Department of Mathematics University of Utah Written Qualifying Exam in Real and Complex Analysis Fall 1995 Instructions

Let \(\mu \) be a positive finite measure on \((X,\mathfrak M)\) and \(g\in L^1(\mu )\). Suppose there exists \(0<M<\infty \) such that \[ \left |\int gs\,d\mu \right |\le M\lVert s\rVert _p\qquad (1\le p<\infty ) \] for all simple measurable functions on \(X\). Then show that \(g\in L^q(\mu )\) where \(\frac {1}{p}+\frac {1}{q}=1\).

Let \((X,\mathfrak M,\mu )\) be a positive measure space such that \(\mu (X)=1\). If \(V\) is a closed subspace of \(L^2(\mu )\) which is contained in \(L^\infty (\mu )\) then show that \(\dim (V)<\infty \) as follows:

(1) Show that there exists \(C>0\) such that \(\lVert f\rVert _\infty \le C\lVert f\rVert _2\) for all \(f\in V\).

(2) Let \(\{f_1,\ldots ,f_n\}\) be any orthonormal set in \(V\) and show that for all \(c_1,\ldots ,c_n\in \mathbb C\), and almost all \(x\in X\), \[ \left |\sum _{i=1}^{n}c_i f_i(x)\right | \le C\left (\sum _{i=1}^{n}|c_i|^2\right )^{1/2}. \]

(3) Choose \(c_i=\overline {f_i(x)}\) in (2) and deduce that \[ \sum _{i=1}^{n}|f_i(x)|^2\le C^2\qquad \text {(for almost all }x\in X\text {)}. \]

(4) Integrate the inequality in (3).

Let \(B\) be a Banach space and \(A\subset B\) a closed subspace. If \(x\in B\) but \(x\notin A\), prove that there is a bounded linear functional \(f\) on \(B\) such that \(f(x)=1\), \(f(y)=0\) for every \(y\in A\) and \(\lVert f\rVert =(\inf \{\lVert x+y\rVert :y\in A\})^{-1}\).

Recall the \(k\)th Fourier coefficient of \(f\in L^1([-\pi ,\pi ])\) is \[ \hat f(k)=\frac {1}{2\pi }\int _{-\pi }^{\pi }f(t)e^{-ikt}\,dt. \]

(1) If \(f(t)=t\) for \(-\pi \le t<\pi \), show that \(\hat f(k)=(-1)^{k+1}\frac {1}{ik}\) for \(k\ne 0\) and \(\hat f(0)=0\);

(2) Show that \(\sum _{k=1}^{\infty }k^{-2}=\frac {\pi ^2}{6}\).

If \(f\) is holomorphic in \(\Delta (0;1)=\{z\in \mathbb C:|z|<1\}\) and \(r<1\) prove that there exists a constant \(M\) so that \[ |f(z)-f(w)|\le M|z-w| \] for all \(z,w\in \Delta (0;r)=\{z\in \mathbb C:|z|<r\}\).

Suppose \(P(z)=a_nz^n+\cdots +a_0\). If \(M(\rho )=\sup _\theta |P(\rho e^{i\theta })|\) then show that \[ \frac {M(r)}{r^n}\ge \frac {M(R)}{R^n} \] if \(0<r<R\).

Let \(f\) be entire, \(|f(t)|=2\) for all \(t\in \mathbb R\) and \(f(i)=1\). Then find \(f(-i)\).

Find \[ \int _{\gamma _R}\frac {dz}{e^z-1} \] where \(\gamma _R\) is the curve \(|z|=R\) traversed counterclockwise and \(R\) is a positive number such that \(\gamma _R\) does not pass through a zero of \(e^z-1\).

How many zeroes (counting multiplicity) does the function \(f(z)=z^4+6z^3+2z^2-z+1\) have inside the region \(|z|<1\).