DEPARTMENT OF MATHEMATICS UNIVERSITY OF UTAH REAL AND COMPLEX ANALYSIS EXAM January 2, 2008

Let \((X,\mu )\) be a measure space and \(f\ge 0\) an integrable function on \(X\). Prove that for \(\epsilon >0\), there exists \(\delta >0\) such that whenever \(\mu (A)<\delta \), then \(\int _A f<\epsilon \).

Let \(K\) be a continuous function on the square \([0,1]\times [0,1]\), and let \([0,1]\) be endowed with the Lebesgue measure. For \(f\in L^2([0,1])\) define \[ Tf(x)=\int _0^1 K(x,y)f(y)\,dy. \] Show that \(Tf\in L^2([0,1])\) and that \(T:L^2\to L^2\) is a bounded operator with \(\lVert T\rVert \le \lVert K\rVert _\infty \).

Let \(H\) be a Hilbert space, and let \(T:H\to H\) be a self adjoint linear operator, with finite dimensional range \(V=T(H)\). Prove that \(T\) is a compact operator, and that \(V=K^\perp \), where \(K\) is the kernel of \(T\).

Let \(f\in L^1([-\pi ,\pi ])\), and let \(c_n\) be the nth Fourier coefficient of \(f\): \[ c_n=\frac {1}{\sqrt {2\pi }}\int _{-\pi }^{\pi } f(x)e^{-inx}\,dx \quad n\in \mathbb Z \] Prove that \(c_n\to 0\) as \(|n|\to \infty \).

Let \(\ell ^\infty \) denote the Banach space of bounded real sequences with norm \(\lVert x\rVert _\infty =\sup _i |x_i|\), and \(\ell ^1\) the Banach space of real sequences with norm \(\lVert x\rVert _1=\sum _i |x_i|<\infty \). For \(x\in \ell ^\infty \) and \(y\in \ell ^1\) define \(Tx(y)=\sum _i x_i y_i\). Prove that \(T:\ell ^\infty \to (\ell ^1)^*\) is a bounded operator that is one to one and onto. (Here \((\ell ^1)^*\) is the dual space of \(\ell ^1\).)

Let \(Q\) be a square in \(\mathbb C\) and \(f:Q\to \mathbb C\) a continuous map that is holomorphic on the interior of \(Q\). Also assume that if \(z\in \partial Q\) then \(|f(z)|=1\). Show that \(f\) extends to a holomorphic map on a neighborhood of \(Q\).

Show that the function \[ f(z)=\frac {\cos z}{z^2} \] is the complex derivative of a holomorphic function \(F\) on \(\mathbb C\setminus \{0\}\). Write down a Laurent series for \(F\).

Let \(\Delta \) be the open unit disk and \(\gamma \) the unit circle in \(\mathbb C\) and \(\phi :\gamma \to \mathbb C\) a continuous function. Let \(g\) be a meromorphic function on \(\mathbb C\) that has a single simple pole at \(0\). Define a function \(f:\Delta \to \mathbb C\) by the formula \[ f(z)=\int _\gamma \phi (w)g(w-z)\,dw. \] Show that \(f\) is holomorphic.

Let \(f\) be a non-constant meromorphic function on \(\mathbb C\). Show that either there exists a sequence \(z_n\to \infty \) with \(f(z_n)\to 0\) as \(n\to \infty \) or there is a \(z\in \mathbb C\) such \(f(z)=0\).

Let \(\Omega \) be an open subset of \(\mathbb C\) and \(f:\Omega \to \mathbb C\) a holomorphic function. Let \(z\) be a point in \(\Omega \) and assume that \(\Omega \) contains a disk \(D\) centered at \(z\) of radius \(R\). Also assume that \(f(D)\) is contained in a disk \(D'\) centered at \(f(z)\) of radius \(r\). Show that \(|f'(z)|\le r/R\).