PhD Preliminary Qualifying Examination Applied Mathematics Aug 19 2014

Let \(T:L^1[0,1]\to L^2[0,1]\) be a bounded linear operator. Let \(a\in L^\infty [0,1]\) and \(g\in L^1[0,1]\).

1. Prove that there exists a constant \(C>0\) such that if \(\lVert a\rVert _{L^\infty }<C\) then the equation \[ f+a(Tf)^2=g \tag {1} \] has a unique solution \(f\) in the set \[ B=\{u\in L^1[0,1]:\lVert u-g\rVert _{L^1}\le 1\}. \] Note: the constant \(C\) may depend on \(T\) and/or \(g\).
2. Suggest an iterative procedure for approximating the unique solution \(f\) to equation (1).

Let \(X\) be a separable Hilbert space and \((e_n)\) be a total orthonormal sequence of \(X\). Let \(T:X\to X\) be a bounded linear operator satisfying \[ \sum _{n=1}^{\infty }\lVert Te_n\rVert ^2<\infty . \tag {2} \] Define the operators \(T_k:X\to X\), \(k=1,2,\ldots \), that applied to an arbitrary \[ x=\sum _{n=1}^{\infty }x_ne_n\in X \qquad \text {give}\qquad T_kx=\sum _{n=1}^k x_nTe_n. \]

1. Are the operators \(T_k\) compact? Explain why or why not.
2. Using equation (2), show that the sequence \((T_k)\) converges uniformly to \(T\), i.e. show that \(\lVert T-T_k\rVert \to 0\) as \(k\to \infty \).
3. Is \(T\) compact? Explain why or why not.

Assume a compact self-adjoint operator \(T:X\to X\), where \(X\) is a separable Hilbert space, satisfies \(\langle Tx,x\rangle \ge 0\) for all \(x\in X\).

1. State the spectral theorem for compact self-adjoint operators.
2. Show that all the eigenvalues of \(T\) are non-negative.
3. Construct an operator \(S:X\to X\) such that \(T=S^2\).

Let \(\ell ^{1'}\) be the dual space of \(\ell ^1\). The objective here is to show that \(\ell ^{1'}\) is isomorphic to \(\ell ^\infty \). You may use that a Schauder basis for \(\ell ^1\) (and \(\ell ^\infty \)) is \((e_k)\), where \(e_k=(\delta _{kj})\), i.e. the sequence with zeroes everywhere except for a \(1\) in the \(k\)–th position.

1. Show that any \(f\in \ell ^{1'}\) corresponds to a sequence \(g=(\gamma _k)\in \ell ^\infty \).
2. Show that any sequence \(g=(\gamma _k)\in \ell ^\infty \) corresponds to some \(f\in \ell ^{1'}\).
3. Parts (a) and (b) define a bijective linear mapping \(G:\ell ^{1'}\to \ell ^\infty \). Prove that \(G\) is an isomorphism by showing \(\lVert f\rVert _{\ell ^{1'}}=\lVert G(f)\rVert _{\ell ^\infty }\) for any \(f\in \ell ^{1'}\).

Let \(K:L^2[0,1]\to L^2[0,1]\) be the linear operator defined by \[ (Kx)(t)=\int _0^1 k(t,\tau )x(\tau )\,d\tau \] where \(k:[0,1]\times [0,1]\to \mathbb {R}\) is continuous.

1. Show that \(K\) is bounded and give an upper bound for \(\lVert K\rVert \) (in terms of \(k\)).
2. Show that there is a constant \(C\) such that if \(\lvert \mu \rvert <C\), the operator \(I+\mu K\) is invertible.
3. Assuming \(\lvert \mu \rvert <C\), where \(C\) is as in part (b), express \((I+\mu K)^{-1}\) as a series. Specify whether the series converges and in what sense.

Find the radius of convergence of the Taylor series with the center at \(x_0=1\) for the following function

1. \[ f(x)=x^{1/3} \]
2. \[ f(z)=\frac {(\sin x+2)^2}{(\sin x-2)^2} \]

(\(x\) is a real variable; you do not need to find the series themselves).

1. Show that any analytic function (not identically equal to zero) can have only isolated zeros inside its analyticity domain.
2. Prove the Uniqueness Theorem: If two functions are analytic in a domain \(D\) and equal on some set of points that has a limiting point inside \(D\), then these functions are identically equal in \(D\).

Evaluate the integrals:

1. \[ I=\int _0^\infty \frac {\sin x}{x}\,dx, \]
2. \[ I=\int _0^\infty \frac {x^\alpha }{2+x}\,dx \qquad (-1<\alpha <0). \]

1. Calculate the Fourier transform of a Gaussian \(f(x)=e^{-x^2}\).
2. Formulate and prove Heisenberg’s uncertainty principle.

Consider integral \[ I(s)=\int _C \frac {e^{s(z^2-1)}}{z-1}\,dz \] with large parameter \(s\to +\infty \); \(C\) is the vertical line \(\operatorname {Im}z=3\), from \(z=3-i\infty \) to \(z=3+i\infty \).

1. What is the saddle point \(z_0\)?

   What is the path \(C_0\) of steepest descent from \(z_0\)?

   What is the path of steepest ascent from \(z_0\)?

2. Find the three-term asymtotic expansion of the integral.