arXiv subjects
Publications and source records attributed to Daryl Geller.
We determine upper asymptotic estimates of Kolmogorov and linear $n$-widths of unit balls in Sobolev and Besov norms in $L_{p}$-spaces on compact Riemannian manifolds. The proofs rely on estimates for the near-diagonal localization of the kernels of elliptic operators. We also summarize some of our previous results about approximations by eigenfunctions of elliptic operators on compact homogeneous manifolds.
We determine upper asymptotic estimates of Kolmogorov and linear $n$-widths of unit balls in Sobolev and Besov norms in $L_{p}$-spaces on smooth compact Riemannian manifolds. For compact homogeneous manifolds, we establish estimates which are asymptotically exact, for the natural ranges of indices. The proofs heavily rely on our previous results such as: estimates for the near-diagonal localization of the kernels of elliptic operators, Plancherel-Polya inequalities on manifolds of bounded geometry, cubature formulas with positive coefficients and uniform estimates on Clebsch-Gordon coefficients on general compact homogeneous manifolds.
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.
The construction of adaptive nonparametric procedures by means of wavelet thresholding techniques is now a classical topic in modern mathematical statistics. In this paper, we extend this framework to the analysis of nonparametric regression on sections of spin fiber bundles defined on the sphere. This can be viewed as a regression problem where the function to be estimated takes as its values algebraic curves (for instance, ellipses) rather than scalars, as usual. The problem is motivated by many important astrophysical applications, concerning for instance the analysis of the weak gravitational lensing effect, i.e. the distortion effect of gravity on the images of distant galaxies. We propose a thresholding procedure based upon the (mixed) spin needlets construction recently advocated by Geller and Marinucci (2008,2010) and Geller et al. (2008,2009), and we investigate their rates of convergence and their adaptive properties over spin Besov balls.
The construction of needlet-type wavelets on sections of the spin line bundles over the sphere has been recently addressed in Geller and Marinucci (2008), and Geller et al. (2008,2009). Here we focus on an alternative proposal for needlets on this spin line bundle, in which needlet coefficients arise from the usual, rather than the spin, spherical harmonics, as in the previous constructions. We label this system mixed needlets and investigate in full their properties, including localization, the exact tight frame characterization, reconstruction formula, decomposition of functional spaces, and asymptotic uncorrelation in the stochastic case. We outline astrophysical applications.
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.
We show that one can characterize the Besov spaces on a smooth compact oriented Riemannian manifold, for the full range of indices, through a knowledge of the size of frame coefficients, using the frames we have constructed in [8].
We consider the statistical analysis of random sections of a spin fibre bundle over the sphere. These may be thought of as random fields that at each point p in $S^2$ take as a value a curve (e.g. an ellipse) living in the tangent plane at that point $T_{p}S^2$, rather than a number as in ordinary situations. The analysis of such fields is strongly motivated by applications, for instance polarization experiments in Cosmology. To investigate such fields, spin needlets were recently introduced by Geller and Marinucci (2008) and Geller et al. (2008). We consider the use of spin needlets for spin angular power spectrum estimation, in the presence of noise and missing observations, and we provide Central Limit Theorem results, in the high frequency sense; we discuss also tests for bias and asymmetries with an asymptotic justification.
In recent years, a rapidly growing literature has focussed on the construction of wavelet systems to analyze functions defined on the sphere. Our purpose in this paper is to generalize these constructions to situations where sections of line bundles, rather than ordinary scalar-valued functions, are considered. In particular, we propose {\em needlet-type spin wavelets} as an extension of the needlet approach recently introduced by Narcowich, Petrushev and Ward, and then considered for more general manifolds by Geller and Mayeli. We discuss localization properties in the real and harmonic domains, and investigate stochastic properties for the analysis of spin random fields. Our results are strongly motivated by cosmological applications, in particular in connection to the analysis of Cosmic Microwave Background polarization data.
Scalar wavelets have been used extensively in the analysis of Cosmic Microwave Background (CMB) temperature maps. Spin needlets are a new form of (spin) wavelets which were introduced in the mathematical literature by Geller and Marinucci (2008) as a tool for the analysis of spin random fields. Here we adopt the spin needlet approach for the analysis of CMB polarization measurements. The outcome of experiments measuring the polarization of the CMB are maps of the Stokes Q and U parameters which are spin 2 quantities. Here we discuss how to transform these spin 2 maps into spin 2 needlet coefficients and outline briefly how these coefficients can be used in the analysis of CMB polarization data. We review the most important properties of spin needlets, such as localization in pixel and harmonic space and asymptotic uncorrelation. We discuss several statistical applications, including the relation of angular power spectra to the needlet coefficients, testing for non-Gaussianity on polarization data, and reconstruction of the E and B scalar maps.
