Department of Mathematics University of Utah Ph.D. Written Qualifying Examination in Real/complex Analysis September 1992

Let \(f\in L^\infty (\mathbb R)\) define a bounded linear operator \(M_f\) from \(L^2(\mathbb R)\) to \(L^2(\mathbb R)\) by \(M_f(g)=fg\).

(a) Prove that \(\lVert M_f\rVert =\lVert f\rVert _\infty \).

(b) Prove that if \(M_f\) is one to one and onto then there exists \(\epsilon >0\) and a set \(E\) of measure zero such that \(|f(x)|\ge \epsilon \) for \(x\in \mathbb R-E\).

Let \((X,\mathcal M,\mu )\) be a measure space. Suppose \(\mu \) is finite and positive, and it satisfies the following property:

(*) for each \(A\in \mathcal M\), such that \(\mu (A)>0\), there exists \(B\in \mathcal M\) such that \(B\subset A\) and \(0<\mu (B)<\mu (A)\).

Prove that for each \(r>0\) there exists \(C\in \mathcal M\) such that \[ 0<\mu (C)<r. \]

Let \((X,\mathcal M,\mu )\) be a measure space, where the measure \(\mu \) is positive. Suppose \(f,f_n\in L^1(X,\mu )\), \(f_n\ge 0\) for each \(n\), \(f_n\to f\) a.e. and \[ \int _X f_n\,d\mu \to \int _X f\,d\mu \] Prove that \(f_n\to f\) in \(L^1(X,\mu )\) norm.

Hint: Show first that \((f-f_n)^+\le f\) for each \(n\) (recall that for real valued function \(g\), \(g^+=\max (g,0)\)).

Let \(X\) be a compact Hausdorff space.

(a) Prove that a complex measure \(\mu \) on \(X\) is positive if and only if \(\mu (X)=|\mu |(X)\).

(b) Let \(A\subset C(X)\) be a linear subspace containing the function \(1\). Let \(S\subset X\) be a compact subset with the property that for each \(f\in A\), \[ \sup \{|f(x)|:x\in X\}=\sup \{|f(x)|:x\in S\} \] Prove that for each \(x\in X\) there exists a positive measure \(\mu _x\) with support contained in \(S\) such that \[ f(x)=\int _S f\,d\mu _x\qquad \text {for }f\in A. \]

In \(L^2((0,2\pi ))\) consider the orthogonal projection onto the subspace spanned by \(\{e^{-it},1,e^{it}\}\). Express this projection as an integral operator (i.e. an operator of the form \(f\to Pf\), where \(Pf(y)=\int _0^{2\pi }f(x)h(x,y)\,dx\)).

Find \[ \int _{|z|=5}\tan (z)\,dz. \]

Show that every continuous function on \(\overline {\Delta }=\{z:|z|\le 1\}\) which is holomorphic on \(\Delta =\{z:|z|<1\}\) can be uniformly approximated on \(\overline {\Delta }\) by holomorphic polynomials.

Show that if \(f\) is a meromorphic function in \(\Delta (0;1+h)=\{z:|z|<1+h\}\), where \(h>0\), and \(|f(z)|=1\) for \(|z|=1\), then \(f\) is a rational function.

Suppose \(f\) is holomorphic in \(\{z:|z|<1\}\) and satisfies \(|f(\frac {1}{n})|\le (\frac {1}{n})^n\) for \(n=2,3,\ldots \). Can you identify \(f\)? Justify your answer.

Let \(H(D)\) be the space of holomorphic functions on an open connected set \(D\subset \mathbb C\). Suppose \(f_n\) is a sequence in \(H(D)\) converging to a function \(f\) uniformly on compact subsets. Suppose also that \(f\) is a nonconstant function. Let \(a\in D\), \(\alpha =f(a)\).

Show there is a sequence \(a_n\in D\) which converges to \(a\), such that \(f_n(a_n)=\alpha \) for each \(n\).