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Gerard Kerkyacharian

Publications and source records attributed to Gerard Kerkyacharian.

12 recordsLinked to original sources
arXiv2021

Gaussian bounds for the heat kernel associated to prolate spheroidal wave functions with applications

Gaussian upper and lower bounds and Hölder continuity are established for the heat kernel associated to the prolate spheroidal wave functions (PSWFs) of order zero. These results are obtained by application of a general perturbation principle using the fact that the PSWF operator is a perturbation of the Legendre operator. Consequently, the Gaussian bounds and Hölder inequality for the PSWF heat kernel follow from the ones in the Legendre case. % As an application of the general perturbation principle, we also establish Gaussian bounds for the heat kernels associated to generalized univariate PSWFs and PSWFs on the unit ball in Rd. Further, we develop the related to the PSWFs of order zero smooth functional calculus, which in turn is the necessary groundwork in developing the theory of Besov and Triebel-Lizorkin spaces associated to the PSWFs. One of our main results on Besov and Triebel-Lizorkin spaces associated to the PSWFs asserts that they are the same as the Besov and Triebel-Lizorkin spaces generated by the Legendre operator.

math.FA↗
arXiv2018

Atomic and molecular decomposition of homogeneous spaces of distributions associated to non-negative self-adjoint operators

We deal with homogeneous Besov and Triebel-Lizorkin spaces in the setting of a doubling metric measure space in the presence of a non-negative self-adjoint operator whose heat kernel has Gaussian localization and the Markov property. The class of almost diagonal operators on the associated sequence spaces is developed and it is shown that this class is an algebra. The boundedness of almost diagonal operators is utilized for establishing smooth molecular and atomic decompositions for the above homogeneous Besov and Triebel-Lizorkin spaces. Spectral multipliers for these spaces are established as well.

math.CA↗
arXiv2018

Gaussian bounds for the weighted heat kernels on the interval, ball and simplex

The aim of this article is to establish two-sided Gaussian bounds for the heat kernels on the unit ball and simplex in $\mathbb{R}^n$, and in particular on the interval, generated by classical differential operators whose eigenfunctions are algebraic polynomials. To this end we develop a general method that employs the natural relation of such operators with weighted Laplace operators on suitable subsets of Riemannian manifolds and the existing general results on heat kernels. Our general scheme allows to consider heat kernels in the weighted cases on the interval, ball, and simplex with parameters in the full range.

math.CA↗
arXiv2015

Regularity of Gaussian Processes on Dirichlet spaces

We are interested in the regularity of centered Gaussian processes (Z_x), x in M indexed by compact metric spaces M. It is shown that the almost everywhere Besov space regularity of such a process is (almost) equivalent to the Besov regularity of the covariance K(x,y) = E(Z_xZ_y) under the assumption that (i) there is an underlying Dirichlet structure on M which determines the Besov space regularity, and (ii) the operator K with kernel K(x,y) and the underlying operator A of the Dirichlet structure commute. As an application of this result we establish the Besov regularity of Gaussian processes indexed by compact homogeneous spaces and, in particular, by the sphere.

math.PR↗
arXiv2014

Heat kernel based decomposition of spaces of distributions in the framework of Dirichlet spaces

Classical and non classical Besov and Triebel-Lizorkin spaces with complete range of indices are developed in the general setting of Dirichlet space with a doubling measure and local scale-invariant Poincaré inequality. This leads to Heat kernel with small time Gaussian bounds and Hölder continuity, which play a central role in this article. Frames with band limited elements of sub-exponential space localization are developed, and frame and heat kernel characterizations of Besov and Triebel-Lizorkin spaces are established. This theory, in particular, allows to develop Besov and Triebel-Lizorkin spaces and their frame and heat kernel characterization in the context of Lie groups, Riemannian manifold, and other settings.

math.FA↗
arXiv2012

Thomas Bayes' walk on manifolds

Convergence of the Bayes posterior measure is considered in canonical statistical settings where observations sit on a geometrical object such as a compact manifold, or more generally on a compact metric space verifying some conditions. A natural geometric prior based on randomly rescaled solutions of the heat equation is considered. Upper and lower bound posterior contraction rates are derived.

