42 research outputs found
Discrete Sampling and Interpolation: Universal Sampling Sets for Discrete Bandlimited Spaces
We study the problem of interpolating all values of a discrete signal f of
length N when d<N values are known, especially in the case when the Fourier
transform of the signal is zero outside some prescribed index set J; these
comprise the (generalized) bandlimited spaces B^J. The sampling pattern for f
is specified by an index set I, and is said to be a universal sampling set if
samples in the locations I can be used to interpolate signals from B^J for any
J. When N is a prime power we give several characterizations of universal
sampling sets, some structure theorems for such sets, an algorithm for their
construction, and a formula that counts them. There are also natural
applications to additive uncertainty principles.Comment: 24 pages, 5 figures, Accepted for publication in IEEE Transactions on
Information Theor
Best M\"obius approximations of convex and concave mappings
We study the best M\"obius approximations (BMA) to convex and concave
conformal mappings of the disk, including the special case of mappings onto
convex polygons. The crucial factor is the location of the poles of the BMAs.
Finer details are possible in the case of polygons through special properties
of Blaschke products and the prevertices of the mapping function.Comment: 11 pages, one figur
Convolution Idempotents with a given Zero-set
We investigate the structure of N-length discrete signals h satisfying h*h=h
that vanish on a given set of indices. We motivate this problem from examples
in sampling, Fuglede's conjecture, and orthogonal interpolation of bandlimited
signals. When N is a prime power, we characterize all such h with a prescribed
zero set in terms of digit expansions of nonzero indices in the inverse DFT of
h
On a theorem of Haimo regarding concave mappings
A relatively simple proof is given for Haimo’s theorem that a meromorphic function with suitably controlled Schwarzian derivative is a concave mapping. More easily verified conditions are found to imply Haimo’s criterion, which is now shown to be sharp. It is proved that Haimo’s functions map the unit disk onto the outside of an asymptotically conformal Jordan curve, thus ruling out the presence of corners
Discrete Sampling: A graph theoretic approach to Orthogonal Interpolation
We study the problem of finding unitary submatrices of the
discrete Fourier transform matrix, in the context of interpolating a discrete
bandlimited signal using an orthogonal basis. This problem is related to a
diverse set of questions on idempotents on and tiling
. In this work, we establish a graph-theoretic approach and
connections to the problem of finding maximum cliques. We identify the key
properties of these graphs that make the interpolation problem tractable when
is a prime power, and we identify the challenges in generalizing to
arbitrary . Finally, we investigate some connections between graph
properties and the spectral-tile direction of the Fuglede conjecture.Comment: Submitted to IEEE Transactions on Information Theor
Fast DFT Computation for Signals with Structured Support
Suppose an length signal has known frequency support of size . Given
sample access to this signal, how fast can we compute the DFT? The answer to
this question depends on the structure of the frequency support.
We first identify some frequency supports for which (an ideal)
complexity is achievable, referred to as homogeneous sets. We give a
generalization of radix-2 that enables computation of signals with
homogeneous frequency support. Using homogeneous sets as building blocks, we
construct more complicated support structures for which the complexity of
is achievable. We also investigate the relationship of DFT
computation with additive structure in the support and provide partial
converses.Comment: 45 pages, 16figure