MATHEMATICS

V. S. ROGOZHIN

ON SOME NEW INTEGRAL REPRESENTATIONS OF ANALYTIC FUNCTIONS

(Presented by Academician P. Ya. Kochina on 8 October 1963)

New representations of analytic functions are established in the article in the form of Cauchy-type integrals whose densities have a special form. The results are used for constructing bases in certain classes of analytic functions.

In what follows, unless otherwise specified, \(\Gamma\) denotes a simple smooth closed contour dividing the plane into two domains: the interior \(D^+\) and the exterior \(D^-\).

We introduce the following classes of functions:

1. \(H(\Gamma)\) is the class of functions of the points \(t=x+iy\) of the contour \(\Gamma\) satisfying on \(t\) the Hölder condition with some exponent \(\alpha\) \((0<\alpha\leqslant 1)\).

2. \(H(D^{(\pm)})\) is the class of functions analytic in the domain \(D^{(\pm)}\) and continuous in its closure \(\overline{D}^{(\pm)}\), whose boundary values on \(\Gamma\) belong to the class \(H(\Gamma)\).

3. \(E_p(D^{(\pm)})\) is the class of functions \(f(z)\), analytic in the domain \(D^{(\pm)}\), for which the inequality
   \[ \int_{\gamma_r} |f(z)|^p |dz| < C, \qquad p>1, \]
   is satisfied, where \(\gamma_r\) is the image of the circle \(|w|=r\) under a conformal mapping of the disk \(|w|<1\) onto the domain \(D^{(\pm)}\) \((C\) depends on \(f(z)\), but does not depend on \(r)\).

4. \(A(D^{(\pm)})\) is the class of functions analytic in the domain \(D^{(\pm)}\).

Further, by \(G(\tau)\) we denote a prescribed function of the points of the contour \(\Gamma\), belonging to the class \(H(\Gamma)\), \(\varkappa=\operatorname{ind}_{\Gamma} G(\tau)\).

Theorem 1. If \(f^+(z)\in H(D^+)\) and \(\varkappa \geqslant 0\), then
\[ f^+(z)=\frac{1}{2\pi i}\int_{\Gamma}\frac{\varphi^+(\tau)}{G(\tau)}\,\frac{d\tau}{\tau-z}, \tag{1} \]
where \(\varphi^+(z)\in H(D^+)\) and is determined by the equalities
\[ \varphi^+(z)=X^+(z)\left[\frac{1}{2\pi i}\int_{\Gamma}\frac{f^+(\tau)}{X^-(\tau)}\,\frac{d\tau}{\tau-z}+P_{\varkappa-1}(z)\right], \]
\[ X^\pm(z)=\exp\left\{\frac{1}{2\pi i}\int_{\Gamma}\frac{\ln\!\left[G(\tau)\tau^{-\varkappa}\right]}{\tau-z}\,d\tau\right\}, \]
\(P_{\varkappa-1}(s)\) is a polynomial of degree \(\varkappa-1\) with arbitrary coefficients. If, however, \(\varkappa<0\), then the representation (1) will hold provided the conditions
\[ \frac{1}{2\pi i}\int_{\Gamma}\frac{f^+(\tau)}{X^-(\tau)}\,\tau^{k-1}\,d\tau=0, \qquad k=1,2,\ldots,|\varkappa|. \tag{2} \]
are fulfilled.

The theorem remains valid if \(f^+(z)\in E_p(D^+)\), and \(\Gamma\) is a Lyapunov curve. In this case \(\varphi^+(z)\in E_p(D^+)\).

