Ultraspherical expansions and pseudo analytic functions

Theorem 2.1. Let $f(z)=\sum_{n=0}^{\infty}a_nz^n$ , where $a_n$ are real, and $\overline{\lim}\ |a_n|^{1/n}=1/R,\,R>1$ . Then $$\lim_{\mu \to 0} L_{\mu}(f) = \operatorname{Re}(f)$$ uniformly on $|z| \leq 1$ . Proof. $$|L_{\mu}(f) - \operatorname{Re}(f)| \leq \sum_{n=0}^{\infty} \left| \frac{2^{2\mu-1} n!}{\mu \Gamma(n+2\mu)} C_n^{\mu}(\cos \theta) - \cos n\theta \; \right| |a_n| \, r^n \; .$$ By the result [23, p. 182] $$C_{\scriptscriptstyle n}^{\,\mu}(\cos\theta) = \sum_{\scriptscriptstyle m=0}^{n} \alpha_{\scriptscriptstyle n} \alpha_{\scriptscriptstyle n-m} \cos{(n\,-\,2m)} \theta$$ , where $$\alpha_m = \frac{\Gamma(\mu + m)}{\Gamma(\mu)m!}.$$ It is easy to see that $$\lim_{\mu \to 0} \frac{\alpha_m \alpha_{n-m}}{\mu} = \begin{cases} 0 & \text{if } m \neq 0, \ n \ 1/n & \text{if } m = 0, \ n \end{cases}.$$ Thus $$\begin{aligned} \left| \frac{2^{2\mu-1}n\,!}{\mu\Gamma(n+2\mu)} C_n^\mu(\cos\theta) - \cos n\theta \, \right| \ & \leq \left| \frac{2^{2\mu-1}n\,!}{\mu\Gamma(n+2\mu)} C_n^\mu(\cos\theta) - \frac{2^{2\mu}n\,!}{\mu\Gamma(n+2\mu)} \alpha_0\alpha_n\cos n\theta \, \right| \ & + \left| \frac{2^{2\mu}n\,!}{\mu\Gamma(n+2\mu)} \alpha_0\alpha_n\cos n\theta - \cos n\theta \, \right| \ & \leq \frac{2^{2\mu-1}}{\mu} \sum_{m=1}^{n-1} \frac{n\,!}{\Gamma(n+2\mu)} \alpha_m\alpha_{n-m} + \left| \frac{2^{2\mu}n\,!\,\alpha_0\alpha_n}{\mu\Gamma(n+2\mu)} - 1 \right| \; , \end{aligned}$$ and for $0 < \mu < 1$ , this is $$\leq n^2 \! \left[ \frac{2^{2\mu-1}}{\mu \varGamma^2(\mu)} + \frac{1}{n^2} \left| \frac{2^{2\mu} \varGamma(n+\mu)}{\mu \varGamma(\mu) \varGamma(n+2\mu)} - 1 \right| \right].$$ Now, since $\Gamma(n+\mu)/\Gamma(n+2\mu) \sim (n+\mu)^{-\mu}$ as $n \to \infty$ , and as $\mu \to 0$ , $\mu\Gamma(\mu) \to 1$ , there exists an $N_1$ such that $$\frac{2^{2\mu-1}}{\mu \Gamma^2(\mu)}+\frac{1}{n^2}\bigg|\frac{2^{2\mu}\Gamma(n+\mu)}{\mu \Gamma(\mu)\Gamma(n+2\mu)}-1\bigg| \le c$$ , for all $n>N_1$ and all $\mu\in[0,1]$ , where c is a constant (eg., 3 will do). Thus given $\varepsilon>0$ , there exists an $N_2>N_1$ such that $$\sum_{n=N_2}^{\infty} \left| \frac{2^{2\mu-1} n!}{\mu \Gamma(n+2\mu)} \, C_n^{\mu}(\cos \theta) - \cos n\theta \, \left| \left| a_n \right| r^n < \varepsilon$$ for all $\mu \in [0, 1]$ , $\theta \in [0, \pi]$ , and $0 \le r \le 1$ . The convergence (2.4) of the first $N_2$ terms then completes the proof. The conjugate V of U given by (2.2) has the series expansion $$(2.5) V(r,\theta) = 2^{2\mu} \sum_{n=1}^{\infty} \frac{n!}{\Gamma(n+2\mu+1)} a_n r^n C_{n-1}^{\mu+1}(\cos\theta) \sin\theta.