Affine synthesis onto LpL^p when 0<p≤10 < p \leq 1

The affine synthesis operator is shown to map the coefficient space $\ell^p$ surjectively onto $L^p$, for $0 < p \leq 1$. Here the synthesizer need satisfy only mild restrictions, for example having nonzero integral or else periodization that is real-valued, nontrivial and bounded below. Consequences include an affine atomic decomposition of $L^p$. Tools include an analysis operator that acts nonlinearly, in contrast to the usual linear analysis operator for $p>1$.Comment: 29 page

The Robin Laplacian - spectral conjectures, rectangular theorems

The first two eigenvalues of the Robin Laplacian are investigated along with their gap and ratio. Conjectures by various authors for arbitrary domains are supported here by new results for rectangular boxes. Results for rectangular domains include that: the square minimizes the first eigenvalue among rectangles under area normalization, when the Robin parameter $\alpha \in \mathbb{R}$ is scaled by perimeter; that the square maximizes the second eigenvalue for a sharp range of $\alpha$-values; that the line segment minimizes the Robin spectral gap under diameter normalization for each $\alpha \in \mathbb{R}$; and the square maximizes the spectral ratio among rectangles when $\alpha>0$. Further, the spectral gap of each rectangle is shown to be an increasing function of the Robin parameter, and the second eigenvalue is concave with respect to $\alpha$. Lastly, the shape of a Robin rectangle can be heard from just its first two frequencies, except in the Neumann case.Comment: 44 pages, 7 figure

Dirichlet eigenvalue sums on triangles are minimal for equilaterals

Among all triangles of given diameter, the equilateral triangle is shown to minimize the sum of the first $n$ eigenvalues of the Dirichlet Laplacian, for each $n \geq 1$. In addition, the first, second and third eigenvalues are each proved to be minimal for the equilateral triangle. The disk is conjectured to be the minimizer among general domains

Minimizing Neumann fundamental tones of triangles: an optimal Poincare inequality

The first nonzero eigenvalue of the Neumann Laplacian is shown to be minimal for the degenerate acute isosceles triangle, among all triangles of given diameter. Hence an optimal Poincar\'{e} inequality for triangles is derived. The proof relies on symmetry of the Neumann fundamental mode for isosceles triangles with aperture less than $\pi/3$. Antisymmetry is proved for apertures greater than $\pi/3$

From Neumann to Steklov and beyond, via Robin: the Weinberger way

The second eigenvalue of the Robin Laplacian is shown to be maximal for the ball among domains of fixed volume, for negative values of the Robin parameter $\alpha$ in the regime connecting the first nontrivial Neumann and Steklov eigenvalues, and even somewhat beyond the Steklov regime. The result is close to optimal, since the ball is not maximal when $\alpha$ is sufficiently large negative, and the problem admits no maximiser when $\alpha$ is positive

Sharp spectral bounds on starlike domains

We prove sharp bounds on eigenvalues of the Laplacian that complement the Faber--Krahn and Luttinger inequalities. In particular, we prove that the ball maximizes the first eigenvalue and minimizes the spectral zeta function and heat trace. The normalization on the domain incorporates volume and a computable geometric factor that measures the deviation of the domain from roundness, in terms of moment of inertia and a support functional introduced by P\'{o}lya and Szeg\H{o}. Additional functionals handled by our method include finite sums and products of eigenvalues. The results hold on convex and starlike domains, and for Dirichlet, Neumann or Robin boundary conditions

Shifted lattices and asymptotically optimal ellipses

Translate the positive-integer lattice points in the first quadrant by some amount in the horizontal and vertical directions. Take a decreasing concave (or convex) curve in the first quadrant and construct a family of curves by rescaling in the coordinate directions while preserving area. Consider the curve in the family that encloses the greatest number of the shifted lattice points: we seek to identify the limiting shape of this maximizing curve as the area is scaled up towards infinity. The limiting shape is shown to depend explicitly on the lattice shift. The result holds for all positive shifts, and for negative shifts satisfying a certain condition. When the shift becomes too negative, the optimal curve no longer converges to a limiting shape, and instead we show it degenerates. Our results handle the $p$-circle $x^p+y^p=1$ when $p>1$ (concave) and also when $0<p<1$ (convex). Rescaling the $p$-circle generates the family of $p$-ellipses, and so in particular we identify the asymptotically optimal $p$-ellipses associated with shifted integer lattices. The circular case $p=2$ with shift $-1/2$ corresponds to minimizing high eigenvalues in a symmetry class for the Laplacian on rectangles, while the straight line case ($p=1$) generates an open problem about minimizing high eigenvalues of quantum harmonic oscillators with normalized parabolic potentials

From Steklov to Neumann and beyond, via Robin: the Szeg\H{o} way

The second eigenvalue of the Robin Laplacian is shown to be maximal for the disk among simply-connected planar domains of fixed area when the Robin parameter is scaled by perimeter in the form $\alpha/L(\Omega)$, and $\alpha$ lies between $-2\pi$ and $2\pi$. Corollaries include Szeg\H{o}'s sharp upper bound on the second eigenvalue of the Neumann Laplacian under area normalization, and Weinstock's inequality for the first nonzero Steklov eigenvalue for simply-connected domains of given perimeter. The first Robin eigenvalue is maximal, under the same conditions, for the degenerate rectangle. When area normalization on the domain is changed to conformal mapping normalization and the Robin parameter is positive, the maximiser of the first eigenvalue changes back to the disk.Comment: 21 pages, 2 figures; strengthened Bossel-type conjecture for first eigenvalue added to Section 1; to appear in the Canadian Journal of Mathematic

Approximately dual frame pairs in Hilbert spaces and applications to Gabor frames

We discuss the concepts of pseudo-dual frames and approximately dual frames, and illuminate their relationship to classical frames. Approximately dual frames are easier to construct than the classical dual frames, and might be tailored to yield almost perfect reconstruction. For approximately dual frames constructed via perturbation theory, we provide a bound on the deviation from perfect reconstruction. An alternative bound is derived for the rich class of Gabor frames, by using the Walnut representation of the frame operator to estimate the deviation from equality in the duality conditions. As illustration of the results, we construct explicit approximate duals of Gabor frames generated by the Gaussian; these approximate duals yield almost perfect reconstruction. Amazingly, the method applies also to certain Gabor frames that are far from being tight.Comment: 23 pages, 5 figure

Affine synthesis and coefficient norms for Lebesgue, Hardy and Sobolev spaces

The affine synthesis operator is shown to map the mixed-norm sequence space $\ell^1(\ell^p)$ surjectively onto L^p(\Rd), 1 \leq p < \infty, assuming the Fourier transform of the synthesizer does not vanish at the origin and the synthesizer has some decay near infinity. Hence the standard norm on f \in L^p(\Rd) is equivalent to the minimal coefficient norm of realizations of $f$ in terms of the affine system. We further show the synthesis operator maps a discrete Hardy space onto H^1(\Rd), which yields a norm equivalence for Hardy space involving convolution with a discrete Riesz kernel sequence. Coefficient norm equivalences are established also for Sobolev spaces, by applying difference operators to the coefficient sequences.Comment: Added references, and improved several proofs. Also added Appendix C, which connects the paper to Banach frame theor