149 research outputs found
Slanted matrices, Banach frames, and sampling
In this paper we present a rare combination of abstract results on the
spectral properties of slanted matrices and some of their very specific
applications to frame theory and sampling problems. We show that for a large
class of slanted matrices boundedness below of the corresponding operator in
for some implies boundedness below in for all . We use
the established resultto enrich our understanding of Banach frames and obtain
new results for irregular sampling problems. We also present a version of a
non-commutative Wiener's lemma for slanted matrices
Iterative actions of normal operators
Let be a normal operator in a Hilbert space , and let
be a countable set of vectors. We investigate
the relations between , , and that makes the system of
iterations complete, Bessel, a
basis, or a frame for . The problem is motivated by the dynamical
sampling problem and is connected to several topics in functional analysis,
including, frame theory and spectral theory. It also has relations to topics in
applied harmonic analysis including, wavelet theory and time-frequency
analysis.Comment: 14 pages, 0 figure
A matrix pencil approach to the existence of compactly supported reconstruction functions in average sampling
The aim of this work is to solve a question raised for average sampling in shift-invariant spaces by using the well-known matrix pencil theory. In many common situations in sampling theory, the available data are samples of some convolution operator acting on the function itself: this leads to the problem of average sampling, also known as generalized sampling. In this paper we deal with the existence of a sampling formula involving these samples and having reconstruction functions with compact support. Thus, low computational complexity is involved and truncation errors are avoided. In practice, it is accomplished by means of a FIR filter bank. An answer is given in the light of the generalized sampling theory by using the oversampling technique: more samples than strictly necessary are used. The original problem reduces to finding a polynomial left inverse of a polynomial matrix intimately related to the sampling problem which, for a suitable choice of the sampling period, becomes a matrix pencil. This matrix pencil approach allows us to obtain a practical method for computing the compactly supported reconstruction functions for the important case where the oversampling rate is minimum. Moreover, the optimality of the obtained solution is established
Response of Autonomic Nervous System to Body Positions: Fourier and Wavelet Analysis
Two mathematical methods, the Fourier and wavelet transforms, were used to
study the short term cardiovascular control system. Time series, picked from
electrocardiogram and arterial blood pressure lasting 6 minutes, were analyzed
in supine position (SUP), during the first (HD1), and the second parts (HD2) of
head down tilt and during recovery (REC). The wavelet transform
was performed using the Haar function of period (,2,,6) to
obtain wavelet coefficients. Power spectra components were analyzed within
three bands, VLF (0.003-0.04), LF (0.04-0.15) and HF (0.15-0.4) with the
frequency unit cycle/interval. Wavelet transform demonstrated a higher
discrimination among all analyzed periods than the Fourier transform. For the
Fourier analysis, the LF of R-R intervals and VLF of systolic blood pressure
show more evident difference for different body positions. For the wavelet
analysis, the systolic blood pressures show much more evident difference than
the R-R intervals. This study suggests a difference in the response of the
vessels and the heart to different body positions. The partial dissociation
between VLF and LF results is a physiologically relevant finding of this work.Comment: RevTex,8 figure
Multipliers for p-Bessel sequences in Banach spaces
Multipliers have been recently introduced as operators for Bessel sequences
and frames in Hilbert spaces. These operators are defined by a fixed
multiplication pattern (the symbol) which is inserted between the analysis and
synthesis operators. In this paper, we will generalize the concept of Bessel
multipliers for p-Bessel and p-Riesz sequences in Banach spaces. It will be
shown that bounded symbols lead to bounded operators. Symbols converging to
zero induce compact operators. Furthermore, we will give sufficient conditions
for multipliers to be nuclear operators. Finally, we will show the continuous
dependency of the multipliers on their parameters.Comment: 17 page
Shift-Orthogonal Wavelet Bases
Shift-orthogonal wavelets are a new type of multiresolution wavelet bases that are orthogonal with respect to translation (or shifts) within one level, but not with respect to dilations across scales. In this paper, we characterize these wavelets and investigate their main properties by considering two general construction methods. In the first approach, we start by specifying the analysis and synthesis function spaces, and obtain the corresponding shift-orthogonal basis functions by suitable orthogonalization. In the second approach, we take the complementary view and start from the digital filterbank. We present several illustrative examples, including a hybrid version of the Battle-Lemarié spline wavelets. We also provide filterbank formulas for the fast wavelet algorithm. A shift-orthogonal wavelet transform is closely related to an orthogonal transform that uses the same primary scaling function; both transforms have essentially the same approximation properties. One experimentally confirmed benefit of relaxing the inter-scale orthogonality requirement is that we can design wavelets that decay faster than their orthogonal counterpart
How to obtain a lattice basis from a discrete projected space
International audienceEuclidean spaces of dimension n are characterized in discrete spaces by the choice of lattices. The goal of this paper is to provide a simple algorithm finding a lattice onto subspaces of lower dimensions onto which these discrete spaces are projected. This first obtained by depicting a tile in a space of dimension n -- 1 when starting from an hypercubic grid in dimension n. Iterating this process across dimensions gives the final result
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