$$ Bers and Gelbart [8, p. 177] have given an integral representation for conjugate functions which yields the definition of a "conjugate" integral operator $L_{\mu}^*$ that transforms the $L_{\mu}$ associate f of U to V. Appealing to this result, the following is immediate: If U is given by the series (2.2), then a function connected with U by the equations (1.5) is $$V(r,\,\theta) = \, -\frac{i}{\mu \varGamma^2(\mu)}\!\!\int_{\scriptscriptstyle 0}^{\scriptscriptstyle \pi} f(r\cos\theta\,+\,ir\sin\theta\cos t)\cos t\sin^{2\mu-1}\!tdt\;.$$ We further verify directly that this expression has the series expansion (2.5) above. An integration by parts shows that $$\begin{split} -\frac{i}{\mu \Gamma^2(\mu)}\!\!\int_0^\pi (r\cos\theta+ir\sin\theta\cos t)^n\cos t\sin^{2\mu-1}\!tdt \ &=\frac{nr\sin\theta}{2\Gamma^2(\mu+1)}\!\!\int_0^\pi (r\cos\theta+ir\sin\theta\cos t)^{n-1}\sin^{2\mu+1}\!tdt \ &=\frac{n}{2}(\mu+1)r\sin\theta L_{\mu+1}(z^{n-1}) \ &=\frac{2^{2\mu}n!}{\Gamma(n+2\mu+1)}r^nC_{n-1}^{\mu+1}(\cos\theta)\sin\theta \; , \end{split}$$ which yields the result. Also, since $$|C_n^{\mu}(x)| \leq \frac{\Gamma(n+2\mu)}{n! \Gamma(2\mu)},$$ the three series (2.2), (2.3), and (2.5) all have the same radius of convergence, i.e., they all converge absolutely and uniformly on compact subsets of |z| < R, where $\overline{\lim} |a_n|^{1/n} = R^{-1}$ . Thus, defining $$(2.7) \qquad L_{\mu}^{\ast}(f) = -\frac{i}{\mu \Gamma^2(\mu)}\!\!\int_0^{\pi} f(r\cos\theta\,+\,ir\sin\theta\sin t)\cos t\sin^{2\mu-1}\!tdt$$ , on interchanging the order of summation and integration we find that if f has the Taylor series expansion (2.3), then $L_{\mu}^{}(f)$ is given by the series (2.5). (Regarding the conjugate operator, see also [14, p. 189]). Further, the $L_{\mu}$ and $L_{\mu}^{}$ integral representations for U and V obtain throughout the common disk |z| < R. In view of Theorem 1, it is natural to expect that $L_{\mu}^{*}(f) \rightarrow \operatorname{Im}(f)$ as $\mu \to 0$ . Now, $$\lim_{n \to 0} C_{n-1}^{n+1}(\cos \theta) = C_{n-1}^{1}(\cos \theta) = \frac{\sin n \, \theta}{\sin \theta}$$ (see [23, p. 187]). Thus (2.8) $$\lim_{n\to 0} L_{n}^{*}(z^{n}) = \lim_{n\to 0} \frac{2^{2n}n!}{\Gamma(n+2\mu+1)} a_{n}C_{n-1}^{n+1}(\cos\theta) \sin\theta$$ $$= r^{n} \sin n\theta$$ $$= \operatorname{Im}(z^{n}).$$ Again, since $|L_{\mu}^*(z^n) - \operatorname{Im}(z^n)| \sim g(n)$ on $|z| \leq 1$ , where $[g(n)]^{1/n} \to 1$ as $n \to \infty$ , we have the desired result for functions which are analytic on a closed disk:

Theorem 2.2. Let $f(z)=\sum_{n=0}^{\infty}a_nz^n$ , where $a_n$ are real and $\overline{\lim}_{n\to\infty}|a_n|^{1/n}=1/R$ , R>1. Then $$\lim_{\mu \to 0} L_{\mu}^*(f) = \operatorname{Im}(f)$$ uniformly on $|z| \leq 1$ . Proof. The generating relation (1.1) yields $$\frac{z \sin \theta}{(1 - 2z \cos \theta + z^2)^{\mu+1}} = \sum_{n=1}^{\infty} z^n C_{n-1}^{\mu+1}(\cos \theta) \sin \theta.$$ Subtracting $$\frac{z\sin\theta}{1-2z\cos\theta+z^2}=\sum_{n=1}^{\infty}z^n\sin n\theta,$$ and noting that each series converges uniformly on compact subsets of |z| < 1, we have for $0 < \rho < 1$ , $n = 1, 2, \cdots$ , $$\begin{split} C_{n-1}^{\prime\prime+1}(\cos\theta) &- \sin n\theta \ &= \frac{1}{2\pi i} \! \int_{|z|=\rho} \frac{z\sin\theta[1-(1-2z\cos\theta+z^2)^\mu]}{(1-2z\cos\theta+z^2)^{\mu+1}} \,\frac{dz}{z^{n+1}} \;. \end{split}$$ Thus $$|C_{n-1}^{n+1}(\cos\theta)\sin\phi - \sin n\theta| \le \frac{C}{\rho^{n-1}}$$ for all $\theta \in [0, 2\pi]$ , and $\mu \in [0, 1]$ , where C is a constant depending only on $\rho$ . Now, $$\begin{split} |L^*(f) - \operatorname{Im}(f)| \\ & \leq \sum_{n=1}^{\infty} \left| \frac{2^{2\mu} n!}{\Gamma(n+2\mu+1)} a_n r^n C_{n-1}^{\mu+1}(\cos\theta) \sin\theta - a_n r^n \sin n\theta \right| \\ & \leq \sum_{n=1}^{\infty} |a_n| r^n \left[ \left| \frac{2^{2\mu} n!}{\Gamma(n+2\mu+1)} C_{n-1}^{\mu+1}(\cos\theta) \sin\theta - \sin n\theta \right| \right. \\ & \left. - C_{n-1}^{\mu+1}(\cos\theta) \sin\theta \right| + |C_{n-1}^{\mu+1}(\cos\theta) \sin\theta - \sin n\theta | \right]. \end{split}$$ Let $\varepsilon > 0$ . Choose $\rho \in (0, 1)$ sufficiently near 1 so that $\sum_{n=0}^{\infty} |a_n| \rho^{-n}$ converges. Then by the result (2.10), there exists an $N_1$ such that $$\sum_{n=N_1}^{\infty} |a_n| r^n |C_{n-1}^{\mu+1}(\cos \theta) \sin \theta - \sin n\theta| < \varepsilon$$ for all $r \in [0, 1]$ , $\theta \in [0, 2\pi]$ . Further, using the bound (2.6), we have $$\begin{aligned} \left| \frac{2^{2\mu}n!}{\Gamma(n+2\mu+1)} C_{n-1}^{\mu+1} (\cos\theta) \sin\theta - C_{n-1}^{\mu+1} (\cos\theta) \sin\theta \right| \ & \leq \left| \frac{2^{2\mu}n!}{\Gamma(n+2\mu+1)} - 1 \right| \frac{\Gamma(n+2\mu+1)}{(n-1)!\Gamma(n+2\mu)} \ & \leq \frac{(2^{2\mu}+1)(n+1)(n+2)}{\Gamma(2\mu+2)} \ . \end{aligned}$$ Thus there exists an $N_2$ such that $$\sum_{n=N_2}^{\infty} |a_n| r^n \left| \frac{2^{2\mu} n!}{\Gamma(n+2\mu+1)} C_{n-1}^{\mu+1}(\cos \theta) \sin \theta - C_{n-1}^{\mu+1}(\cos \theta) \sin \theta \right| < \varepsilon.$$ The convergence (2.8) of the first $N = \max(N_1, N_2)$ terms then completes the proof. Define $$\mathscr{L}_{\mu} = L_{\mu} + iL_{\mu}^{*}$$ . If U and V are conjugate functions given by (2.2) and (2.5), then F = U + iV is a $\mu$ -pseudo analytic function of the first kind, and $$\begin{align} F(r,\theta) &= \mathscr{L}_{\mu}(f) \ &= \frac{1}{\mu \varGamma^2(\mu)} \!