V. S. Rogozhin

THE RIEMANN BOUNDARY-VALUE PROBLEM IN THE SPACE OF GENERALIZED FUNCTIONS AND FABER POLYNOMIALS

(Presented by Academician V. I. Smirnov on 6 V 1963)

The paper sets forth a theory of generalized functions defined on a basic space consisting of infinitely differentiable functions of points of a sufficiently smooth contour. On the basis of this theory it is possible to solve, in closed form, the Riemann boundary-value problem whose free term is a generalized function, and to interpret the solution as the limiting value of a piecewise analytic function. The formulas giving the solution of the problem can be put in the form that they have in the classical case \((^{1,2})\). The paper continues investigations carried out by various authors in \((^{3-7})\).

§ 1. Let \(L\) be a rectifiable Jordan curve. Denote by \(w=\Phi(z)\) the function mapping the exterior of the curve \(L\)—the domain \(D^{-}\)—onto the exterior of the unit circle \(|w|>1\), under the conditions \(\Phi(\infty)=\infty\) and \(\Phi'(\infty)>0\), and let \(z=\Psi(w)\) be the function inverse to \(w=\Phi(z)\), while \(\Phi_n(z)\) are the Faber polynomials for the domain \(D^{+}\) bounded by the curve \(L\).

Theorem 1. If \(L\) is a closed Jordan curve such that \(\Psi'(e^{i\theta})\) satisfies, with respect to \(\theta\), the Lipschitz condition

\[ |\Psi'(e^{i\theta_1})-\Psi'(e^{i\theta_2})|<A|\theta_1-\theta_2|,\qquad A=\mathrm{const}, \]

then every function \(\varphi(t)\) of points of the contour \(L\) satisfying the Hölder condition

\[ |\varphi(t_1)-\varphi(t_2)|<K|t_1-t_2|^\alpha,\qquad K=\mathrm{const},\qquad 0<\alpha\le 1, \]

can be expanded in the uniformly convergent series

\[ \varphi(t)=\sum_{k=0}^{\infty}\varphi_k\Phi_k(t)+\sum_{k=0}^{\infty}\widetilde{\varphi}_k\Psi_k(t), \tag{1} \]

where \(\Phi_k(t)\) are the Faber polynomials for the domain \(D^{+}\), \(\Psi_k(t)=\Phi'(t)/\Phi^{k+1}(t)\), and the coefficients \(\varphi_k\) and \(\widetilde{\varphi}_k\) are determined by the formulas

\[ \varphi_k=\frac{1}{2\pi i}\int_L \varphi(t)\Psi_k(t)\,dt,\qquad \widetilde{\varphi}_k=\frac{1}{2\pi i}\int_L \varphi(t)\Phi_k(t)\,dt. \]

The proof of this theorem is based on the results of S. Ya. Al’per \((^{8})\), who proved the expandability of a broad class of functions analytic in a closed domain in Faber series uniformly convergent in this closed domain.

The formulas for the coefficients follow from the orthogonality of the system of functions \(\Phi_k(t)\), \(\Psi_k(t)\) on the contour \(L\):

\[ \frac{1}{2\pi}\int_L \Phi_k(t)\Phi_n(t)\,dt=0,\qquad \frac{1}{2\pi i}\int_L \Psi_k(t)\Psi_n(t)\,dt=0, \]

\[ \frac{1}{2\pi i}\int_L \Phi_k(t)\Psi_l(t)\,dt= \begin{cases} 1, & k=l,\\ 0, & k\ne l. \end{cases} \]

We now consider on \(L\) a set \(S\) consisting of infinitely differentiable functions \(\varphi(t)\). We shall say that a sequence \(\{\varphi_k\}\), \(\varphi_k\in S\), tends to zero as \(k\to\infty\), and write \(\lim\limits_{k\to\infty}\varphi_k(t)=0\) or \(\varphi_k(t)\to0\),

if, for any \(s=0,1,2,\ldots\),
\[ \lim_{k\to\infty}\max \left|\frac{d^s\varphi_k}{dt^s}\right|=0. \]
In what follows we shall call \(S\) the basic space.

The subspace \(S^+\), by definition, is formed by those functions \(\varphi^+\in S\) which are boundary values of functions analytic inside \(L\). The subspace \(S^-\) consists of basic functions that are boundary values of functions analytic outside \(L\) and vanishing at infinity. We shall call generalized functions (g.f.) linear functionals defined on the space \(S\) and having the property that \((f,\varphi_k)\to 0\) if \(\varphi_k(t)\to 0\) in the sense indicated above.* If the functional \((f,\varphi^+)=0\) for \(\varphi^+\in S^+\), then \(f\) is called a g.f. of plus type and is denoted by the plus sign \((f^+)\). Similarly, g.f. of minus type \((f^-)\) are introduced.

