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Topic: Kernel (integral operator)

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Integral transform

Convolution

Fourier transform

	Springer Online Reference Works (Site not responding. Last check: 2007-11-06)
		The systematic study of self-adjoint differential operators of the second order on a finite interval dates from 1830 (the Sturm–Liouville problem) and was the subject of intensive study in the 19th century, in particular in connection with the theory of special functions.
		The theory of singular differential operators began in 1909–1910, when the spectral decomposition of a self-adjoint unbounded differential operator of the second order with an arbitrary spectral structure was discovered, and when, in principle, the concept of a deficiency index was introduced, and the first results in the theory of extensions were obtained.
		The systematic investigation of non-self-adjoint singular differential operators began in 1950, when the foundations of the theory of operator pencils were expounded and a method was found for proving the completeness of the system consisting of the eigenfunctions of a differential operator and of their associates.
	eom.springer.de /s/s086530.htm (2063 words)

	Torsten Ehrhardt
		Another part of his research is devoted to the spectral theory of structured matrices and convolution-type operators (such as Toeplitz, Hankel, Wiener-Hopf, and singular integral operators).
		His research has shown that an invertibility theory can be established for a larger class of operators (namely, for Toeplitz-plus-Hankel and generalized singular integral operators) by embarking on a different kind of factorization theory [6,7,8].
		This is a simply stated, but non-trivial problem, at the conjunction of the areas of complex analysis, operator theory, and the theory of Banach algebras [11,12].
	www.math.ucsc.edu /faculty/ehrhardt.html (555 words)

	Abstracts: Advances in Kernel Methods - Support Vector Learning
		A basic result of the invariance analysis is that directly imposing a symmetry on the class of kernels effectively results in a preprocessing step, in which the preprocessed data lies in a space whose dimension is reduced by the number of generators of the symmetry group.
		These numbers, which characterize the degree of compactness of the operator, can be bounded in terms of the eigenvalues of an integral operator induced by the kernel function used by the machine.
		We present a new optimization procedure and set of kernels that guarantee the estimate to be a density (be non-negative everywhere and have an integral of 1).
	www.kernel-machines.org /nips97/book-abstract.html (2741 words)

	Fredholm integral equation - Wikipedia, the free encyclopedia
		In mathematics, the Fredholm integral equation is an integral equation whose solution gives rise to Fredholm theory, the study of Fredholm kernels and Fredholm operators.
		One of the principal results is that the kernel K is a compact operator, known as the Fredholm operator.
		As an operator, it has a spectral theory that can be understood in terms of a discrete spectrum of eigenvalues that tend to 0.
	en.wikipedia.org /wiki/Fredholm_integral_equation (306 words)

	Integral equation - Wikipedia, the free encyclopedia
		In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign.
		There is a close connection between differential and integral equations, and some problems may be formulated either way.
		Problems in which integral equations are encountered include radiative energy transfer and the oscillation of a string, membrane, or axle.
	en.wikipedia.org /wiki/Integral_equation (305 words)

	Singular operators in multiwavelet bases
		We note that this count is somewhat different from that obtained by estimating the number of arithmetic operations necessary to obtain a solution with a given accuracy, where the discretization of the operator depends on accuracy as well.
		The modern fast multipole method (e.g., for the Poisson kernel 1/r) is based on essentially this type of approximation, with the functions being plane waves with the region of validity in selected directions [8–10].
		In [5] a multiresolution definition of the regularization of singular and hypersingular homogeneous integrals was introduced using wavelets.
	www.research.ibm.com /journal/rd/482/fann.html (3279 words)

	Re: Density operator in second quantization (via CobWeb/3.1 planetlab2.cs.virginia.edu) (Site not responding. Last check: 2007-11-06)
		If f is a 1-particle operator given by an integral operator with kernel f(x,y) (the general case follows by taking limits), the formula = integral dx dy f(x,y) defines the 1-particle density matrix Rho.
		The form of f in second quantization is fhat = integral dx dy f(x,y) a^(x) a(y) (exercise: check that it has indeed the desired action on an N-particle state!), hence one has = integral* dx dy f(x,y) .
		Among the common operators, the second quantized density > operators are unique in having this vacuum-to-vacuum operator as a > factor.
	www.lns.cornell.edu.cob-web.org:8888 /spr/2006-07/msg0074955.html (719 words)

	Hierarchical Matrices: Literature
		For classical integral operators, the existence of suitable nested systems is obvious, but for more general situations, a careful analysis is required.
		A popular approach to constructing low-rank approximations of discretized integral operators is based on the application of cross approximation techniques [Tyrtyshnikov2000], [Bebendorf/Rjasanow2003] to entries of the discrete matrix.
		This thesis gives an introduction to the theory of hierarchical matrices, covering the approximation of integral operators, the arithmetic operations and very general complexity estimates based on the concepts of sparsity and idempotence of block cluster trees.
	www.hlib.org /literature.html (1596 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Homework 3 Multidimensional Fredholm integral equation, resolvent operators and eigenvalues of compact integral operators, non-self adjoint compact operator, and solvability of Ku=f.
		Spectral representations of Operators A LaTeX file for two theorems on the spectral representation of bounded and unbounded operators.
		Separable Fredholm Integral Equations Separable integral equations are equivalent to matrix equations with appropriate definitions.
	www.math.montana.edu /~pernarow/M560/1998/M560.html (663 words)