\! \int_0^{\pi} f(r\cos\theta + ir\sin\theta\cos t) \; (1+\cos t)\sin^{2\mu-1}\!tdt \; , \end{split}$$ where $f(z) = \sum_{n=0}^{\infty} a_n z^n$ . If F is regular on |z| < 1 (i.e., U and V are $C^2$ solutions of the system (1.5)), then the $\mathcal{L}_{\mu}$ associate f of F is analytic on |z| < 1, and the integral representation (2.11) holds throughout the disk |z| < 1. As an immediate consequence of Theorems 1 and 2, we have the COROLLARY. Let $f(z)=\sum_{n=0}^{\infty}a_nz^n$ , where $a_n$ are real, and $\overline{\lim} |a_n|^{1/n}=1/R$ , R>1. Then $$\lim_{\mu \to 0} \mathscr{L}_{\mu}(f) = f$$ uniformly on $|z| \leq 1$ . Thus analytic functions of a single complex variable appear as a limiting case of $\mu$ -pseudo analytic functions. With the exception of an explicit representation for the inverse transform $(L_{\mu}^{})^{-1}$ , our discussion of integral operators which generalize the operations Re and Im, and examination of their relationship with pseudo analytic functions is complete. The development of an explicit representation for $(L_{\mu}^{})^{-1}$ will be postponed to the next section, since (as will be shown) both this transform and $L_{\mu}^{-1}$ are closely related to the Poisson integral. 3. Function theoretic results on a disk. Since ultraspherical expansions and their conjugates enjoy the same relation with $\mu$ -pseudo analytic functions that is found between trigonometric series and analytic functions, it is natural to expect analogs of classical function theoretic results on a disk. Such results have been obtained by Muckenhaupt and Stein [19] and include, in particular, a Poisson integral formula for U, conjugate kernel, analog of Fatou's theorem, Riese's theorem ( $U \rightarrow V$ is a bounded operator on $L^p$ ), and $H^p$ theory. Our objectives in this section are to examine the relationship between the Poisson integral formula and the transform $L^{-1}_{\mu}$ , obtain new integral representations for the Poisson kernel and conjugate kernel, develop an explicit integral representation for $(L^*_{\mu})^{-1}$ , and to give an extension of Privaloff's theorem [20] to ultraspherical expansions and their conjugates. If $U(\theta)$ has ultraspherical expansion $$U(\theta)=\frac{2^{2\mu-1}}{\mu}\sum_{n=0}^{\infty}\frac{n\,!}{\Gamma(n+2\mu)}\,a_{n}C_{n}^{\,\mu}\left(\cos\theta\right)$$ , then the Abel sum of this series (3.1) $$U(r, \theta) = \frac{2^{2\mu-1}}{\mu} \sum_{n=0}^{\infty} \frac{n!}{\Gamma(n+2\mu)} a_n r^n C_n (\cos \theta)$$ is given by the "Poisson integral" $$U(r,\,\theta)=\int_0^\pi\!U(\phi)P(r,\,\theta,\,\phi)\sin^{2\mu}\!\phi d\phi$$ , where the "Poisson kernel" P is $$\begin{aligned} P(r,\theta,\phi) &= \frac{\Gamma^2(\mu) 2^{2\mu-1}}{\pi} \sum_{n=0}^{\infty} \frac{(n+\mu) n\,!}{\Gamma(n+2\mu)} r^n C_n^{\,\mu} (\cos\theta) C_n^{\,\mu} (\cos\phi) \ &= \frac{\mu (1-r^2)}{\pi} \!\!\int_0^{\pi} \!