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# Topic: Orthogonal functions

###### In the News (Thu 18 Jul 19)

 Orthogonality - One Language   (Site not responding. Last check: 2007-11-01) These functions are orthogonal with respect to a unit weight function on the interval from −1 to 1. are orthogonal with respect to Lebesgue measure on the interval from 0 to 2π. The Legendre polynomials are orthogonal with respect to the uniform distribution on the interval from −1 to 1. www.onelang.com /encyclopedia/index.php/Orthogonality   (1035 words)

 Empirical orthogonal functions -- In statistics and signal processing, the ...   (Site not responding. Last check: 2007-11-01) In statistics and signal processing, the method of empirical orthogonal functions is a decomposition of a signal or data set in terms of orthogonal basis functions which are determined from the data. The ith basis function is chosen to be orthogonal to the basis functions from the first through i − 1, and to minimize the residual variance. The method of empirical orthogonal functions is similar in spirit to harmonic analysis, but harmonic analysis typically uses predetermined orthogonal functions, for example, sine and cosine functions at fixed frequencies. empirical-orthogonal-functions.en.tracking24.net   (247 words)

 Orthogonal matrix   (Site not responding. Last check: 2007-11-01) In three dimesions, the orthogonal matrices with determinant 1 correspond to proper rotation s and those with determinant −1 to improper rotations. The set of all orthogonal matrices whose determinant is 1 is a subgroup of O(n) of index 2, the special orthogonal group SO(n). The complex analog to orthogonal matrices are the unitary matrices. www.serebella.com /encyclopedia/article-Orthogonal_matrix.html   (1095 words)

 Empirical orthogonal functions   (Site not responding. Last check: 2007-11-01) i 'th basis function is chosen to be to the basis functions from the first i -1 and to minimize the residual variance. The method of empirical orthogonal functions is in spirit to harmonic analysis but harmonic analysis typically uses predetermined functions for example sine and cosine functions fixed frequencies. The basis functions are typically found by the eigenvectors of the covariance matrix of the data set. www.freeglossary.com /Empirical_orthogonal_functions   (472 words)

 CORFU project Orthogonal rational functions (ORF) generalize the theory of polynomials orthogonal with respect to a positive measure on the real line or the complex unit circle. Hereby orthogonality was considered with respect to the Lebesgue measure (weight equal to 1) and only one real or a complex conjugate pair of repeated poles is used. At first sight, these applications seem to be quite different, but, although they have each their own specific problems, the fundamental properties of the orthogonal functions involved that have to be investigated are the same: an optimal choice of the poles, properties of the zeros, convergence theory. www.cs.kuleuven.ac.be /~nalag/research/projects/CORFU.shtml   (1937 words)

 Orthogonal functions | TutorGig.co.uk Encyclopedia   (Site not responding. Last check: 2007-11-01) Whether or not two particular functions are orthogonal depends on how their inner product has been defined. Orthogonal Functions in Systems and Control (Advanced Series in Electr.. Christoffel Functions and Orthogonal Polynomials for Exponential Weigh.. www.tutorgig.co.uk /encyclopedia/getdefn.jsp?keywords=Orthogonal_functions   (249 words)

 Method of access control in a communication system using orthogonal functions - Patent 4583217   (Site not responding. Last check: 2007-11-01) A pulse function from a system of orthogonal or substantially orthogonal pulse functions is used, which, by mutual agreement, are recognized by both the central unit and the subscriber stations. Transmitting the orthogonal pulse functions is effected during the total duration of the authorized time slot and is no longer limited to the exceptional case in which a fixed character is transmitted in one of the different call slots. Such orthogonal pulse functions and their evaluation are known (NTG, Signalverarbeitung, Proceedings of the symposium held in Erlangen from Apr. 4-6, 1973 under the same name, published by W. Schussler, a paper on signal theory by Hans Dieter Luke). www.freepatentsonline.com /4583217.html   (2304 words)

 Orthogonal Polynomials and Special Function - Summer Course Hadamard´s Theorem on the radii of m-meromorphy of an analytic function. Polynomials orthogonal with respect to a perturbation of certain classical weight functions by the addition of mass points at the end points of the interval $\Omega\subset\mathbb{R}$ are considered. Using a functional analytic method, based on the three terms recurrence relations that the q-associated polynomials satisfy, we present some monotonicity results and inequalities of the zeros of the orthogonal polynomials under consideration. www.mat.uc.pt /~ajplb/opsf2.htm   (3102 words)

 Functions in TutorGig Encyclopedia Soil functions are general capabilities of soil s that are important for various agriculture agricultural... In mathematics, several function mathematics function s or groups of functions are important enough to deserve their own names. is chosen to be orthogonal to the basis functions from the first through i &minus 1, and to minimize the residual variance. www.tutorgig.com /es/Functions   (786 words)

 Legendre Polynomials Generating functions are available for most orthogonal polynomials, but only in the Legendre case does the generating function have a clear and simple meaning. If you work the functions out explicitly, you will find that the functions for +m and -m are essentially the same, as might be expected, and differ at most by a factor of -1. The functions are not orthogonal for different values of m; orthogonality of the spherical harmonics in this case depends on the φ functions. www.du.edu /~jcalvert/math/legendre.htm   (1164 words)

 Math Forum - Ask Dr. Math   (Site not responding. Last check: 2007-11-01) Two functions are said to be orthogonal over a given interval if integrating their product (over that interval) equals zero. Now, the zero function will be orthogonal to all functions in exactly the same sense that the zero vector is orthogonal to all vectors; the dot product is zero because the length is zero, but it really says nothing about the direction of the other vector, since the zero vector has no direction. Likewise, the zero function is trivially orthogonal to all functions just because it has no "direction". www.mathforum.org /library/drmath/view/52141.html   (333 words)