	Kernel Principal Component Analysis - Sch, Smola, uller (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		A new method for performing a nonlinear form of Principal Component Analysis is proposed.
		By the use of integral operator kernel functions, one can efficiently compute principal components in high-- dimensional feature spaces, related to input space by some nonlinear map; for instance the space of all possible d--pixel products in images.
		H1 as a smoothing functional the kernel K 1 should not be centered in the feature space, as in usual kernel CCA [BJ01] and kernel PCA [
	citeseer.ist.psu.edu /25296.html (549 words)

	XIII. Geometry and integral operators
		You may wish to review the notion of a linear operator by referring to the Maple worksheet or the Mathematica notebook for Chapter I.
		Often an integral equation can be converted into a differential equation or vice versa, so many of the laws of nature which we think of as differential equations might just as well have been developed as integral equations initially.
		In this course integral operators will be called upon to solve differential equations, and this is one of their main uses.
	www.mathphysics.com /pde/ch13wr.html (1783 words)

	Geometry Kernels
		While in the internal homogeneous representation, an integral number type is sufficient, rational numbers must sometimes be used outside the internal representation, for example, when the squared length of a vector is computed.
		Each kernel object is provided as both a stand-alone class, which is parameterized by a kernel class (Geo_object_D), and as a type in the kernel class (K::Geo_object_D).
		Similarly, each operation and construction in the kernel is provided via a function object class in the kernel class and also as either a member function or a global function; developers should use the function object classes to gain access to the functionality.
	www.cgal.org /DManual/html/Developers_manual/Chapter_kernels.html (1114 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-11-06)
		A completely-continuous operator is nowadays usually called a compact operator.
		A Fredholm kernel is a bivalent tensor (cf.
		The concept of a Fredholm kernel can also be generalized to the case of the tensor product of several locally convex spaces.
	eom.springer.de /F/f041440.htm (308 words)

	FuncAna
		Remark: the kernel of a bounded operator is always closed while the range may be not closed.
		Norm of a self-adjoint operator is equal to the supremum of the absolute value of its quadratic form on the unit sphere.
		Reduction to an integral equation and proof of the spectral theorem.
	www.math.ttu.edu /~vshubov/FuncAna/FuncAna.html (947 words)

	New Page 1
		As in GP regression, analytical expressions for the mean and variance of the estimated target function are obtained from which the solution to the operator equation follows by a manipulation of the kernel.
		Projection operators are used to illustrate the relations between canonical vectors and variates.
		A new topographic clustering algorithm is proposed, which -- by the use of integral operator kernel functions -- efficiently estimates the centers of clusters in a high-dimensional feature space, which is related to data space by some non linear map.
	research.microsoft.com /~thoreg/pubs.aspx (8164 words)

	dD Kernel
		Almost all the kernel objects (and the corresponding functions) are templates with a parameter that allows the user to choose the representation of the kernel objects.
		Algorithms and data structures in the basic library of CGAL are parameterized by a traits class that subsumes the objects on which the algorithm or data structure operates as well as the operations to do so.
		For some algorithms you even do not have to specify the kernel; it is detected automatically using the types of the geometric objects passed to the algorithm.
	cgal.org /Manual/3.2/doc_html/cgal_manual/Kernel_d/Chapter_main.html (2931 words)

	Tutorial for Green's Functions, Materials Reliability Division, N.I.S.T
		For a discussion of the concept of self-adjoint and non self-adjoint differential operators please refer, for example, to the text by Morse and Feshbach.
		Since L is a differential operator, it is reasonable to expect its inverse to be an integral operator.
		The integral is an isotropic Fourier integral since it depends only on the magnitude of q, which is q, but does depend on the direction of q.
	www.boulder.nist.gov /div853/greenfn/tutorial.html?.html (1037 words)

	[No title]
		Riemann integrals are approximated for the derivative of the integrand belonging to a variety of norms.
		If the one-dimensional integrals are not known, they themselves can be approximated by using a suitable rule, to produce a cubature rule consisting only of sampling points.
		This is employed to approximate double integrals using one dimensional integrals and function evaluations at the boundary and interior points.
	rgmia.vu.edu.au /monographs/Ostrowski.html (956 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		We first show that the canonical solution operator to d-bar restricted to (0,1)-forms with holomorphic coefficients can be expressed by an integral operator using the Bergman kernel.
		This result is used to prove that in the case of the unit disc in C the canonical solution operator to d-bar restricted to (0,1)-forms with holomorphic coefficients is a Hilbert-Schmidt operator.
		In the sequel we give a direct proof of the last statement using orthonormal bases and show that in the case of the polydisc and the unit ball in C^n, n>1, the corresponding operator fails to be a Hilbert-Schmidt operator.
	www.mat.univie.ac.at /~has/artikel/paper3.html (119 words)

