arXiv subjects
Publications and source records attributed to Isaac Z. Pesenson.
The paper contains Boas-type formulas for trajectories of one-parameter groups of operators in Banach spaces. The results are illustrated using one-parameter groups of operators which appear in representations of Lie groups.
In the chapter "Multiresolution Analysis on Compact Riemannian Manifolds" Isaac Pesenson describes multiscale analysis, sampling, interpolation and approximation of functions defined on manifolds. His main achievements are: construction on manifolds of bandlimited and space-localized frames which have Parseval property and construction of variational splines on manifolds. Such frames and splines enable multiscale analysis on arbitrary compact manifolds, and they already found a number of important applications (statistics, CMB, crystallography) related to such manifolds as two-dimensional sphere and group of its rotations.
It is shown that eigenvalues of Laplace-Beltrami operators on compact Riemannian manifolds can be determined as limits of eigenvalues of certain finite-dimensional operators in spaces of polyharmonic functions with singularities. In particular, a bounded set of eigenvalues can be determined using a space of such polyharmonic functions with a fixed set of singularities. It also shown that corresponding eigenfunctions can be reconstructed as uniform limits of the same polyharmonic functions with appropriate fixed set of singularities.
By using Bernstein-type inequality we define analogs of spaces of entire functions of exponential type in $L_{p}(X), 1\leq p\leq \infty$, where $X$ is a symmetric space of non-compact. We give estimates of $L_{p}$-norms, $1\leq p\leq \infty$, of such functions (the Nikolskii-type inequalities) and also prove the $L_{p}$- Plancherel-Polya inequalities which imply that our functions of exponential type are uniquely determined by their inner products with certain countable sets of measures with compact supports and can be reconstructed from such sets of "measurements" in a stable way.
In this paper we consider the so-called crystallographic Radon transform (or crystallographic $X$-ray transform) and totally geodesic Radon transform on the group of rotations SO(3). As we show both of these transforms naturally appear in texture analysis, i.e. the analysis of preferred crystallographic orientation. Although we discuss only applications to texture analysis both transforms have other applications as well. In section 2 we start with motivations and applications. In sections 3 and 4 we develop a general framework on compact Lie groups. In section 5 we give a detailed analysis of the totally geodesic Radon transform on SO(3). In section \ref{relations} we compare crystallographic Radon transform on SO(3) and Funk transform on $S^{3}$. In section \ref{1} we show non-invertibility of the crystallographic transform. In section 8 we describe an exact reconstruction formula for bandlimited functions, which uses only a finite number of samples of their Radon transform. Some auxiliary results for this section are collected in Appendix.
We extend some results about sampling of entire functions of exponential type to Banach spaces. By using generator $D$ of one-parameter group $e^{tD}$ of isometries of a Banach space $E$ we introduce Bernstein subspaces $\mathbf{B}_σ(D),\>\>σ>0,$ of vectors $f$ in $E$ for which trajectories $e^{tD}f$ are abstract-valued functions of exponential type which are bounded on the real line. This property allows to reduce sampling problems for $e^{tD}f$ with $f\in \mathbf{B}_σ(D)$ to known sampling results for regular functions of exponential type $σ$.
Let $X$ be a symmetric space of the noncompact type. The goal of the paper is to construct in the space $L_{2}(X)$ nearly Parseval frames consisting of functions which simultaneously belong to Paley-Wiener spaces and to Schwartz space on $X$. We call them Paley-Wiener-Schwartz frames in $L_{2}(X)$. These frames are used to characterize a family of Besov spaces on $X$. As a part of our construction we develop on $X$ the so-called average Shannon-type sampling.
The goal of the present paper is to construct bandlimited highly localized and nearly tight frames on domains with smooth boundaries in Euclidean spaces. These frames are used do describe corresponding Besov spaces.
Our article is a summary of some results for Riemannian manifolds that were obtained in \cite{gpes}-\cite{Pesssubm}. To the best of our knowledge these are the pioneering papers which contain the most general results about frames, Shannon sampling, and cubature formulas on compact and non-compact Riemannian manifolds. In particular, the paper \cite{gpes} gives an "end point" construction of tight localized frames on homogeneous compact manifolds. The paper \cite{Pessubm} is the first systematic development of localized frames on compact domains in Euclidean spaces.
