DEPARTMENT OF MATHEMATICS University of Utah Ph.D. PRELIMINARY EXAMINATION IN ANALYSIS August 2009

Let \(\{f_n\}\), \(f\) be measurable functions on a measure space \((X,\mu )\) such that \[ f_1\ge f_2\ge \cdots \ge f\ge 0 \] Suppose that \(\int f_n\,d\mu \to \int f\,d\mu \) and that \(f_1\) is integrable. Prove that \(f_n\to f\) almost everywhere. Show by example, that this conclusion may be false if \(f_1\) is not integrable.

Recall that the Fourier transform of a function \(f\in L^1(\mathbb R)\) (with respect to Lebesgue measure) is given by \[ \hat f(x)=\int _{-\infty }^{\infty } e^{-2\pi ixy}f(y)\,dy \] Show that \(\hat f\) is a bounded, continuous function on \(\mathbb R\).

Let \(H\) be a Hilbert Space, and suppose that \(\{x_n\}\) is a sequence in \(H\) such that \(\langle x_n,y\rangle \) is convergent for each \(y\in H\). Prove that there exists \(x\in H\) such that \(\langle x_n,y\rangle \to \langle x,y\rangle \) for all \(y\in H\).

Let \((X,\mu )\) be a finite measure space, and let \(f\in L^\infty (X)\). For \(g\in L^1(X)\), define \[ Tg=\int fg\,d\mu \] Show that \(T\) is a bounded linear functional on \(L^1(X)\), and that \(\lVert T\rVert =\lVert f\rVert _\infty \).

Either prove the following statement if true, or prove that it is false by providing a counterexample. Let \(\{f_n\}\) be a sequence in \(L^2([0,1])\) such that \(\lVert f_n\rVert _2\to 0\). Then \(f_n\to 0\) almost everywhere.

Prove Schwarz’s lemma: If \(f:\Delta \to \Delta \) is a holomorphic function from the open unit disk \(\Delta \) to itself with \(f(0)=0\) show that \(|f(z)|\le |z|\), \(|f'(0)|\le 1\) and if \(|f(z)|=|z|\) for some \(z\ne 0\) or \(|f'(0)|=1\) then \(f(z)=\lambda z\) with \(|\lambda |=1\).

(Hint: Apply the maximum principle to the function \(f(z)/z\).)

Calculate \[ \int _\gamma (1+z^2)e^{1/z}\,dz \] where \(\gamma \) is the unit circle traversed in the counter-clockwise direction.

Let \(f:\mathbb C\to \mathbb C\) be a function such that:

(a) \(f\) is continuous;

(b) \(f\) is a holomorphic at all points \(z\) not on the real axis;

(c) if \(z\) is on the real axis then \(f(z)\) is real.

Show that \(f(\bar z)=\overline {f(z)}\) and that \(f\) is holomorphic on all of \(\mathbb C\).

Let \(f\) be a function holomorphic for \(z\) in the annulus \(r<z<R\). Show that if for some \(\rho \) with \(r<\rho <R\), \[ \int _{|z|=\rho } z^n f(z)\,dz=0 \] for all integers \(n\) with \(n\ge 0\) then \(f\) extends to a holomorphic function on the disk \(|z|<R\).

Show that all zeroes of \(z^4-6z-3\) lie inside the circle \(|z|=2\).