Proof. Let $\varphi^{-}(z)$ be the value of the integral in formula (1) for $z \in D^{-}$; then from (1) there follows the relation
$f^{+}(t)-\varphi^{-}(t)=\varphi^{+}(t)/G(t)$, $t\in\Gamma$, which may be regarded as the boundary condition of the Riemann problem in the class of functions satisfying the Hölder condition ($^{1,2}$), or in the class $L_p(\Gamma)$ ($^3$), depending on the class to which $f^{+}(z)$ belongs, $H(D^{+})$, or $E_p(D^{+})$. Hence we conclude that $\varphi^{+}(z)$ is indeed determined by formula (2) and belongs to the class $H(D^{+})$ if $f^{+}(z)\in H(D^{+})$. If, however, $f^{+}(z)\in E_p(D^{+})$, then $\varphi^{+}(t)\in L_p(\Gamma)$, and the gluing theorems show that $\varphi^{+}(z)$ will also belong to the class $E_p(D)^{+}$. The theorem is proved.

Let the function $\alpha(t)$, $t\in\Gamma$, map one-to-one and with preservation of the direction of traversal the Lyapunov contour $\Gamma$ onto some, generally speaking, other Lyapunov contour $\widetilde{\Gamma}$, bounding the domain $\widetilde{D}^{+}$, with $\alpha'(t)\ne0$ and $\alpha'(t)\in H(\Gamma)$.

Theorem 2. If $f^{+}(z)\in H(D^{+})\ (E_p(D^{+}))$ and $\varkappa\ge0$, then

\[ f^{+}(z)=\frac{1}{2\pi i}\int_{\Gamma}\frac{\varphi^{+}[\alpha(\tau)]}{G(\tau)}\,\frac{d\tau}{\tau-z}, \tag{3} \]

where $\varphi^{+}(z)\in H(\widetilde{D}^{+})\ (E_p(\widetilde{D}^{+}))$. For $\varkappa=0$, $\varphi^{+}(z)$ is determined by $f^{+}(z)$ uniquely; if $\varkappa>0$, then $\varphi^{+}(z)$ contains linearly $\varkappa$ arbitrary complex constants.

Proof. The justification of the representation (3) reduces to the study of the solvability of the problem of linear conjugation with shift

\[ f^{+}(t)-\varphi^{-}(t)=\frac{\varphi^{+}[\alpha(t)]}{G(t)},\qquad t\in\Gamma,\quad \alpha(t)\in\widetilde{\Gamma}, \tag{4} \]

studied in ($^4$). In that work it is proved that problem (4) reduces to an analogous problem in the case when the contour is the circle $\Gamma_0$, and the unknown functions are analytic inside and outside $\Gamma_0$, respectively, while $\alpha(t)$ maps $\Gamma_0$ onto itself and has the properties indicated above.

We then argue as in the proof of Theorem 1, using the results of ($^{5,6}$).

Similarly one proves

Theorem 3. If $\Gamma$, $\widetilde{\Gamma}$, and $\alpha(t)$ satisfy the conditions of Theorem 2 and $f^{-}(z)\in H(D^{-})$ $(f^{-}(\infty)=0)$, then for $\varkappa\ge1$

\[ f^{-}(z)=\frac{1}{2\pi i}\int_{\Gamma}\frac{\overline{\varphi^{+}[\alpha(\tau)]}}{G(\tau)}\,\frac{d\tau}{\tau-z}, \tag{5} \]

where $\varphi^{+}(z)\in H(\widetilde{D}^{+})\ (E_p(\widetilde{D}^{+}))$. For $\varkappa=1$, $\varphi^{+}(z)$ is determined uniquely, while for $\varkappa>1$ it depends linearly on $\varkappa-1$ arbitrary complex constants.

Let us now consider a function $G(z)$ analytic and different from zero in the curvilinear annulus $K$ bounded by the closed Jordan curves $\Gamma$ and $\Gamma_1$ ($\Gamma_1$ lies inside $\Gamma$). Clearly, the index of the function $G(z)$ on any contour lying inside $K$ and enclosing $\Gamma_1$ will be one and the same. Consequently, one may speak of the index $\varkappa$ of the function $G(z)$ in the annulus $K$: $\varkappa=\operatorname{ind}_{K}G(z)$.