	Wave Guiding Stuctures (Site not responding. Last check: 2007-11-06)
		The guiding region is usually immersed in a layered background environment (e.g., conductor/ substrate/cover for microstrip) and the necessary Green’s function kernel for the integral operator must be expressed in Sommerfeld-type integral representation.
		Properties of the latter Green’s function have a strong influence on the propagation-mode spectrum of the waveguide structure.
		Contributions include the first unambiguous conceptualization for the continuous spectrum and the categorization of various proper and leaky discrete modes based upon properties of the Green’s function kernel.
	www.egr.msu.edu /em/facilities/index_files/wave_guide.html (361 words)

	Kernel Traffic #251 For 9 Feb 2004 (Site not responding. Last check: 2007-11-06)
		However, as long as the kernel keeps using C as it's language, keeps being GPL'ed, and keeps exporting a module interface that is defined by some prototypes in some C include files, I don't see how this could lead to any trouble for them.
		Kernel Traffic is grateful to be developed on a computer donated by Professor Greg Benson and Professor Allan Cruse in the Department of Computer Science at the University of San Francisco.
		Kernel Traffic is hosted by the generous folks at kernel.org.
	www.kerneltraffic.org /kernel-traffic/kt20040209_251.html (8526 words)

	The Kernel Trick
		K(x,w) is a kernel in a reproducing kernel Hilbert space or rushed to the library or to the guy next door to find out, and probably very soon after that said aha!, K(x,w) is the kernel of a RKHS.
		The kernel function becomes useful for choosing the classification boundary but even that could be empirically approximated.
		The point is that it is obvious that a choice of kernel function is an ad-hoc way of sweeping under the rug prior information into the problem, indutransductibly (!) ducking the holy Bayes Theorem.
	omega.albany.edu:8008 /machine-learning-dir/notes-dir/ker1/ker1-l.html (907 words)

	[No title]
		France} \bigskip \centerline {Abstract} \medskip We consider a $d$-dimensional quantum anharmonic crystal, where the interaction between the ions satisfies hypotheses, based on the idea that the ions are not too far from the points $\Z ^d$, and that the interaction between them decreases exponentially with their distance.
		Under these conditions, we study carefully the heat kernel of the Hamiltonian related to each finite set of $\Z^d$, with constants in the inqualities that are independent of this set, (thus, improving an earlier result of Sj\"ostrand).
		The decay of correlation relies on the study of the heat kernel, and is itself used for estimating the rate of convergence in the thermodynamic limit.
	www.ma.utexas.edu /mp_arc/e/03-541.tex.mime (8466 words)

	Amazon.com: Hypersingular Integrals and Their Applications (Analytical Methods and Special Functions): Books: Stefan ... (Site not responding. Last check: 2007-11-06)
		Hypersingular integrals arise as constructions inverse to potential-type operators and are realized by the methods of regularization and finite differences.
		Hypersingular Integrals and Their Applications also presents recent results and applications that will prove valuable to graduate students and researchers working in mathematical analysis.
		approximative inverses, potential type operators, hypersingular integrals, fractional smoothness, hypersingular integral operator, hypersingular operator, multidimensional integral equations, positive fractional powers, centred difference, identity approximation, continuity modulus, approximation kernel, fractional differentiation, test function space, rotation rot, nice functions, generalized differences, neutral type, analytical continuation, limiting operator, fractional integrals, potential operator, fractional integration, odd type, operator inverse
	www.amazon.com /Hypersingular-Integrals-Applications-Analytical-Functions/dp/0415272688 (866 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		% % Set up a discretization of a convolution integral operator K with a % Gaussian kernel.
		Generate a true solution and convolve it with the % kernel.
		kernel = (1/sqrt(pi)/sig) * exp(-(x-h/2).^2/sig^2); K = toeplitz(kernel)h; % Set up true solution f_true and data d = Kf_true + error.
	www.math.montana.edu /~vogel/Book/Codes/Ch1/Gen_data.m (275 words)

	Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps -- Coifman et al. ...
		The original geometry (b) is mapped as a "butterfly" set, in which the red (higher conductivity) and blue phases are organized according to the diffusion they generate: the cord length between two points in the diffusion space measures the quantity of heat that can travel between these points.
		operator on the manifold, the backward Fokker–Planck operator
		are the integral curves of the field orthogonal to the gradient
	www.pnas.org /cgi/content/full/102/21/7426 (3429 words)

	Citebase - Distributions on partitions, point processes, and the hypergeometric kernel
		The kernel can be expressed through the Gauss hypergeometric function; we call it the hypergeometric kernel.
		The integral operator corresponding to the Whittaker kernel is an integrable operator as defined by Its, Izergin, Korepin, and Slavnov.
		This fact is parallel to the degeneration of the Whittaker kernel to the Christoffel-Darboux kernel for Laguerre polynomials.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/9904010 (261 words)

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