Department of Mathematics University of Utah\ 1.5 ] Written Qualifying Examination in APPLIED MATHEMATICS\ 0.5 ] September 23, 1997

Let \(x\) and \(y\) be vectors in an inner product space.

1. Verify that the choice \[ \alpha =\frac {\langle x,y\rangle }{\lVert y\rVert ^2} \] makes \(\lVert x-\alpha y\rVert ^2\) as small as possible.
2. Show that \(|\langle x,y\rangle |^2=\lVert x\rVert ^2\lVert y\rVert ^2\) if and only if \(x\) and \(y\) are linearly dependent.

Suppose \(A\in \mathbb {R}^{n\times m}\) and \(b\in \mathbb {R}^n\).

1. Show that the equation \[ A^*Ax=A^*b \] has a solution. Under what conditions on \(A\) is the solution unique?
2. Show that the eigenvalues of \(A^*A\) are non-negative and that its positive eigenvalues are the same as the positive eigenvalues of \(AA^*\).

Let \(H\) be a Hilbert space and \(L:H\to H\).

1. Find and prove the condition that ensures \[ Lu=\int _0^1 k(x,y)u(y)\,dy \] is a bounded linear operator in \(L^2[0,1]\).
2. Under what condition does the adjoint operator \(L^*\) exist? Assuming that this condition is satisfied, give \(L^*\).

Use a Green’s function to find the representation of a possible solution to \[ y''+\omega ^2y=f(x),\qquad y(0)=y'(1)=0 \] for arbitrary values of the parameter \(\omega \). For what values of \(\omega \) does the solution exist? If the solution does not exist find the least square solution. Explain the meaning of the least square solution.

Derive the boundary value problem for a minimizer \(w_0\) of the functional \[ \min _{\substack {w\in C^2[0,1]\\ w(0)=0}} \frac {\int _0^1\left (\phi (x)(w'')^2+w^2\right )\,dx} {\int _0^1\phi (x)(w')^2\,dx+\beta w(1)^2} \] where \(\phi (x)>0\), \(\forall x\in [0,1]\), \(\beta \) is a real parameter.

Evaluate the integral \[ I=\int _0^\infty \frac {x^2\cos x}{x^4+a^4}\,dx \] by contour integration. Use the Jordan lemma.

Define and find the Green’s function for the discrete operator \(Lu=-\Delta u-\lambda u\) defined on the space of doubly infinite sequences \(u=\{u_n\}\), \(-\infty <n<\infty \), with \(\sum _{n=-\infty }^{\infty }u_n^2<\infty \), and with \((\Delta u)_n=u_{n+1}-2u_n+u_{n-1}\).

Prove that the Hermite polynomials form a complete set on \(-\infty <x<\infty \) with the inner product \(\langle u,v\rangle =\int _{-\infty }^{\infty }u(x)v(x)e^{-x^2/2}\,dx\).

Hint: The following steps will prove useful:

1. Show that if the Hermite-Bessel coefficients of \(f(x)\) are all zero, then \(\int _{-\infty }^{\infty }e^{-x^2/2}x^nf(x)\,dx=0\) for all \(n\).
2. Show that the Fourier transform \(F(z)\) of the function \(f(x)e^{-x^2/2}\) has no singularities in the complex plane and is therefore an entire function.
3. Calculate the Taylor series of \(F(z)\) and show that it is zero. What conclusion can you draw about \(f(x)\) and what does this say about Hermite polynomials?