\! \frac{\sin^{2\mu-1}\! t}{[1-2r(\cos\theta\cos\phi+\sin\theta\sin\phi\cos t)+r^2]^{\mu+1}} d\,t \end{aligned}$$ (see [19, p. 25]). Gilbert's inverse transform $L_{\mu}^{-1}$ is easily obtained from the Poisson integral formula (3.2). Letting $\theta=0$ in (3.2) and analytically continuing r to complex values z throughout the disk |z|<1 yields (3.3) $$U(z, 0) = \int_0^{\pi} U(\phi) P(z, 0, \phi) \sin^{2\mu} \phi d\phi,$$ where $$\begin{align} P(z,\,0,\,\phi) &= \frac{\mu}{\pi}\,\frac{1-z^2}{(1-2z\cos\phi+z^2)^{\mu+1}}\!\int_0^{\pi}\sin^{2\mu-1}\!tdt \ &= \frac{\mu \Gamma(\mu)\Gamma(1/2)}{\pi\Gamma(\mu+1/2)}\frac{1-z^2}{(1-2z\cos\phi+z^2)^{\mu+1}} \;. \end{gathered}$$ Since $C_n^{\mu}(1) = \Gamma(n+2\mu)/n!\Gamma(2\mu)$ , we have $$U(z,\,0)=\frac{2^{2\mu-1}}{\mu \Gamma(2\mu)}\sum_{n=0}^{\infty}a_{n}z^{n}\,=\frac{2^{2\mu+1}}{\mu \Gamma(2\mu)}f(z)$$ , where f is the $L_{\mu}$ associate of $U(r, \theta)$ . Thus (3.3) yields for |z| < 1, $$f(z) = \frac{ \varGamma^2 (\mu + 1)}{\pi} \! \int_0^\pi U(\phi) \! \frac{1 - z^2}{(1 - 2z\cos\phi + z^2)^{\mu + 1}} \sin^{2\mu}\!\!\phi d\phi$$ , which is the expression (1.9) for $L_{\mu}^{-1}$ in the case where the integral is taken over the unit half circle. Conversely, one obtains the Poisson integral formula by applying the $L_{\mu}$ transform to the expression for $L_{\mu}^{-1}$ . Letting r=1 in (1.9), we have for |z|<1 $$(3.4) \begin{aligned} L_{\mu}[f(z)] &= L_{\mu}\!\!\left[\frac{\varGamma^2(\mu+1)}{\pi}\!\int_0^{\pi}U(1,\,\phi)\frac{1-z^2}{(1-2z\cos\phi+z^2)^{\mu+1}}\sin^{2\mu}\!\phi d\phi \right. \ &= \frac{\varGamma^2(\mu+1)}{\pi}\!\int_0^{\pi}U(1,\,\phi)L_{\mu}\!\left[\frac{1-z^2}{(1-2z\cos\phi+z^2)^{\mu+1}}\right]\!\sin^{2\mu}\!\phi d\phi \;. \end{aligned}$$ Now, $$(3.5) \qquad \frac{1-z^2}{(1-2z\cos\phi+z^2)^{\mu+1}} = \sum_{n=0}^{\infty} \frac{n+\mu}{\mu} z^n C_n^{\mu}(\cos\phi) ,$$ and the series converges uniformly for $|z| \leq \rho < 1$ , $\phi \in [0, \pi]$ . Applying the $L_{\mu}$ transform termwise to this series yields $$\begin{split} L_{\mu} & \left[ \frac{1-z^2}{(1-2z\cos\phi+z^2)^{\mu+1}} \right] = \frac{2^{2\mu-1}}{\mu^2} \sum_{n=0}^{\infty} \frac{(n+\mu)n\,!}{\Gamma(n+2\mu)} r^n C_n^{\,\mu}(\cos\phi) C_n^{\,\mu}(\cos\theta) \ & = \frac{\pi}{\Gamma^2(\mu+1)} P(r,\,\theta,\,\phi) \;. \end{split}$$ Since $U(r, \theta) = L_{\mu}[f(z)]$ , (3.4) is the Poisson integral formula for U. Note that this also yields an additional integral representation for the Poisson kernel: $$P(r,\,\theta,\,\phi) = \frac{\mu}{\pi} \int_0^\pi \frac{1 - (x + iy\cos t)^2}{[1 - 2(x + iy\cos t)\cos\phi + (x + iy\cos t)^2]^{\mu+1}} \sin^{2\mu-1}t dt$$ , where $(x, y) = (r \cos \theta, r \sin \theta)$ . Again let $U(r, \theta)$ be given by the series (3.1). Then the conjugate $V(r, \theta)$ of U has on the unit disk the representation $$V(r,\,\theta)=\int_0^\pi U(1,\,\phi)Q(r,\,\theta,\,\phi)\sin^{2\mu}\!