372 research outputs found
Orthogonal polynomials and Riesz bases applied to the solution of Love's equation
In this paper we reinvestigate the structure of the solution of a well-known Love’s problem, related to the electrostatic field generated by two circular coaxial conducting disks, in terms of orthogonal polynomial expansions, enlightening the role of the recently introduced class of the Lucas–Lehmer polynomials. Moreover we show that the solution can be expanded more conveniently with respect to a Riesz basis obtained starting from Chebyshev polynomials
Moving and oblique observations of beams and plates
We study the observability of the one-dimensional Schr{\"o}dinger equation
and of the beam and plate equations by moving or oblique observations. Applying
different versions and adaptations of Ingham's theorem on nonharmonic Fourier
series, we obtain various observability and non-observability theorems. Several
open problems are also formulated at the end of the paper
Weighted frames of exponentials and stable recovery of multidimensional functions from nonuniform Fourier samples
In this paper, we consider the problem of recovering a compactly supported
multivariate function from a collection of pointwise samples of its Fourier
transform taken nonuniformly. We do this by using the concept of weighted
Fourier frames. A seminal result of Beurling shows that sample points give rise
to a classical Fourier frame provided they are relatively separated and of
sufficient density. However, this result does not allow for arbitrary
clustering of sample points, as is often the case in practice. Whilst keeping
the density condition sharp and dimension independent, our first result removes
the separation condition and shows that density alone suffices. However, this
result does not lead to estimates for the frame bounds. A known result of
Groechenig provides explicit estimates, but only subject to a density condition
that deteriorates linearly with dimension. In our second result we improve
these bounds by reducing the dimension dependence. In particular, we provide
explicit frame bounds which are dimensionless for functions having compact
support contained in a sphere. Next, we demonstrate how our two main results
give new insight into a reconstruction algorithm---based on the existing
generalized sampling framework---that allows for stable and quasi-optimal
reconstruction in any particular basis from a finite collection of samples.
Finally, we construct sufficiently dense sampling schemes that are often used
in practice---jittered, radial and spiral sampling schemes---and provide
several examples illustrating the effectiveness of our approach when tested on
these schemes
Stability of exponential bases on d- dimensional domains
We find explicit stability bounds for exponential Riesz bases on domains of
R^d. Our results generalize Kadec theorem and other stability theorems in the
literature.Comment: We have discovered an error in Theorem 1.
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