DEPARTMENT OF MATHEMATICS UNIVERSITY OF UTAH REAL AND COMPLEX PRELIMINARY EXAM, 2003

Prove that a non-decreasing function from \(\mathbb R\) to \(\mathbb R\) is measurable with respect to the Borel \(\sigma \)-algebra.

Let \(\mu \) be a positive measure on a measure space \((X,\Sigma )\) and let \(f\) be a complex valued function on \(X\) which is integrable with respect to \(\mu \). For each \(n\), set \[ E_n=\{x\in X:1/n\le |f(x)|\le n\}. \] Then each \(E_n\) is a measurable set on which \(|f|\) is bounded. Prove that

(a) \(\mu (E_n)\) is finite for each \(n\); and

(b) \(\displaystyle \lim _{n\to \infty }\int _{E_n} f(x)\,d\mu (x) =\int _X f(x)\,d\mu (x)\).

Prove that if \(X\) is a Banach space with a closed unit ball which is compact, then \(X\) must be finite dimensional. Hint: consider the family of sets of the form \(\{x\in X:\|x\|=1,\text { and }f(x)=0\}\) where \(f\) is a bounded nonzero linear functional on \(X\).

Consider \(L^1(T)\) to be the space of periodic functions on \(\mathbb R\) of period \(2\pi \) which are integrable on \([0,2\pi ]\). Let \(M\) be a proper closed translation invariant subspace of \(L^1(T)\).

(a) Prove that there exists a non-zero \(h\in L^\infty (T)\) such that \(f*h(x)=0\) for all \(f\in M\), \[ \text {where } f*h(x)=\frac {1}{2\pi }\int _0^{2\pi } f(x-y)h(y)\,dy; \]

(b) use part (a) to prove there must be an \(n\in \mathbb Z\) such that \(\hat f(n)=0\) for every \(f\in M\), where \(\hat f(n)\) is the \(n\) th Fourier coefficient of \(f\).

Let \(A\) be a closed subspace of \(C([0,1])\) which is also closed in \(L^2([0,1])\). Prove that the orthogonal projection \(P\) which projects \(L^2([0,1])\) onto \(A\) is continuous as a linear transformation from \(L^2([0,1])\) into \(C([0,1])\). Note that \(C([0,1])\) has the topology determined by the norm \(\|\cdot \|_\infty \), while \(L^2([0,1])\) has the topology determined by the norm \(\|\cdot \|_2\).

Let \(f\) be holomorphic on the unit disc \(U\). Let \(h(z,w)\) be the function on \(U\times U\) defined by \[ h(z,w)= \begin {cases} \dfrac {f(z)-f(w)}{z-w} & \text {if } z\ne w,\\[0.6ex] f'(z) & \text {if } z=w \end {cases} \] Prove that \(f\) is continuous as a function of two variables on \(U\times U\). You need only check the continuity at points of the diagonal. At other points it is obvious.

Let \(\Omega \) be an unbounded domain in the plane with the property that there exists a sequence \(\{h_n\}\) of functions holomorphic on a neighborhood of \(\overline {\Omega }\) such that each \(h_n\) vanishes at infinity on \(\Omega \), \(|h_n|\le 1\) on \(\overline {\Omega }\) for each \(n\), and \(\{h_n\}\) converges uniformly to \(1\) on each compact subset of \(\overline {\Omega }\). Prove that if \(f\) is a function which is bounded and holomorphic in a neighborhood of \(\overline {\Omega }\) and if \(|f|\le 1\) on \(\partial \Omega \), then \(|f|\le 1\) on all of \(\overline {\Omega }\).

Suppose \(f\) is continuous on \(U=\{z\in \mathbb C: |z|<1\}\) and holomorphic on the complement in \(U\) of the interval \((-1,1)\). Prove that \(f\) is actually holomorphic on \(U\).

Determine how many zeroes, counting multiplicity, the polynomial \[ p(x)=z^5-6z^4+z^3+2z-1 \] has in the unit disc \(U=\{z\in \mathbb C: |z|<1\}\). Prove that your answer is correct.

Suppose \(a\) is a real number greater than \(1\). Compute \[ \int _0^{2\pi }\frac {d\theta }{a+\sin \theta } \] using residues. Hint: \(\sin \theta =\frac {z-z^{-1}}{2i}\) where \(z=e^{i\theta }\).