\!\phi d\phi$$ , where Q is the "conjugate Poisson kernel" $$Q(r,\,\theta,\,\phi) = \frac{2^{2\mu}\mu \Gamma^2(\mu)}{\pi} \sum_{n=0}^{\infty} \frac{n\,!\,(n+\mu)}{\Gamma(n+2\mu+1)} r^n C_n^{\,\mu}(\cos\phi) C_{n-1}^{\,\mu+1}(\cos\theta) \sin\theta$$ (see [19, p. 35]). On applying the $L_{\mu}^{*}$ transform to the generating relation (3.5), a routine computation yields an integral representation for Q similar to that found above for P: $$Q(r,\,\theta,\,\phi) = \frac{-i\mu}{\pi}\!\!\int_0^\pi \frac{1\!-\!(x\!+\!iy\cos t)}{[1\!-\!2(x\!+\!iy\cos t)\cos\phi\!+\!(x\!+\!iy\cos t)^2]^{\mu+1}}\!\!\cos t\sin^{2\mu-1}\!tdt \,.$$ Expressed in terms of the $L_{\mu}$ , $L_{\mu}^{*}$ integral operators, the representations (3.6) and (3.7) become $$P(r,\,\theta,\,\phi)=\frac{\varGamma(\mu+1)}{\pi}\,L_{\mu}\bigg[\frac{1-z^2}{(1-2z\cos\phi+z^2)^{\mu+1}}\bigg]$$ , and $$Q(r,\,\theta,\,\phi) = \frac{\varGamma(\mu+1)}{\pi} L_{\mu}^{\ast} \bigg[ \frac{1-z^2}{(1-2z\cos\phi+z^2)^{\mu+1}} \bigg] \,.$$ Thus, for $0 \le r < 1$ , $\theta \in [0, 2\pi]$ , (3.8) $$F(r, \theta) = \int_0^{\pi} U(1, \phi) [P(r, \theta, \phi) + iQ(r, \theta, \phi)] \sin^{2\mu}\!\!\phi d\phi$$ , where F = U + iV, and $$P(r,\,\theta,\,\phi) + iQ(r,\,\theta,\,\phi) = \frac{\varGamma(\mu+1)}{\pi} \mathscr{L}_{\mu} \bigg[ \frac{1-z^2}{(1-2z\cos\phi+z^2)^{\mu+1}} \bigg] \,.$$ This result provides an analog of the Schwartz formula (3.9) $$f(z) = \frac{1}{2\pi} \int_{0}^{2\pi} u(e^{it}) \frac{e^{it} + z}{e^{it} - z} dt$$ which recovers an analytic function f from its real part u. Notice that on restricting $\theta=0$ in (3.8) and analytically continuing r to complex values $\zeta$ in $|\zeta|<1$ , the integral reduces to the expression for the inverse transform $L_{\mu}^{-1}$ . This integral representation for $L_{\mu}^{-1}$ might also be viewed as analogous to Schwartz's formula. For $L_{\mu}$ provides a transformation from analytic functions to solutions of an elliptic equation—in fact, is a continuous extension of the operation Re. And Schwartz's formula is nothing more than an explicit integral representation for the transform Re^-1. This observation leads to the following analog of Privaloff's theorem [20]:

Theorem 3. If $$U(\theta) = \frac{2^{2 p - 1}}{\mu} \sum_{n=0}^{\infty} \frac{n\,!}{\Gamma(n\,+2\mu)} a_{n} C_{n}^{\,\prime\prime}(\cos\theta)$$ satisfies a Lipschitz condition of order $\alpha$ , $0<\mu<\alpha<1$ , on $[0,\pi]$ , then its conjugate $$V(\theta)=2^{2\mu}\sum_{n=0}^{\infty}\frac{n\,!}{\varGamma(n+2\mu+1)}a_{n}C_{n-1}^{\mu+1}(\cos\theta)\sin\theta$$ satisfies a Lipschizt condition of order $\alpha - \mu$ on $[0, \pi]$ .

