DEPARTMENT OF MATHEMATICS UNIVERSITY OF UTAH REAL AND COMPLEX ANALYSIS EXAM 0.5 August 13, 2007

Let \(f\) be integrable on \(\mathbb R\) with respect to Lebesgue measure, and define \[ F(x)=\int _{-\infty }^{x} f(t)\,dt \] Prove that \(F\) is continuous.

Assume that \(\mathbb R\) has the Lebesgue measure. Let \(f\in L^\infty (\mathbb R)\). For \(g\in L^2(\mathbb R)\) define \[ M(g)=fg \] Prove that \(M:L^2(\mathbb R)\to L^2(\mathbb R)\) is a bounded linear operator, and that \(\lVert M\rVert =\lVert f\rVert _\infty \).

Let \(H\) be a complex Hilbert space, and let \(T:H\to H\) be a bounded self adjoint operator (so \(T=T^*\)).

a. Prove that if \(\lambda \) is an eigenvalue of \(T\), then \(\lambda \) is real.

b. Show that eigenvectors corresponding to different eigenvalues are orthogonal.

c. Assume in addition that \(T\) is compact. Show that if \(\{\lambda _n\}\) is a sequence of distinct eigenvalues with \(\lambda _n\to \lambda \), then \(\lambda =0\).

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

Let \(I=[0,1]\) be the unit interval with the Lebesgue measure, and let \(I\times I\) be endowed with the product measure. Assume that \(K\in L^2(I\times I)\) and \(f\in L^2(I)\). Define \[ g(x)=\int _0^1 K(x,y)f(y)\,dy \] Show that \(g\in L^2(I)\).

Let \(f\) be an entire function such that \(|f(z)|\le K|z|^n\) where \(K\) is a positive real constant and \(n\) is a positive integer. Show that \(f\) is a polynomial of degree \(\le n\).

Let \(H=\{z\in \mathbb C:\operatorname {Im}z>0\}\) be the upper half plane and let \(\Delta =\{z\in \mathbb C:|z|<1\}\) be the unit disk.

a. Find a conformal map from \(H\) to \(\Delta \) that takes \(i\) to \(0\).

b. Let \(f:H\to H\) be a holomorphic map with \(f(i)=i\). Show that \(|f'(i)|\le 1\) and that if \(|f'(i)|=1\) then \(f\) is of the form \(f(z)=\frac {az+b}{cz+d}\) with \(a,b,c,d\in \mathbb R\).

Let \(f\) be an entire function such that for all \(x\in \mathbb R\), \(f(ix)\) and \(f(1+ix)\) are in \(\mathbb R\). Show that \(f\) is periodic with period \(2\). That is show that \(f(z)=f(z+2)\) for all \(z\in \mathbb C\).

Let \(\Omega \) be an open connected subset of \(\mathbb C\) and \(\mathcal F\) a family of holomorphic functions with domain \(\Omega \) such that for every compact subset \(K\) of \(\Omega \) there is a positive real constant \(M_K\) such that for all \(f\in \mathcal F\) and \(z\in K\) we have \(|f(z)|<M_K\). Show that for every \(z\in \Omega \) there exists a positive constant \(B_z\) such that \(|f'(z)|<B_z\) for all \(f\in \mathcal F\).

Calculate the integral \[ \int _{-\infty }^{\infty }\frac {\cos x}{x^2+4}\,dx. \]