Department of Mathematics University of Utah Ph.D. Preliminary Examination in Real/Complex Analysis Fall 1993 INSTRUCTIONS

Let \(H=L^2(\mathbb R)\) be the Hilbert space with the inner product \(\int _{\mathbb R}f(x)\bar g(x)\,dx\), where \(dx\) is the Lebesgue measure. Let \(T:H\to M\) be the orthogonal projection of \(H\) onto a finite dimensional subspace \(M\) of \(H\).

Show that there exists a function \(h:\mathbb R^2\to \mathbb C\) such that \[ Tf(x)=\int _{\mathbb R}h(x,y)f(y)\,dy \] for all \(f\in L^2(\mathbb R)\).

(Hint: Let \(u_1,\ldots ,u_n\) be an orthonormal basis of \(M\). You may express \(h\) in terms of \(u_1,\ldots ,u_n\).)

Let \(X=[a,b]\subset \mathbb R\) be a closed interval. Assume \(a<b\). Regard \(X\) as a measure space with respect to the usual Lebesgue measure.

(a) Show directly that the space \(C(X)\) of continuous functions on \(X\) is not dense in \(L^\infty (X)\).

(b) Using (a), if necessary, show that there exists a bounded linear functional on \(L^\infty (X)\) which vanishes on \(C(X)\).

Let \(X\) be as in Problem 2. Give an example of a sequence of measurable, nonnegative functions \(f_n:X\to \mathbb R\), \(f_1\ge f_2\ge \cdots \ge f_n\ge \cdots \), converging pointwise to zero, such that \(\int _X f_n(x)\,dx\) does not converge to zero.

Can any of the integrals \(\int _X f_n(x)\,dx\) be finite? Justify your answer.

For \(f:\mathbb R\to \mathbb C\), and \(t\in \mathbb R\) define \[ f_t(x)=f(x-t). \] Suppose \(f\) is continuous with compact support. Prove directly that \(t\to f_t\) is a continuous map of \(\mathbb R\) into \(L^1(\mathbb R)\).

(a) Let \(D\) be an open connected set in \(\mathbb C\), and \(f\) a holomorphic function on \(D\). Show that if either the real or imaginary part of \(f\) is constant, then \(f\) is constant.

(b) Let \(f\) be an entire function. Show that if either the real or imaginary part of \(f\) is a bounded function then \(f\) is a constant.

Find a conformal transformation of the disc \(\Delta =\{z:|z|<1\}\) onto itself, which maps \(0\) to \(-\frac {1}{2}\).

Let \(D\) be an open connected set in \(\mathbb C\). Let \(\mathcal F\) be a collection of holomorphic functions defined on \(D\), which is uniformly bounded on each circle contained in \(D\). Show that \(\mathcal F\) forms a normal family.

Evaluate \[ \int _{\Gamma }\frac {e^z}{z^3}\,dz \] where \(\Gamma \) is a positively oriented simple unit circle \(\{z:|z|=1\}\).

Suppose \(f\) is an entire function and there exist \(M>0\), \(R>0\) and a number \(0<k<1\) such that \(|f(z)|\le M|z|^k\) for \(|z|\ge R\). Show \(f\) is a constant.