Theorem 4.1. If U is a generalized axisymmetric potential which is regular on an axiconvex region $\Omega$ , then its conjugate V is also regular on $\Omega$ . The following result is also immediate.

Theorem 4.2. F is \(\mu\text{-pseudo}\) analytic on an axiconvex region $\Omega$ if and only if its $\mathscr{L}_{\mu}$ associate f is analytic throughout $\Omega$ .

Theorem 5.1. Let $\Omega$ be a bounded axiconvex region. If F is $\mu$ -pseudo analytic on $\mathrm{Cl}(\Omega)$ , then there exist $\mu$ -pseudo analytic polynomials $$W_n(z) = \sum_{k=0}^n \alpha_k W^{(k)}(z)$$ such that $$\|F-W_n\| \leq \frac{M}{R^n}$$ , where $\|\cdot\|$ denotes the sup norm over $\operatorname{Cl}(\Omega)$ , M is a constant independent of n, and R > 1.

Theorem 5.2. Let $\Omega$ be a bounded axiconvex region, $w = \phi(z) = z + a_0 + a_1 z^{-1} + \cdots$ map the compliment of $\Omega$ onto the circular region $|w| > \rho$ , and let $\psi = \phi^{-1}$ . If F(x, y) is $\mu$ -pseudo analytic on the region $B_R$ bounded by $\Gamma_R = \{z: |\phi(z)| = R\}$ , then F can be expanded into a series of $\mu$ -pseudo analytic Faber polynomials (5.3) $$F(x, y) = \sum_{n=0}^{\infty} a_n F_n(x, y),$$ where $$(5.4) \hspace{1cm} a_{\scriptscriptstyle n} = \frac{\mu \Gamma(2\mu)}{2^{2\mu} \pi i} \int_{|z| = r} F(\psi(z), \, 0) z^{-k-1} dz, \; \rho < r < R \; .$$ Further $$\lim_{n \to \infty} \sup |a_n|^{\scriptscriptstyle 1/n} = 1/R$$ , and the series (4.3) converges uniformly on $\operatorname{Cl}(\Omega)$ at a geometric rate.

Theorem 5.3. Let $\Omega$ be a bounded axiconvex region and $C_n^*$ maximize $|V(C_n)|$ over $\operatorname{Cl}(\Omega)$ . Then $$\lim_{n\to\infty} L_n(z; C_n^*; F) = F(z)$$ uniformly on $Cl(\Omega)$ for every function F which is $\mu$ -pseudo analytic on $Cl(\Omega)$ . Further, the convergence is at a geometric rate, i.e., $$\|F-L_n\| \leq \frac{(n+2)M}{R^n}$$ where $\|\cdot\|$ denotes the sup norm over $\Omega$ and the constants M, R>1 are as given in Theorem 5.1.