\beta+\gamma+\beta_0}}, \]
where
\[ A(x)=\prod_{k=1}^{p}(x-a_k)^{\alpha_k},\quad B(x)=\prod_{j=1}^{q}(x-b_j)^{\beta_j},\quad C(x)=\prod_{l=1}^{h}(x-c_l)^{\gamma_l}; \]
\(a_k, b_j, \alpha_k, \beta_j, \Omega(x), T(x)\) satisfy the previous conditions; moreover,
\[ a_k\ne b_j,\quad \operatorname{Im}c_l=0,\quad \gamma_l\ge 0 \text{ are integers},\quad \sum_{l=1}^{h}\gamma_l=\gamma,\quad \max_l\{\gamma_l\}\le n_0^*. \]
The boundary-value problem by which the generalized functions \(\Psi^{\pm}\) are determined has the form
\[ \frac{A(x)\Omega(x)}{(x+i)^{\alpha-\beta_0}}\Psi^{+} = \frac{B(x)T(x)}{(x+i)^{\beta-\alpha_0}}\Psi^{-}+G_1, \tag{11} \]
where
\[ G_1=\frac{B(x)T(x)}{(x+i)^{\beta-\alpha_0}}G^{+} -\frac{A(x)\Omega(x)}{(x+i)^{\alpha-\beta_0}}G^{-}, \]
and is analogous to problem (4). If
\[ G^{+}\in L_2(r_1,n-n_0,\rho_2(x)),\quad G^{-}\in L_2(r_2,n-n_0,\rho_2(x)),\quad r_1\le m-\alpha_0, \]
\[ r_2\le m-\beta_0,\quad \text{where}\quad \rho_2(x)= \left| \frac{\prod_{k=1}^{p_2}(x-\tilde b_k)} {(x+i)^{p_2}} \right|^{\varepsilon_2}, \quad 0<\varepsilon_2<1, \]
\(\tilde b_k\) are those among \(a_k,b_j,c_l\) for which \(\alpha_k=n_0,\ \beta_j=n_0,\ \gamma_l=n_0\), then for \(\chi\ge 0\), \(\chi=\chi_1+\beta+m-\alpha_0\) is the index of the problem, it is solvable in the indicated classes of generalized functions.
Solving successively the boundary-value problem (11) and the system (10), by the inverse Fourier transform of the solution \(F\) of the latter one can find the solution \(f=V^{-1}F\) of the original system of integral equations (9).
Let us note that the solution \(F\) of system (10) is determined up to a functional
\[ \sum_{l=1}^{h}\sum_{j=0}^{\gamma_l-1} C_{lj}\delta^{(j)}(x-c_l), \]
where \(C_{lj}\) are arbitrary constants.
We formulate the final result for “even” equations. The homogeneous system of equations (9), when the functions \(K_1(x)\) and \(K_2(x)\) have common zeros \(c_l\), respectively of multiplicities \(\gamma_l,\ 1\le l\le h\), is unconditionally solvable in the space \(L_2(n,m)\); for \(\chi>0\) it has \(\chi+\gamma\) linearly independent solutions, while for \(\chi\le 0\) the number of its solutions is equal to
\[ \gamma=\sum_{l=1}^{h}\gamma_l\; **. \]
The nonhomogeneous system (9), under the restrictions indicated above on the right-hand side, is unconditionally solvable in the space \(L_2(n,m)\) if \(\chi\ge 0\); for its solvability in the case \(\chi<0\) the following \(|\chi|\) additional conditions imposed on the right-hand side are necessary and sufficient:
\[ \left( \left(\frac{x-i}{x+i}\right)^{\chi_1} \frac{B(x)}{(x+i)^{\beta+m-\alpha_0}} \frac{1}{X^{-}(x)}G^{+} - \frac{A(x)}{(x+i)^{\alpha+m-\beta_0}} \frac{1}{X^{+}(x)}G^{-}, \frac{1}{(x+i)^k} \right)=0, \]
\[ k=1,2,\ldots,|\chi|. \]
The well-known Wiener–Hopf equation of the first kind, which is a special case \((k_2(x)\equiv0,\ G^{-}\equiv0)\) of the equations considered, in one subclass of generalized functions from the space \(L_2(n,m)\) was studied in work (5).
The normal case of the problem of linear conjugation and of equations (1), (2) in the space \(L_2(n,m)\) was investigated by Yu. I. Cherskii (3), under whose supervision the present work was carried out.
Rostov State University
Received
5 VI 1964
REFERENCES
- N. I. Muskhelishvili, Singular Integral Equations, Moscow, 1962.
- F. D. Gakhov, V. I. Smagina, Izv. Akad. Nauk SSSR, Ser. Mat., 26, No. 3 (1962).
- Yu. I. Cherskii, Dokl. Akad. Nauk SSSR, 125, No. 3 (1959).
- V. L. Goncharov, Theory of Interpolation and Approximation of Functions, Moscow, 1954.
- V. V. Ivanov, Dokl. Akad. Nauk SSSR, 151, No. 3 (1953).
* Equalities \(c_{l_0}=a_{k_0}\) \((c_{l_0}=b_{j_0})\) are admissible, but it is then required that
\[ \gamma_{l_0}+\alpha_{k_0}\le n_0\quad (\gamma_{l_0}+\beta_{j_0}\le n_0). \]
** In the absence of common zeros of the coefficients \(K_j(x),\ j=1,2\), the number of linearly independent solutions of the homogeneous system is determined by the index \(\chi\).