Theorem 2. If the contour \(L\) is such that \(\Psi(e^{i\theta})\) has derivatives of arbitrary order with respect to the variable \(\theta\), then every g.f. \(\nu\), defined on the basic space \(S\), expands into the series
\[ \nu=\sum_{k=0}^{\infty}\nu_k\Phi_k(t)+\sum_{k=0}^{\infty}\widetilde{\nu}_k\Psi_k(t), \tag{2} \]
convergent in the sense of convergence in the space of g.f.; i.e., for every \(\varphi(t)\in S\)
\[ \lim_{\substack{m\to\infty\\ n\to\infty}} \left( \nu-\sum_{k=0}^{m}\nu_k\Phi_k(t)-\sum_{k=0}^{n}\widetilde{\nu}_k\Psi_k(t),\ \varphi(t) \right)=0. \]
Here
\[ \nu_k=\frac{1}{2\pi i}(\nu,\Psi_k(t)),\qquad \widetilde{\nu}_k=\frac{1}{2\pi i}(\nu,\Phi_k(t)). \]

Corollary. If \(\varphi(t)\in S\), and \(\nu\) is a g.f. on \(S\), with the contour \(L\) satisfying the conditions of Theorem 2, then an analogue of Parseval’s equality holds
\[ \frac{1}{2\pi i}(\nu,\varphi)=\sum_{k=0}^{\infty}\nu_k\widetilde{\varphi}_k+ \sum_{k=0}^{\infty}\widetilde{\nu}_k\varphi_k, \]
where \(\nu_k,\widetilde{\varphi}_k,\widetilde{\nu}_k,\varphi_k\) are defined as above.

Proof follows directly from Theorems 1 and 2. Let us note that g.f. of plus type, and only they, have an expansion into a series of the form
\[ f^+(t)=\sum_{k=0}^{\infty} f_k\Phi_k(t), \]
whereas g.f. of minus type, and only they, have one of the form
\[ f^-(t)=\sum_{k=0}^{\infty}\widetilde{f}_k\Psi_k(t). \]

Theorem 3. If the contour \(L\) satisfies the conditions of Theorem 1, then the expansion
\[ \frac{1}{t-z}=\sum_{k=0}^{\infty}\Phi_k(z)\Psi_k(t), \]
holds.

* It is easy to see that this requirement is equivalent to the following formally stronger condition: there exists an \(r\), depending on \(f\), such that from the equalities
\[ \lim_{k\to\infty}\max|\varphi_k^{(s)}(t)|=0,\qquad 0\le s\le r, \]
there follows the equality
\[ \lim_{k\to\infty}(f,\varphi_k)=0. \]
The proof of the equivalence of these requirements, which define the continuity of the functional, is given in [9], p. 21, for the case of linear functionals on an interval, but it remains valid in our case as well.

uniformly convergent inside \(D^+\) for \(t \in L\), and the expansion

\[ \frac{1}{t-z}=-\sum_{k=0}^{\infty}\Phi_k(t)\Psi_k(z), \]

uniformly convergent inside \(D^-\) for \(t \in L\).

The proof is based on results obtained in P. K. Suetin’s paper \((^{10})\).

If the contour \(L\) satisfies the conditions of Theorem 2 and \(\nu\) is a generalized function defined on \(S\), then from Theorem 3 there follow the expansions, uniformly convergent inside the corresponding domains, of “functionals of Cauchy type” into series:

\[ \text{for } z \in D^+:\qquad \frac{1}{2\pi i}\left(\nu,\frac{1}{t-z}\right)=\sum_{k=0}^{\infty}\nu_k\Phi_k(z), \]

\[ \text{for } z \in D^-:\qquad \frac{1}{2\pi i}\left(\nu,\frac{1}{t-z}\right)=-\sum_{k=0}^{\infty}\widetilde{\nu}_k\Phi_k(z). \]

Let us also point out an interesting representation of the Dirac \(\delta\)-function in the form of a series:

\[ \delta(t-x)=\frac{1}{2\pi i}\left[ \sum_{k=0}^{\infty}\Phi_k(x)\Psi_k(t) + \sum_{k=0}^{\infty}\Phi_k(t)\Psi_k(x) \right], \qquad t,x\in L. \]

Theorem 4. If the contour \(L\) satisfies the conditions of Theorem 2, then every generalized function of plus type

\[ f^+(t)=\sum_{k=0}^{\infty} f_k\Phi_k(t) \]

is analytically continued from the contour into the domain \(D^+\) in the sense that the series

\[ \sum_{k=0}^{\infty} f_k\Phi_k(z) \]

converges in \(D^+\). More precisely, the sequence of partial sums of the series

\[ \sum_{k=0}^{\infty} f_k\Phi_k(z) \]

converges uniformly on every closed subset lying inside \(D^+\), and on the contour it has as its limit (in the sense of convergence in the space of generalized functions) the generalized function \(f^+\).