Theorem 4. If $f^{+}(z)\in A(D^{+})$, then for $\varkappa\ge0$

\[ f^{+}(z)=\frac{1}{2\pi i}\int_{\gamma}\frac{\varphi^{+}(\tau)}{G(\tau)}\,\frac{d\tau}{\tau-z}, \tag{6} \]

where $\varphi^{+}(z)\in A(D^{+})$ and depends linearly on $\varkappa$ arbitrary real constants, $\gamma$ is an arbitrary rectifiable contour lying in the annulus $K$ and enclosing $\Gamma_1$.

The proof of Theorem 4 follows immediately from the analysis of the relation
$\varphi^{+}(t)=-G(t)\varphi^{-}(t)+f^{+}(t)G(t)$, $t\in\gamma$, equivalent to equality (6).*

* In Theorems 2 and 4 one could also have considered the case $\varkappa<0$, and in Theorem 3 the case $\varkappa<1$, by analogy with Theorem 1.

Below we consider bases in the classes of functions \(H(D^{(\pm)})\), \(E_p(D^{(\pm)})\), \(A(D^{(\pm)})\). By a basis in the class \(H(D^{(\pm)})\) \(\bigl(E_p(D^{(\pm)}), A(D^{(\pm)})\bigr)\) we mean a system of functions belonging to the corresponding class such that every function of the class under consideration is uniquely expanded in the series

\[ f(z)=\sum_{k=1}^{\infty} c_k f_k(z), \tag{7} \]

where, if the class \(A(D^{(\pm)})\) \(\bigl(H(D^{(\pm)})\bigr)\) is taken, the series (7) must converge uniformly in the domain \(D^{(\pm)}\) (the closed domain \(\overline{D^{(\pm)}}\)); if, however, the class \(E_p(D^{(\pm)})\) is considered, then convergence of the series (7) on the contour \(\Gamma\) in the norm of the space \(L_p(\Gamma)\) must hold.

With the aid of Theorems 1—4 the following propositions are easily proved.

Theorem 5. If \(\{\varphi_k^+(z)\}\) is a basis in \(H(D^+)\) \(\bigl(E_p(D^+)\bigr)\), then:

a) the functions

\[ f_k^+(z)=\frac{1}{2\pi i}\int_\Gamma \frac{\varphi_k^+(\tau)}{G(\tau)}\,\frac{d\tau}{\tau-z} \]

for \(\varkappa=0\) also form a basis in \(H(D^+)\) \(\bigl(E_p(D^+)\bigr)\);

b) for \(\varkappa>0\), any function from \(H(D^+)\) \(\bigl(E_p(D^+)\bigr)\) can be expanded in the series (7), but the expansion may fail to be unique;

c) for \(\varkappa<0\), the functions \(f_k^+(z)\) form a basis among the functions of the class under consideration that satisfy condition (2).

For brevity of exposition, in the theorems below we restrict ourselves to the cases \(\varkappa=0\) and \(\varkappa=1\).

Theorem 6. If \(\{\varphi_k^+(\xi)\}\) is a basis in \(H(\widetilde D^+)\) \(\bigl(E_p(\widetilde D^+)\bigr)\) and \(\varkappa=0\), then the functions

\[ f_k^+(z)=\frac{1}{2\pi i}\int_\Gamma \frac{\varphi_k^+[\alpha(\tau)]}{G(\tau)}\,\frac{d\tau}{\tau-z} \]

also form a basis in \(H(\widetilde D^+)\) \(\bigl(E_p(\widetilde D^+)\bigr)\).

Theorem 7. If \(\{\varphi_k^+(\xi)\}\) is a basis in \(H(\widetilde D^+)\) \(\bigl(E_p(\widetilde D^+)\bigr)\) and \(\varkappa=1\), then the functions

\[ f_k^-(z)=\frac{1}{2\pi i}\int_\Gamma \frac{\overline{\varphi_k^+[\alpha(\tau)]}}{G(\tau)}\,\frac{d\tau}{\tau-z} \]

form a basis among the functions of the class \(H(D^-)\) \(\bigl(E_p(D^-)\bigr)\) that vanish at infinity.