math.ST↗
arXiv2012

Heat kernel generated frames in the setting of Dirichlet spaces

Wavelet bases and frames consisting of band limited functions of nearly exponential localization on Rd are a powerful tool in harmonic analysis by making various spaces of functions and distributions more accessible for study and utilization, and providing sparse representation of natural function spaces (e.g. Besov spaces) on Rd. Such frames are also available on the sphere and in more general homogeneous spaces, on the interval and ball. The purpose of this article is to develop band limited well-localized frames in the general setting of Dirichlet spaces with doubling measure and a local scale-invariant Poincaré inequality which lead to heat kernels with small time Gaussian bounds and Hölder continuity. As an application of this construction, band limited frames are developed in the context of Lie groups or homogeneous spaces with polynomial volume growth, complete Riemannian manifolds with Ricci curvature bounded from below and satisfying the volume doubling property, and other settings. The new frames are used for decomposition of Besov spaces in this general setting.

math.FA↗
arXiv2012

Radon needlet thresholding

We provide a new algorithm for the treatment of the noisy inversion of the Radon transform using an appropriate thresholding technique adapted to a well-chosen new localized basis. We establish minimax results and prove their optimality. In particular, we prove that the procedures provided here are able to attain minimax bounds for any $\mathbb {L}_p$ loss. It s important to notice that most of the minimax bounds obtained here are new to our knowledge. It is also important to emphasize the adaptation properties of our procedures with respect to the regularity (sparsity) of the object to recover and to inhomogeneous smoothness. We perform a numerical study that is of importance since we especially have to discuss the cubature problems and propose an averaging procedure that is mostly in the spirit of the cycle spinning performed for periodic signals.

math.ST↗
arXiv2011

Concentration Inequalities and Confidence Bands for Needlet Density Estimators on Compact Homogeneous Manifolds

Let $X_1,...,X_n$ be a random sample from some unknown probability density $f$ defined on a compact homogeneous manifold $\mathbf M$ of dimension $d \ge 1$. Consider a 'needlet frame' $\{ϕ_{j η}\}$ describing a localised projection onto the space of eigenfunctions of the Laplace operator on $\mathbf M$ with corresponding eigenvalues less than $2^{2j}$, as constructed in \cite{GP10}. We prove non-asymptotic concentration inequalities for the uniform deviations of the linear needlet density estimator $f_n(j)$ obtained from an empirical estimate of the needlet projection $\sum_ηϕ_{j η} \int f ϕ_{j η}$ of $f$. We apply these results to construct risk-adaptive estimators and nonasymptotic confidence bands for the unknown density $f$. The confidence bands are adaptive over classes of differentiable and H\"{older}-continuous functions on $\mathbf M$ that attain their Hölder exponents.

math.ST↗
arXiv2008

Spin Needlets for Cosmic Microwave Background Polarization Data Analysis

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.

astro-ph↗
arXiv2005

Thresholding in Learning Theory

In this paper we investigate the problem of learning an unknown bounded function. We be emphasize special cases where it is possible to provide very simple (in terms of computation) estimates enjoying in addition the property of being universal : their construction does not depend on a priori knowledge on regularity conditions on the unknown object and still they have almost optimal properties for a whole bunch of functions spaces. These estimates are constructed using a thresholding schema, which has proven in the last decade in statistics to have very good properties for recovering signals with inhomogeneous smoothness but has not been extensively developed in Learning Theory. We will basically consider two particular situations. In the first case, we consider the RKHS situation. In this case, we produce a new algorithm and investigate its performances in $L\_2(\hatρ\_X)$. The exponential rates of convergences are proved to be almost optimal, and the regularity assumptions are expressed in simple terms. The second case considers a more specified situation where the $X\_i$'s are one dimensional and the estimator is a wavelet thresholding estimate. The results are comparable in this setting to those obtained in the RKHS situation as concern the critical value and the exponential rates. The advantage here is that we are able to state the results in the $L\_2(ρ\_X)$ norm and the regularity conditions are expressed in terms of standard Hölder spaces.

math.ST↗