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	Orthonormal basis - Wikipedia, the free encyclopedia
		In mathematics, an orthonormal basis of an inner product space V (i.e., a vector space with an inner product), or in particular of a Hilbert space H, is a set of elements whose span is dense in the space, in which the elements are mutually orthogonal and normal, that is, of magnitude 1.
		An orthonormal basis is not generally a "basis", i.e., it is not generally possible to write every member of the space as a linear combination of finitely many members of an orthonormal basis.
		Note that in the infinite-dimensional case, an orthonormal basis will not be a basis in the sense of linear algebra; to distinguish the two, the latter bases are also called Hamel bases.
	en.wikipedia.org /wiki/Orthonormal_basis (576 words)

	PlanetMath: orthonormal (Site not responding. Last check: 2007-11-07)
		A standard application is finding an orthonormal basis for a vector space, such as by Gram-Schmidt orthonormalization.
		Orthonormal bases are computationally simple to work with.
		This is version 7 of orthonormal, born on 2002-01-04, modified 2003-04-19.
	planetmath.org /encyclopedia/Orthonormal.html (119 words)

	Orthonormality - Wikipedia, the free encyclopedia
		In linear algebra, two vectors v and w are said to be orthonormal if they are both orthogonal (according to a given inner product) and normalized.
		A basis which forms an orthonormal set is called an orthonormal basis.
		An equivalent formulation of the two conditions is done by using the Delta function.
	en.wikipedia.org /wiki/Orthonormal (146 words)

	Orthonormal basis -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-07)
		For finite-dimensional spaces the condition of a dense span is the same as 'span', as used in (The part of algebra that deals with the theory of linear equations and linear transformation) linear algebra.
		An orthonormal basis of a vector space V makes no sense unless V is given an inner product; (Click link for more info and facts about Banach space) Banach spaces do not generally have orthonormal bases.
		This is fundamental to the study of (The sum of a series of trigonometric expressions; used in the analysis of periodic functions) Fourier series.
	www.absoluteastronomy.com /encyclopedia/o/or/orthonormal_basis.htm (603 words)

	STA TURBULENCE
		Finally, it is demonstrated that orthonormal wavelet and wavelet packets thresholding are insensitive to the analyzing wavelet.
		The role of energy-containing eddy motion in the deviations between the measured and predicted \beta is considered using orthonormal wavelet expansion in conjunction with a wavelet shrinkage approach.
		Abstract: Orthonormal wavelet expansions were derived and applied to atmospheric surface layer turbulence measurements of temperature and vapor concentration under unstable and stable atmospheric stability conditions.
	www.isye.gatech.edu /~brani/.public_html/gaby1.html (2130 words)

	Orthonormal Ridgelets and Linear Singularities
		We construct a new orthonormal basis for $L^2({\Bbb R}^2)$, whose elements are angularly integrated ridge functions---{\it orthonormal ridgelets}.
		Orthonormal ridgelet expansions expose an interesting phenomenon in nonlinear approximation: they give very efficient approximations to objects such as $1_{\{ x_1\cos\theta+ x_2\sin\theta > a\}} \ e^{-x^2_1-x^2_2}$ which are smooth away from a discontinuity along a line.
		Orthonormal ridgelets make available the machinery of orthogonal decompositions, which is not available for ridge functions as they are not in $L^2({\Bbb R}^2)$.
	epubs.siam.org /sam-bin/dbq/article/34440 (390 words)

	Orthonormal matrix - Wikipedia, the free encyclopedia
		In linear algebra, an orthonormal matrix is a (not necessarily square) matrix with real or complex entries whose columns, treated as vectors in R
		The real n-by-k orthonormal matrices are precisely the matrices that result from deleting n-k columns from an orthogonal matrix; the complex n-by-k orthonormal matrices are precisely the matrices that result from deleting n-k columns from a unitary matrix.
		In particular, unitary and orthogonal matrices are themselves orthonormal.
	www.wikipedia.org /wiki/Orthonormal_matrix (141 words)

	The Wavelet Digest :: View topic - Thesis: Optimal Orthonormal Subband Coding and Lattice...
		The Wavelet Digest :: View topic - Thesis: Optimal Orthonormal Subband Coding and Lattice...
		Subject: Thesis: Optimal Orthonormal Subband Coding and Lattice...
		orthonormal filter bank to decompose the signal into components that
	www.wavelet.org /phpBB2/viewtopic.php?t=4049 (422 words)

	Orthonormal Basis Functions for Modelling Continuous-Time Systems - Akcay, Ninness (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		The contribution of the paper is to establish that the obtained model sets are complete in all of the Hardy spaces H p (); 1 < p < 1 and the right half plane algebra A() provided that a mild condition on the choice of basis poles is satis ed.
		6 Construction of generalised orthonormal bases in H (context) - Bokor, Gianone et al.
		1 A uni ed approach to the synthesis of orthonormal exponentia..
	citeseer.ist.psu.edu /akcay99orthonormal.html (699 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		Then any orthonormal subset T of G will meet the sum of E2 and E3 only in E3, and hence the projection of T to E2 will be countable; but the projection of G onto E2 has dense image.
		Let H_2 be the non-separable Hilbert space with orthonormal basis {y_b: b in B} indexed by B. Let K be the linear span of the vectors {x_b + y_b: b in B} in the direct sum H_1 + H_2.
		Then there is no orthonormal set in K whose span is dense in K. In fact: 1) Any orthonormal set in K is countable.
	www.math.niu.edu /~rusin/known-math/00_incoming/innerprod (534 words)

