729 research outputs found
Schur functions and their realizations in the slice hyperholomorphic setting
we start the study of Schur analysis in the quaternionic setting using the
theory of slice hyperholomorphic functions. The novelty of our approach is that
slice hyperholomorphic functions allows to write realizations in terms of a
suitable resolvent, the so called S-resolvent operator and to extend several
results that hold in the complex case to the quaternionic case. We discuss
reproducing kernels, positive definite functions in this setting and we show
how they can be obtained in our setting using the extension operator and the
slice regular product. We define Schur multipliers, and find their co-isometric
realization in terms of the associated de Branges-Rovnyak space
Pontryagin de Branges Rovnyak spaces of slice hyperholomorphic functions
We study reproducing kernel Hilbert and Pontryagin spaces of slice
hyperholomorphic functions which are analogs of the Hilbert spaces of analytic
functions introduced by de Branges and Rovnyak. In the first part of the paper
we focus on the case of Hilbert spaces, and introduce in particular a version
of the Hardy space. Then we define Blaschke factors and Blaschke products and
we consider an interpolation problem. In the second part of the paper we turn
to the case of Pontryagin spaces. We first prove some results from the theory
of Pontryagin spaces in the quaternionic setting and, in particular, a theorem
of Shmulyan on densely defined contractive linear relations. We then study
realizations of generalized Schur functions and of generalized Carath'eodory
functions
Vector splines on the sphere with application to the estimation of vorticity and divergence from discrete, noisy data
Vector smoothing splines on the sphere are defined. Theoretical properties are briefly alluded to. The appropriate Hilbert space norms used in a specific meteorological application are described and justified via a duality theorem. Numerical procedures for computing the splines as well as the cross validation estimate of two smoothing parameters are given. A Monte Carlo study is described which suggests the accuracy with which upper air vorticity and divergence can be estimated using measured wind vectors from the North American radiosonde network
Realizations of slice hyperholomorphic generalized contractive and positive functions
We introduce generalized Schur functions and generalized positive functions
in setting of slice hyperholomorphic functions and study their realizations in
terms of associated reproducing kernel Pontryagin spacesComment: Revised version,to appear in the Milan Journal of Mathematic
Sampling Theorem and Discrete Fourier Transform on the Riemann Sphere
Using coherent-state techniques, we prove a sampling theorem for Majorana's
(holomorphic) functions on the Riemann sphere and we provide an exact
reconstruction formula as a convolution product of samples and a given
reconstruction kernel (a sinc-type function). We also discuss the effect of
over- and under-sampling. Sample points are roots of unity, a fact which allows
explicit inversion formulas for resolution and overlapping kernel operators
through the theory of Circulant Matrices and Rectangular Fourier Matrices. The
case of band-limited functions on the Riemann sphere, with spins up to , is
also considered. The connection with the standard Euler angle picture, in terms
of spherical harmonics, is established through a discrete Bargmann transform.Comment: 26 latex pages. Final version published in J. Fourier Anal. App
A Primer on Reproducing Kernel Hilbert Spaces
Reproducing kernel Hilbert spaces are elucidated without assuming prior
familiarity with Hilbert spaces. Compared with extant pedagogic material,
greater care is placed on motivating the definition of reproducing kernel
Hilbert spaces and explaining when and why these spaces are efficacious. The
novel viewpoint is that reproducing kernel Hilbert space theory studies
extrinsic geometry, associating with each geometric configuration a canonical
overdetermined coordinate system. This coordinate system varies continuously
with changing geometric configurations, making it well-suited for studying
problems whose solutions also vary continuously with changing geometry. This
primer can also serve as an introduction to infinite-dimensional linear algebra
because reproducing kernel Hilbert spaces have more properties in common with
Euclidean spaces than do more general Hilbert spaces.Comment: Revised version submitted to Foundations and Trends in Signal
Processin
A constructive theory of sampling for image synthesis using reproducing kernel bases
Sampling a scene by tracing rays and reconstructing an image from such pointwise samples is fundamental to computer graphics. To improve the efficacy of these computations, we propose an alternative theory of sampling. In contrast to traditional formulations for image synthesis, which appeal to nonconstructive Dirac deltas, our theory employs constructive reproducing kernels for the correspondence between continuous functions and pointwise samples. Conceptually, this allows us to obtain a common mathematical formulation of almost all existing numerical techniques for image synthesis. Practically, it enables novel sampling based numerical techniques designed for light transport that provide considerably improved performance per sample. We exemplify the practical benefits of our formulation with three applications: pointwise transport of color spectra, projection of the light energy density into spherical harmonics, and approximation of the shading equation from a photon map. Experimental results verify the utility of our sampling formulation, with lower numerical error rates and enhanced visual quality compared to existing techniques
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