1,455 research outputs found
Affine synthesis onto when
The affine synthesis operator is shown to map the coefficient space
surjectively onto , for . 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 .
Tools include an analysis operator that acts nonlinearly, in contrast to the
usual linear analysis operator for .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
is scaled by perimeter; that the square maximizes the
second eigenvalue for a sharp range of -values; that the line segment
minimizes the Robin spectral gap under diameter normalization for each ; and the square maximizes the spectral ratio among rectangles
when . 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 .
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 eigenvalues of the Dirichlet Laplacian, for
each . 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 . Antisymmetry is proved for apertures
greater than
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
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 is sufficiently large
negative, and the problem admits no maximiser when 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 -circle when (concave) and also
when (convex). Rescaling the -circle generates the family of
-ellipses, and so in particular we identify the asymptotically optimal
-ellipses associated with shifted integer lattices.
The circular case with shift corresponds to minimizing high
eigenvalues in a symmetry class for the Laplacian on rectangles, while the
straight line case () 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 , and
lies between and . 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
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
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
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