Real and Complex Variables Preliminary Exam - 2001

Let \(X\) be a Banach space and let \(T\) be a function which assigns to each real number \(s\in \mathbb R\) a bounded linear operator \(T(s):X\to X\). The derivative \(T'(s)\) of \(T\) at \(s\) is the operator defined by \[ T'(s)x=\lim _{h\to 0}h^{-1}(T(s+h)-T(s))x \] where the domain of \(T'(s)\) is the set of all \(x\in X\) for which this limit exists. Prove that if the domain of \(T'(s)\) is all of \(X\) for a given \(s\), then \(T'(s)\) is a bounded linear operator on \(X\).

Prove that if \(f,g\in L^2(\mathbb R)\) and the convolution product \(f*g\) is defined by \[ f*g(x)=\int f(y)g(x-y)\,dy, \] then \(f*g(x)\) is a continuous function which vanishes at infinity.

Let \(X\) be a compact Hausdorff space and denote by \(C(X)\) the Banach space of continuous complex valued functions on \(X\) and by \(P(X)\) the cone consisting of all \(f\in C(X)\) such that \(\operatorname {Re}(f)\ge 0\). If \(H\) is a linear subspace of \(C(X)\) such that \(H\cap P(X)=(0)\), prove that

(a) \(\|\lambda +h\|_\infty \ge |\lambda |\) for all \(\lambda \in \mathbb C\), \(h\in H\) (hint: show this is equivalent to \(\|1-h\|_\infty \ge 1\) for all \(h\in H\));

(b) there exists a positive Borel measure \(\mu \) on \(X\) such that \(\mu (X)=1\) and \[ \int h\,d\mu =0 \quad \text {for all } h\in H \] (recall that a measure \(\mu \) on \(X\) is positive if \(|\mu |=\mu (X)\)).

Let \(H_n\) be the linear span in \(L^2([-\pi ,\pi ])\) of the set \(\{e^{ikt}\}_{k=-n}^n\). The orthogonal projection \(P_n\) of \(L^2([-\pi ,\pi ])\) onto \(H_n\) may be written in the form \[ P_n f(x)=h_n*f(x)=\frac {1}{2\pi }\int _{-\pi }^{\pi }h_n(x-t)f(t)\,dt. \] Find the function \(h_n\) and prove that it has this property.

Prove that

(a) \(\log (1+e^t)\le t+\log (2)\) for all \(t\ge 0\);

(b) if \(f\) is a real valued function in \(L^1([0,1])\) then \[ \lim _{n\to \infty }\frac {1}{n}\int _0^1 \log (1+e^{n f(x)})\,dx =\int _0^1 f^+(x)\,dx, \] where \(f^+(x)=f(x)\) if \(f(x)\ge 0\) and \(f^+(x)=0\) if \(f(x)<0\).

Suppose that \(f\) is holomorphic in the open disc of radius \(r\), centered at \(z_0\), but does not have an analytic extension to any neighborhood of the closure of this disc. Show that the radius of convergence of the power series expansion for \(f\) centered at \(z_0\) is \(r\).

Let \(f\) be a holomorphic function defined on a neighbourhood of the closed disc \(|z|\le 1\). Assume that \(f(0)=1\), and \(|f(z)|>1\) if \(|z|=1\). Must \(f\) have a zero in the open disc \(|z|<1\)? Justify your answer.

Determine the number of zeroes of the polynomial \[ z^5+4z^2+2z+1 \] in the region \(|z|<1\).

Suppose \(f\) is holomorphic in \(r<|z|<R\), and suppose that for some \(\rho \), \(r<\rho <R\), \[ \int _{|z|=\rho } f(z)z^n\,dz=0 \] for all negative integers \(n\). Show that \(f\) has a holomorphic extension to the set \(|z|>r\).

Assume that \(f\) is a one-to-one conformal mapping of the open disc \(|z|<1\) onto a regular \(n\)-gon with center at \(0\), and \(f(0)=0\). Prove that \(f(\rho z)=\rho f(z)\) where \(\rho \) is any \(n\)-th root of \(1\). Remember that conformal maps preserve angles.