	No Title (Site not responding. Last check: 2007-11-07)
		An orthonormal basis is an orthogonal basis with the additional property that all of the basis vectors are normalized.
		is an orthonormal basis that you should all be very familiar with.
		An orthonormal basis for a vector space is very easy to work with, because only dot products are needed to determine the coordinates for any vector in the space, relative to the basis.
	www.sci.wsu.edu /math/faculty/genz/220v/lessons/l11/l11.html (241 words)

	Perpendicularity in Vector Spaces
		Such a basis is called orthonormal: ortho for orthogonal (perpendicular) and normal for normalized (length 1).
		To discuss orthonormal bases for other vector spaces, we must extend the idea of dot product to what is called an inner product.
		The fact that orthonormal sets are always linearly independent makes them nice to work with when we are finding bases for vector spaces with inner products.
	distance-ed.math.tamu.edu /Math640/chapter3/node13.html (470 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		Andy Tjon, Project 2, CAP5725, Computer Graphics I Part A of this project is to compute a right-handed orthonormal basis uvw from a single vector a such that w is parallel to a.
		Part B of this project is to compute a right-handed orthonormal basis uvw from two vectors a and b such that u is parallel to a and w is normal to the planes defined by a and b.
		One of the entered vectors is the first orthonormal basis vector.
	longwood.cs.ucf.edu /~tjie/cap5725/Proj2.txt (335 words)

	Orthonormal Bases in Rn
		that is not orthonormal, and want to find a basis for the space that is orthonormal, we follow the following process.
		First, if we can find an orthogonal basis, we can always divide each of the basis vectors by their magnitudes to arrive at an orthonormal basis.
		If we want to find an orthonormal basis we just divide each of these vectors by their magnitudes.
	www.ltcconline.net /greenl/courses/203/Vectors/orthonormalBases.htm (316 words)

	Orthonormal shift-invariant wavelet packet decomposition and representation
		The proposed library, generated via a generalized version of the multiresolution decomposition, is cast into a binary tree configuration, in which the nodes represent subspaces with different time-frequency localization characteristics.
		Recently, several authors proposed independently to extend the library of bases, in which the best representations are searched for, by introducing additional degrees of freedom that adjust the time-localization of the basis functions.
		It was proved that the proposed modifications of the wavelet transform and wavelet packet decomposition lead to orthonormal best-basis representations which are shift invariant and are characterized by lower information costs.
	www.uh.edu /~ywang4/Project2.htm (811 words)

	SIPG (Site not responding. Last check: 2007-11-07)
		This paper addresses the question "what makes a good orthonormal wavelet for image compression?", by considering objective and subjective measurements of quality.
		We propose a new metric for wavelet design: a balanced uncertainty; that is filters with their time and frequency spreads in a fixed ratio.
		Perhaps the most important comparison is with the Daubechies calss of 'maximum regularity' orthonormal wavelets.
	www.bath.ac.uk /elec-eng/pages/sipg/research/wavelet/owbu.htm (506 words)

	II. Geometry of Functions
		This one is the basis of the usual Fourier series, and is perhaps the most important of all our orthonormal sets.
		It is often convenient to have orthonormal, or at least orthogonal sets.
		Although we shall first concentrate on the set (2.8) as a basis for a vector space of functions, the other sets of orthonormal functions (2.5)-(2.7) and (2.9) will be useful later for the same purpose.
	www.mathphysics.com /pde/ch2wr.html (3085 words)

	Orthonormal basistransformations
		When all new basis vectors have unit length (measured in the old system), the term orthonormal transformation is used.
		R is the rotation matrix and l is a diagonal scaling matrix, containing the scaling factors for each coordinate in the diagonal.
		In the orthonormal case l is the identity matrix.
	www.diku.dk /undervisning/1999f/f99.134/Orthonormal_basistransformations.html (1366 words)

	Abstract:Symmetric Orthonormal Scaling Functions (Site not responding. Last check: 2007-11-07)
		It is well known that in the univariate case, up to an integer shift and possible sign change, there is no dyadic compactly supported symmetric orthonormal scaling function except for the Haar function.
		In this paper we are concerned with the construction of symmetric orthonormal scaling functions with dilation factor $d=4$.
		In particular, two examples of $C^1$ orthonormal scaling functions, which are symmetric about $0$ and $\frac{1}{6}$, respectively, are presented.
	www.ualberta.ca /~bhan/abstracts/1997symd4wav.abs.html (120 words)

