94,073 research outputs found
Multiscale Computation with Interpolating Wavelets
Multiresolution analyses based upon interpolets, interpolating scaling
functions introduced by Deslauriers and Dubuc, are particularly well-suited to
physical applications because they allow exact recovery of the multiresolution
representation of a function from its sample values on a finite set of points
in space. We present a detailed study of the application of wavelet concepts to
physical problems expressed in such bases. The manuscript describes algorithms
for the associated transforms which, for properly constructed grids of variable
resolution, compute correctly without having to introduce extra grid points. We
demonstrate that for the application of local homogeneous operators in such
bases, the non-standard multiply of Beylkin, Coifman and Rokhlin also proceeds
exactly for inhomogeneous grids of appropriate form. To obtain less stringent
conditions on the grids, we generalize the non-standard multiply so that
communication may proceed between non-adjacent levels. The manuscript concludes
with timing comparisons against naive algorithms and an illustration of the
scale-independence of the convergence rate of the conjugate gradient solution
of Poisson's equation using a simple preconditioning, suggesting that this
approach leads to an O(n) solution of this equation.Comment: 33 pages, figures available at
http://laisla.mit.edu/muchomas/Papers/nonstand-figs.ps . Updated: (1) figures
file (figs.ps) now appear with the posting on the server; (2) references got
lost in the last submissio
Holographic Calculation for Large Interval R\'enyi Entropy at High Temperature
In this paper, we study the holographic R\'enyi entropy of a large interval
on a circle at high temperature for the two-dimensional conformal field theory
(CFT) dual to pure AdS gravity. In the field theory, the R\'enyi entropy is
encoded in the CFT partition function on -sheeted torus connected with each
other by a large branch cut. As proposed by Chen and Wu [Large interval limit
of R\'enyi entropy at high temperature, arXiv:1412.0763], the effective way to
read the entropy in the large interval limit is to insert a complete set of
state bases of the twist sector at the branch cut. Then the calculation
transforms into an expansion of four-point functions in the twist sector with
respect to . By using the operator product expansion of
the twist operators at the branch points, we read the first few terms of the
R\'enyi entropy, including the leading and next-to-leading contributions in the
large central charge limit. Moreover, we show that the leading contribution is
actually captured by the twist vacuum module. In this case by the Ward identity
the four-point functions can be derived from the correlation function of four
twist operators, which is related to double interval entanglement entropy.
Holographically, we apply the recipe in [T. Faulkner, The entanglement R\'enyi
entropies of disjoint intervals in AdS/CFT, arXiv:1303.7221] and [T. Barrella
et al., Holographic entanglement beyond classical gravity, J. High Energy Phys.
09 (2013) 109] to compute the classical R\'enyi entropy and its one-loop
quantum correction, after imposing a new set of monodromy conditions. The
holographic classical result matches exactly with the leading contribution in
the field theory up to and , while the holographical
one-loop contribution is in exact agreement with next-to-leading results in
field theory up to and as well.Comment: minor corrections, match with the published versio
Uncertainty principles for orthonormal bases
In this survey, we present various forms of the uncertainty principle (Hardy,
Heisenberg, Benedicks). We further give a new interpretation of the uncertainty
principles as a statement about the time-frequency localization of elements of
an orthonormal basis, which improves previous unpublished results of H.
Shapiro. Finally, we show that Benedicks' result implies that solutions of the
Shr\"{o}dinger equation have some (appearently unnoticed) energy dissipation
property
Discrete Hilbert transforms on sparse sequences
Weighted discrete Hilbert transforms from to a weighted space are studied, with
a sequence of distinct points in the complex plane and
a corresponding sequence of positive numbers. In the special case
when grows at least exponentially, bounded transforms of this kind
are described in terms of a simple relative to the Muckenhoupt
condition. The special case when is restricted to another sequence
is studied in detail; it is shown that a bounded transform satisfying
a certain admissibility condition can be split into finitely many surjective
transforms, and precise geometric conditions are found for invertibility of
such two weight transforms. These results can be interpreted as statements
about systems of reproducing kernels in certain Hilbert spaces of which de
Branges spaces and model subspaces of are prime examples. In particular,
a connection to the Feichtinger conjecture is pointed out. Descriptions of
Carleson measures and Riesz bases of normalized reproducing kernels for certain
"small" de Branges spaces follow from the results of this paper
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