687 research outputs found
A general wavelet-based profile decomposition in the critical embedding of function spaces
We characterize the lack of compactness in the critical embedding of
functions spaces having similar scaling properties in the
following terms : a sequence bounded in has a subsequence
that can be expressed as a finite sum of translations and dilations of
functions such that the remainder converges to zero in as
the number of functions in the sum and tend to . Such a
decomposition was established by G\'erard for the embedding of the homogeneous
Sobolev space into the in dimensions with
, and then generalized by Jaffard to the case where is a Riesz
potential space, using wavelet expansions. In this paper, we revisit the
wavelet-based profile decomposition, in order to treat a larger range of
examples of critical embedding in a hopefully simplified way. In particular we
identify two generic properties on the spaces and that are of key use
in building the profile decomposition. These properties may then easily be
checked for typical choices of and satisfying critical embedding
properties. These includes Sobolev, Besov, Triebel-Lizorkin, Lorentz, H\"older
and BMO spaces.Comment: 24 page
Gabor frames and asymptotic behavior of Schwartz distributions
We obtain characterizations of asymptotic properties of Schwartz distribution by using Gabor frames. Our characterizations are indeed Tauberian theorems for shift asymptotics (S-asymptotics) in terms of short-time Fourier transforms with respect to windows generating Gabor frames. For it, we show that the Gabor coefficient operator provides (topological) isomorphisms of the spaces of tempered distributions and distributions of exponential type onto their images
Wavelet frame bijectivity on Lebesgue and Hardy spaces
We prove a sufficient condition for frame-type wavelet series in , the
Hardy space , and BMO. For example, functions in these spaces are shown to
have expansions in terms of the Mexican hat wavelet, thus giving a strong
answer to an old question of Meyer.
Bijectivity of the wavelet frame operator acting on Hardy space is
established with the help of new frequency-domain estimates on the
Calder\'on-Zygmund constants of the frame kernel.Comment: 23 pages, 7 figure
A Guide to Localized Frames and Applications to Galerkin-like Representations of Operators
This chapter offers a detailed survey on intrinsically localized frames and
the corresponding matrix representation of operators. We re-investigate the
properties of localized frames and the associated Banach spaces in full detail.
We investigate the representation of operators using localized frames in a
Galerkin-type scheme. We show how the boundedness and the invertibility of
matrices and operators are linked and give some sufficient and necessary
conditions for the boundedness of operators between the associated Banach
spaces.Comment: 32 page
Besov priors for Bayesian inverse problems
We consider the inverse problem of estimating a function from noisy,
possibly nonlinear, observations. We adopt a Bayesian approach to the problem.
This approach has a long history for inversion, dating back to 1970, and has,
over the last decade, gained importance as a practical tool. However most of
the existing theory has been developed for Gaussian prior measures. Recently
Lassas, Saksman and Siltanen (Inv. Prob. Imag. 2009) showed how to construct
Besov prior measures, based on wavelet expansions with random coefficients, and
used these prior measures to study linear inverse problems. In this paper we
build on this development of Besov priors to include the case of nonlinear
measurements. In doing so a key technical tool, established here, is a
Fernique-like theorem for Besov measures. This theorem enables us to identify
appropriate conditions on the forward solution operator which, when matched to
properties of the prior Besov measure, imply the well-definedness and
well-posedness of the posterior measure. We then consider the application of
these results to the inverse problem of finding the diffusion coefficient of an
elliptic partial differential equation, given noisy measurements of its
solution.Comment: 18 page
- …