The Radon transform Rf of functions f on SO(3) has recently been applied extensively in texture analysis, i.e. the analysis of preferred crystallographic orientation. In practice one has to determine the orientation probability density function f \in L2(SO(3)) from Rf \in L2(S2\times S2) which is known only on a discrete set of points. Since one has only partial information about Rf the inversion of the Radon transform becomes an ill-posed inverse problem. Motivated by this problem we define a new notion of the Radon transform Rf of functions f on general compact Lie groups and introduce two approximate inversion algorithms which utilize our previously developed generalized variational splines on manifolds. Our new algorithms fit very well to the application of Radon transform on SO(3) to texture analysis.
We prove Poincaré and Plancherel-Polya inequalities for weighted {\ell}p -spaces on weighted graphs in which the constants are explicitly expressed in terms of some geometric characteristics of a graph. We use Poincaré type inequality to obtain some new relations between geometric and spectral properties of the combinatorial Laplace operator. Several well known graphs are considered to demonstrate that our results are reasonably sharp. The Plancherel-Polya inequalities allow for application of the frame algo- rithm as a method for reconstruction of Paley-Wiener functions on weighted graphs from a set of samples. The results are illustrated by developing Shannon- type sampling in the case of a line graph. Our work has potential applications to data mining and learning theory on graphs.
In this paper we address sampling and approximation of functions on combinatorial graphs. We develop filtering on graphs by using Schrödinger's group of operators generated by combinatorial Laplace operator. Then we construct a sampling theory by proving Poincare and Plancherel-Polya-type inequalities for functions on graphs. These results lead to a theory of sparse approximations on graphs and have potential applications to filtering, denoising, data dimension reduction, image processing, image compression, computer graphics, visualization and learning theory.
The goal of the paper is to describe essentially optimal cubature formulas on compact Riemannian manifolds which are exact on spaces of band- limited functions.
We develop an approximation theory in Hilbert spaces that generalizes the classical theory of approximation by entire functions of exponential type. The results advance harmonic analysis on manifolds and graphs, thus facilitating data representation, compression, denoising and visualization. These tasks are of great importance to machine learning, complex data analysis and computer vision.
Recent and forthcoming advances in instrumentation, and giant new surveys, are creating astronomical data sets that are not amenable to the methods of analysis familiar to astronomers. Traditional methods are often inadequate not merely because of the size in bytes of the data sets, but also because of the complexity of modern data sets. Mathematical limitations of familiar algorithms and techniques in dealing with such data sets create a critical need for new paradigms for the representation, analysis and scientific visualization (as opposed to illustrative visualization) of heterogeneous, multiresolution data across application domains. Some of the problems presented by the new data sets have been addressed by other disciplines such as applied mathematics, statistics and machine learning and have been utilized by other sciences such as space-based geosciences. Unfortunately, valuable results pertaining to these problems are mostly to be found only in publications outside of astronomy. Here we offer brief overviews of a number of concepts, techniques and developments, some "old" and some new. These are generally unknown to most of the astronomical community, but are vital to the analysis and visualization of complex datasets and images. In order for astronomers to take advantage of the richness and complexity of the new era of data, and to be able to identify, adopt, and apply new solutions, the astronomical community needs a certain degree of awareness and understanding of the new concepts. One of the goals of this paper is to help bridge the gap between applied mathematics, artificial intelligence and computer science on the one side and astronomy on the other.
In the last decade, methods based on various kinds of spherical wavelet bases have found applications in virtually all areas where analysis of spherical data is required, including cosmology, weather prediction, and geodesy. In particular, the so-called needlets (=band-limited Parseval frames) have become an important tool for the analysis of Cosmic Microwave Background (CMB) temperature data. The goal of the present paper is to construct band-limited and highly localized Parseval frames on general compact homogeneous manifolds. Our construction can be considered as an analogue of the well-known phi-transform on Euclidean spaces.