	TOS pub: 8.2
		In this report we present a collection of results concerning two families of rational orthonormal functions, one on the unit circle, and another on the imaginary axis.
		Special cases of these rational orthonormal functions include the Laguerre and Kautz orthonormal functions, as well as the orthonormal functions recently introduced by Heuberger, Van den Hof and Bosgra.
		Among the results presented herein are completeness and uniform boundedness conditions, and their respective proofs, for the above mentioned families of rational orthonormal functions, as well as some interpolatory properties of truncated orthonormal expansions based on these functions.
	www.ieeta.pt /~tos/bib/8.2.html (435 words)

	Clearing up the market cycle... best Orthonormal Set (Site not responding. Last check: 2007-11-07)
		If Periodia value is 26, that means that from the vary last maximum of its passed 26 ticks of time and we have "period length"/2-26 ticks of time to reverse point.
		An orthogonal set S of unit vectors is an orthonormal set.
		An orthonormal S is an orthonormal basis for Span(S).
	ascot.pl /th/Fourier5/Orthonormal-Set.htm (345 words)

	Abstract: Symmetric real-valued orthonormal scaling functions (Site not responding. Last check: 2007-11-07)
		It is well known that there is no continuous real-valued orthonormal scaling function $\phi\in L_2(\bR)$ such that $\phi$ has compact support and $\phi$ is symmetric about $x=c$ for some $c\in \frac{1}{2}\bZ$.
		More precisely, we prove that there is no {\it continuous} real-valued orthonormal scaling function $\phi\in L_2(\bR^s)$ such that $\phi$ is compactly supported and $\phi$ is symmetric about all the superplanes $x_j=c_j, j=1, \cdots, s$ for some $c_1, \cdots, c_s\in \frac{1}{2}\bZ$.
		Finally, we discuss some properties of real-valued compactly supported orthonormal scaling functions $\phi\in L_2(\bR^s)$ such that $\phi$ is symmetric about a point $x=c$ for some $c\in \frac{1}{2}\bZ^s$.
	www.ualberta.ca /~bhan/papers/symorth2d.abs.html (129 words)

	Inverting an Orthogonal Matrix (Site not responding. Last check: 2007-11-07)
		Therefore the determinant of an orthonormal matrix is
		For instance, write down the x y and z unit vectors, in that order, and obtain a 3×3 matrix with ones down the main diagonal.
		This is an orthonormal matrix with determinant 1.
	www.mathreference.com /la-det,iorth.html (258 words)

	Interrelation between Two-Parametric Models of Random Signals with Limited Energy in Different Orthonormal Bases - ... (Site not responding. Last check: 2007-11-07)
		An interrelation between two-parametric models of random signals with limited energy in different orthonormal bases is established.
		These interrelations are obtained by using the mathematical apparatus of the Hilbert random process spaces over the Hilbert space of samples of the Hilbert space of their weighted samples.
		The energy characteristics of two-parametric models of random signals in arbitrary orthonormal bases are introduced and their interrelation is determined.
	www.begellhouse.com /journals/0632a9d54950b268,73fc5e5373ce7d6e,774c4f134fee31e9.html (140 words)

	Efficient design of orthonormal wavelet bases (Site not responding. Last check: 2007-11-07)
		The efficient representation of a signal as a linear combination of elementary ``atoms'' or building blocks is central to much signal processing theory and many applications.
		In this paper, we develop an efficient method for selecting an orthonormal wavelet which is matched to a given signal, in the sense that the squared error between the signal and some finite resolution wavelet representation of it is minimized.
		Since the squared error is not an explicit function of the design parameters, some form of approximation of this objective is required if conventional optimization techniques are to be used.
	www.ece.mcmaster.ca /~davidson/pubs/wavelet_design.html (223 words)

	Orthonormal Basis Functions for Modelling Continuous-Time Systems (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		Abstract: This paper studies continuous-time system model sets that are spanned by fixed pole orthonormal bases.
		The nature of these bases is such as to generalise the well known Laguerre and two--parameter Kautz bases.
		8 A unified approach to the synthesis of orthonormal exponenti..
	citeseer.ist.psu.edu /198025.html (644 words)

	R: Derive Orthonormal Basis from Wavelet Packet Tree (Site not responding. Last check: 2007-11-07)
		An orthonormal basis for the discrete wavelet transform may be characterized via a disjoint partitioning of the frequency axis that covers [0,1/2).
		This subroutine produces an orthonormal basis from a full wavelet packet tree.
		Boolean vector describing the orthonormal basis for the DWPT.
	www.matematik.lu.se /help/R/.R/library/waveslim/html/ortho.basis.html (141 words)

	Selection of Best Orthonormal Rational Basis
		The model structure is parameterized by a prespecified set of poles representing a finite-dimensional subspace of ${\cal H}^2$.\ Given this structure and experimental data, a model can be estimated using linear regression techniques.
		Since the variance of the estimated model increases with the number of estimated parameters, one objective is to find coordinates, or a basis, for the finite-dimensional subspace giving as compact or parsimonious a system representation as possible.
		In this paper, a best basis algorithm and a coefficient decomposition scheme are derived for the generalized orthonormal rational bases.
	epubs.siam.org /sam-bin/dbq/article/32818 (177 words)

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