Let $\bf M$ be a smooth compact oriented Riemannian manifold, and let $Δ$ be the Laplace-Beltrami operator on ${\bf M}$. Say $0 \neq f \in \mathcal{S}(\RR^+)$, and that $f(0) = 0$. For $t > 0$, let $K_t(x,y)$ denote the kernel of $f(t^2 Δ)$. Suppose $f$ satisfies Daubechies' criterion, and $b > 0$. For each $j$, write ${\bf M}$ as a disjoint union of measurable sets $E_{j,k}$ with diameter at most $ba^j$, and comparable to $ba^j$ if $ba^j$ is sufficiently small. Take $x_{j,k} \in E_{j,k}$. We then show that the functions $ϕ_{j,k}(x)=[μ(E_{j,k})]^{1/2} \bar{K_{a^j}}(x_{j,k},x)$ form a frame for $(I-P)L^2({\bf M})$, for $b$ sufficiently small (here $P$ is the projection onto the constant functions). Moreover, we show that the ratio of the frame bounds approaches 1 nearly quadratically as the dilation parameter approaches 1, so that the frame quickly becomes nearly tight (for $b$ sufficiently small). Moreover, based upon how well-localized a function $F \in (I-P)L^2$ is in space and in frequency, we can describe which terms in the summation $F \sim SF = \sum_j \sum_k < F,ϕ_{j,k} > ϕ_{j,k}$ are so small that they can be neglected. If $n=2$ and $\bf M$ is the torus or the sphere, and $f(s)=se^{-s}$ (the "Mexican hat" situation), we obtain two explicit approximate formulas for the $ϕ_{j,k}$, one to be used when $t$ is large, and one to be used when $t$ is small. Finally we explain in what sense the kernel $K_t(x,y)$ should itself be regarded as a continuous wavelet on ${\bf M}$, and characterize the Hölder continuous functions on ${\bf M}$ by the size of their continuous wavelet transforms, for Hölder exponents strictly between 0 and 1.
Let $\bf M$ be a smooth compact oriented Riemannian manifold, and let $Δ_{\bf M}$ be the Laplace-Beltrami operator on ${\bf M}$. Say $0 \neq f \in \mathcal{S}(\RR^+)$, and that $f(0) = 0$. For $t > 0$, let $K_t(x,y)$ denote the kernel of $f(t^2 Δ_{\bf M})$. We show that $K_t$ is well-localized near the diagonal, in the sense that it satisfies estimates akin to those satisfied by the kernel of the convolution operator $f(t^2Δ)$ on $\RR^n$. We define continuous ${\cal S}$-wavelets on ${\bf M}$, in such a manner that $K_t(x,y)$ satisfies this definition, because of its localization near the diagonal. Continuous ${\cal S}$-wavelets on ${\bf M}$ are analogous to continuous wavelets on $\RR^n$ in $\mathcal{S}(\RR^n)$. In particular, we are able to characterize the H$\ddot{o}$lder continuous functions on ${\bf M}$ by the size of their continuous ${\mathcal{S}}-$wavelet transforms, for H$\ddot{o}$lder exponents strictly between 0 and 1. If $\bf M$ is the torus $\TT^2$ or the sphere $S^2$, and $f(s)=se^{-s}$ (the ``Mexican hat'' situation), we obtain two explicit approximate formulas for $K_t$, one to be used when $t$ is large, and one to be used when $t$ is small.
We produce, on general homogeneous groups, an analogue of the usual Hörmander pseudodifferential calculus on Euclidean space, at least as far as products and adjoints are concerned. In contrast to earlier works, we do not limit ourselves to analogues of classical symbols, nor to the Heisenberg group. The key technique is to understand ``multipliers'' of any given order j, and the operators of convolution with their inverse Fourier transforms, which we here call convolution operators of order j. (Here a ``multiplier'' is an analogue of a Hörmander-type symbol a(x,ξ), which is independent of x.) Specifically, we characterize the space of inverse Fourier transforms of multipliers of any order j, and use this characterization to show that the composition of convolution operators of order j_1 and j_2 is a convolution operator of order j_1+j_2.
Let G be a stratified Lie group and L be the sub-Laplacian on G. Let 0 \neq f\in S(R^+). We show that Lf(L)δ, the distribution kernel of the operator Lf(L), is an admissible function on G. We also show that, if ξf(ξ) satisfies Daubechies' criterion, then L f(L)δgenerates a frame for any sufficiently fine lattice subgroup of G.