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2 votes
1 answer
112 views

On compactly supported functions with prescribed sparse coordinates

Let $\{\phi_n\}_{n=1}^{\infty}$ be an orthonormal basis for $L^2((0,1))$ consisting of Dirichlet eigenfunctions for the operator $-\partial^2_x + q(x)$ where $q \in C^{\infty}_c((0,1))$ is fixed. ...
0 votes
1 answer
122 views

Existence of an eigenpair for d-bar operator in the unit disck

Let $\overline{\partial}=\frac{1}{2}(\partial_{x}+\textrm{i} \,\partial_y)$ and let $D$ be the unit disc in the complex plane. For each $\lambda \in \mathbb C$, consider the problem: $$ \overline{\...
7 votes
1 answer
204 views

Are $\log(\sigma(A(z))$ subharmonic functions?

Let $A$ be a matrix-valued entire function. It is then well-known that $\log \Vert A(z)\Vert$ is subharmonic. In particular, the operator norm is just the largest singular value of $A$. Is it ...
4 votes
1 answer
390 views

Existence of periodic solution to ODE

We shall consider the matrix-valued differential operator $$(L u)(x) :=u'(x) - \begin{pmatrix} 0 & \sin(2\pi x-\frac{\pi}{6})\\ - 2\sin(2\pi x+\frac{\pi}{6}) & 0 \end{pmatrix} u(x).$$ This is ...
0 votes
1 answer
185 views

Meromorphic solutions to Legendre's equation

I just saw the following question that was asked yesterday on math overflow on meromorphic solutions to ODEs Although, I understand the answers and comments to the questions, I did not understand how ...
0 votes
1 answer
152 views

When can two Cauchy transforms intersect?

Given two polynomials $p$ and $q$ over reals and being guaranteed that both have all roots real I want to know if there is any characterization of the solutions of the equation $\frac{p'}{p} = \frac{q'...