The proof is based on the representation of the series

\[ \sum_{k=0}^{\infty} f_k\Phi_k(z) \]

by means of a “functional of Cauchy type”

\[ \sum_{k=0}^{\infty} f_k\Phi_k(z) = \frac{1}{2\pi i}\left(f^+,\frac{1}{t-z}\right). \]

An analogous theorem also holds for generalized functions of minus type.

§ 2. We now proceed to the solution of the Riemann boundary-value problem

\[ f^+=Gf^-+g \tag{3} \]

under the assumption that \(G(t)\in S\) on \(L\), and \(g(t)\) is a generalized function. The unknowns are the generalized functions \(f^+(z)\) and \(f^-(z)\), respectively of plus type and of minus type.

Theorem 5. The Riemann problem in the class of generalized functions on \(S\) with \(\operatorname{ind}G(t)=\chi\ge 0\) is unconditionally solvable. The number of arbitrary constants entering the solution is equal to \(\chi\).

Proof. Let \(G(t)=X^+(t)/X^-(t)\), where \(X(z)\) is the canonical function of the homogeneous problem \((^{1,2})\). Then the boundary condition can be given the form

\[ f^+[X^+]^{-1}=f^-[X^-]^{-1}+g[X^+]^{-1}. \]

Since \(X^{+}(t)\) belongs to \(S\) and is nonzero on \(L\), \([X^{+}(t)]^{-1}\) also belongs to \(S\), and, consequently, \(g[X^{+}(t)]^{-1}\) will be a generalized function on \(S^{*}\) and, by Theorem 2, it can be expanded in the series

\[ g[X^{+}]^{-1} = \sum_{k=0}^{\infty} h_k \Phi_k(t) + \sum_{k=0}^{\infty} \widetilde{h}_k \Psi_k(t). \]

Applying to the boundary condition, rewritten in the form

\[ g[X^{+}]^{-1} = \sum_{k=0}^{\infty} h_k \Phi_k(t) + \sum_{k=0}^{\infty} \widetilde{h}_k \Psi_k(t), \]

Liouville’s theorem for generalized functions \((^{5})\), we obtain for the unknown analytic functions the formulas

\[ f^{+}(z) = \left[ \frac{1}{2\pi i} \left( g[X^{+}]^{-1}, \frac{1}{t-z} \right) + P_{\varkappa-1}(z) \right]X^{+}(z), \]

\[ f^{-}(z) = \left[ \frac{1}{2\pi i} \left( g[X^{+}]^{-1}, \frac{1}{t-z} \right) + P_{\varkappa-1}(z) \right]X^{-}(z), \tag{4} \]

where \(P_{\varkappa-1}\) is a polynomial of degree \(\varkappa-1\) with arbitrary coefficients.

These formulas differ from the classical ones, derived for the case when \(G(t)\) and \(g(t)\) satisfy the Hölder condition \((^{1,2})\), in that instead of integrals there are written “Cauchy-type functionals,” defined by virtue of the fact that in them the role of the fundamental function is played by

\[ \frac{1}{t-z}, \]

a function which, for \(z \in D^{+}\) or \(z \in D^{-}\), belongs to the fundamental space.

Analogously to what was done above, one can establish that in the case of a negative index the problem, generally speaking, has no solutions, and derive the conditions for its solvability.

Received
3 V 1963

CITED LITERATURE

\(^{1}\) N. I. Muskhelishvili, Singular Integral Equations, 1962.
\(^{2}\) F. D. Gakhov, Boundary Value Problems, 1959.
\(^{3}\) O. S. Parasyuk, DAN, 110, No. 6 (1956).
\(^{4}\) Yu. I. Cherskii, DAN, 125, No. 3 (1959).
\(^{5}\) V. S. Rogozhin, Siberian Math. Journal, No. 5 (1961).
\(^{6}\) H. G. Tillman, Math. Zs., 76, No. 1 (1961).
\(^{7}\) H. J. Bremermann, Z. Durand, J. Math. Phys., 2, No. 2 (1961).
\(^{8}\) S. Ya. Al’per, Izv. AN SSSR, ser. matem., 19, No. 5 (1955).
\(^{9}\) I. Gelferin, Introduction to the Theory of Generalized Functions, 1954.
\(^{10}\) P. K. Suetin, DAN, 58, No. 1 (1959).
\(^{11}\) I. M. Gel’fand, P. E. Shilov, Generalized Functions, Vol. 1, 1958.

* The product of the fundamental function \(\Psi(t)\) by a generalized \(f\) is defined by the equality
\[ (f\Psi, \varphi) = (f, \Psi\varphi) \quad (^{11}). \]