Theorem 8. If \(\{\varphi_k^+(z)\}\) is a basis in the class \(A(D^+)\), then for \(\varkappa=0\) the functions

\[ f_k^+(z)=\frac{1}{2\pi i}\int_\gamma \frac{\varphi_k^+(\tau)}{G(\tau)(\tau-z)}\,d\tau, \]

where \(\gamma\) is any rectifiable curve lying inside the ring \(K\), also form a basis in \(A(D^+)\).

We note that, in the proof of Theorems 5—7 in the part concerning the class \(H(D^+)\), the following proposition plays an essential role:

Lemma. If \(\varphi_k^+(z)\in H(D^+)\), \(k=1,2,\ldots\), \(a(t)\in H(\Gamma)\), and \(\lim\limits_{k\to\infty}\max |\varphi_k(t)|=0\), then \(\lim\limits_{k\to\infty}\max |h_k(t)|=0\), where

\[ h_k(t)=\frac{1}{\pi i}\int_\Gamma \frac{a(\tau)\varphi_k^+(\tau)}{\tau-t}\,d\tau,\qquad t\in\Gamma . \]

Below we formulate some corollaries of Theorems 5–7. Let \(\Delta^{-}(z)\) be a given function belonging to the class \(H(D^{-})\) and different from zero in \(\overline{D}^{-}\).

Corollary 1 (see Theorem 5). Let \(P_k(z)\), \(k=1,2,\ldots\), be a polynomial basis in \(H(D^{+})\). Then the functions \(Q_k(z)\)—the collection of the terms with nonnegative powers of \(z\) in the Laurent expansion of the functions \(\Delta^{-}(z)P_k(z)\) in a neighborhood of the point \(z=\infty\)—will form a basis.

Let \(\Phi_0(z)\) be the function mapping \(D^{-}\) onto the exterior of the unit disk; \(\Phi_1(z)\), the function mapping \(D^{-}\) onto \(\widetilde D^{-}\); and \(\Phi_k(z)\) \((\widetilde{\Phi}_k(z))\), the Faber polynomials for the domain \(D^{+}\) \((\widetilde D^{+})\).

Corollary 2 (see Theorem 6). Each of the systems of functions

\[ {}^{(1)}\Psi_k(z)=\frac{1}{2\pi i}\int_\Gamma \frac{\Phi_0^k(\tau)}{G(\tau)}\,\frac{d\tau}{\tau-z}, \qquad {}^{(2)}\Psi_k(z)=\frac{1}{2\pi i}\int_\Gamma \frac{\widetilde{\Phi}_k[\Phi_1(\tau)]}{G(\tau)}\,\frac{d\tau}{\tau-z} \]

forms a basis in the class \(H(D^{+})\) \((E_p(D^{+})\), if \(\Gamma\) is a Lyapunov curve).

Corollary 3 (see Theorem 7). Each of the systems of functions

\[ {}^{(1)-}\Omega_k(z)=\frac{1}{2\pi i}\int_\Gamma \frac{\Phi_0^k(\tau)}{G(\tau)}\,\frac{d\tau}{\tau-z}, \qquad {}^{(2)-}\Omega_k(z)=\frac{1}{2\pi i}\int_\Gamma \frac{\widetilde{\Phi}_k[\Phi_1(\tau)]}{G(\tau)}\,\frac{d\tau}{\tau-z}, \]

\[ {}^{(3)-}\Omega_k(z)=\frac{1}{2\pi i}\int_\Gamma \frac{\overline{\Phi_k(\tau)}}{G(\tau)}\,\frac{d\tau}{\tau-z} \qquad (\chi=1) \]

forms a basis among the functions of the class \(H(D^{-})\) \((E_p(D^{-})\), if \(\Gamma\) is a Lyapunov curve) that vanish at infinity.

Rostov-on-Don
State University

Received
21 IX 1963

REFERENCES

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