 Orthogonal   (Site not responding. Last check: 2007-11-01) ORTHOGONAL SOFTWARE CORPORATION PRESENTS: Orthogonal Toolbox v1.5 is a free tool created to assist in the design and development of orthogonal software. Orthogonal polynomials are classes of polynomials {p_n(x)} defined over a range [a,b] that obey an... Related: orthogonal :: orthogonal matrix :: orthogonality :: orthogonal frequency division multiplexing :: orthogonal projection :: orthogonal definition :: orthogonal vectors :: orthogonal array :: orthogonal matrices :: orthogonal vector www.logicjungle.com /wiki/Orthogonal   (261 words)

 Orthogonal group   (Site not responding. Last check: 2007-11-01) In mathematics, the orthogonal group of degree ''n over a field F (written as O(n'',''F)) is the group of n -by- n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. Every orthogonal matrix has determinant either 1 or -1. The orthogonal n -by- n matrices with determinant 1 form a normal subgroup of O(n'',''F) known as the special orthogonal group SO(n'',''F). www.serebella.com /encyclopedia/article-Orthogonal_group.html   (1044 words)

 Abstract: Symmetric multivariate orthogonal refinable functions   (Site not responding. Last check: 2007-11-01) In this paper, we shall investigate the symmetry property of a multivariate orthogonal $M$-refinable function with a general dilation matrix $M$. For an orthogonal $M$-refinable function $\phi$ such that $\phi$ is symmetric about a point (centro-symmetric) and $\phi$ provides approximation order $k$, we show that $\phi$ must be an orthogonal $M$-refinable function that generates a generalized coiflet of order $k$. Next, we show that there does not exist a real-valued compactly supported orthogonal $2 I_s$-refinable function $\phi$ in any dimension such that $\phi$ is symmetric about a point and $\phi$ generates a classical coiflet. www.ualberta.ca /%7Ebhan/abstracts/2003symorth.abs.html   (178 words)

 Statistical Analysis of Chilean Precipitation Anomalies associated with ENSO The cumulative probability distribution functions, derived from the bootstrap analysis, are used to estimate the probability of experiencing a wetter, normal, or drier condition for a specific station, during a given season and ENSO categorization. These cumulative probability distribution functions can be used to estimate the probability of experiencing a wetter, normal, or drier condition for a specific station, during a given season and ENSO categorization. The cumulative probability distribution functions are used to estimate the probability of experiencing a wetter, normal, or drier condition for a specific station, during a given season and ENSO categorization. www.coaps.fsu.edu /papers/statistical_analysis_southern_oscillation   (3652 words)

 Sines and Cosines are Orthogonal   (Site not responding. Last check: 2007-11-01) The fourier transform employs an orthogonal basis, making it easy for a computer to derive the fourier coefficients. Remember, two functions are orthogonal if their dot product is 0, and the dot product of two functions is the integral of their product. The various two dimensional functions, sin(mx)×sin(ny), cos(mx)×sin(ny), cos(mx)×cos(ny), sin(mx)×cos(ny), are all pairwise orthogonal. www.mathreference.com /la-xf-four,orth.html   (330 words)

 Ideas for HF Modulation and Coding   (Site not responding. Last check: 2007-11-01) Plain mark/space FSK is 2-ary orthogonal signalling, assuming the shift is large enough for the signalling rate. In m-ary orthogonal signalling, you have m possible channel states, where m is usually a power of 2. This property means that the rows (and the columns, for that matter) of a Hademard matrix form a set of mutually orthogonal functions. www.qsl.net /n9zia/pkarn.html   (1909 words)

 [No title]   (Site not responding. Last check: 2007-11-01) Clark: Word recognition by means of orthogonal functions. According to 8587 Harmuth, 3, he cites John Barrett as inventor of the transposition of conductors via Walsh functions. "The usefulness of Walsh functions for nonlinear processes was recognized by Weiser in the early sixties but his untimely death delayed the use of his work for several years." (8587 Harmuth) C. felix.unife.it /Root/d-Mathematics/d-Analysis/b-Walsh-functions-applications   (274 words)

 Dover Phoenix Editions Topics include symmetric functions, theory of the compositions of numbers, distributions upon a chessboard, and partition... The axioms that form the basis for the laws of physics, variational principles are usually discussed individually; this compact but far-reaching work surveys the variational principles of several areas and examines their relationships. Topics include symmetric functions, theory of the compositions of numbers, distributions upon a chessboard, and partitions of multipartite numbers. store.doverpublications.com /doverpublications/by-subject-science-and-mathematics-dover-phoenix-editions.html   (1024 words)

 1948 The proof substitutes the sum with a sum of independent random variables that could be in a certain sense considered as first order functional approximations to the multivariate functions. The sum of approximations goes to a Gaussian limit because of central limit theorem and the approximation converges to zero faster than the speed of convergence in the central limit theorem. If the operator formed by covariances of a sequence of functions is bounded then the conditions for a.s. www.io.com /~slava/history/1948.htm   (163 words)

 Graduate Course Descriptions: ENCS   (Site not responding. Last check: 2007-11-01) Functions of one variable, Taylor's series expansion, review of differentiation, integration and solution of ordinary differential equations. Functions of several variables, partial derivatives, multiple integrals, introduction to partial differential equations, wave equation and diffusion equation. Matrix and vector analysis, characteristic value problems, orthogonal functions; introduction to statistics and numerical methods. bcee.concordia.ca /programs/courselists/gradcourseistencs.htm   (725 words)

 SELECTED PUBLICATIONS   (Site not responding. Last check: 2007-11-01) On the Convergence of Rational Approximation of Caratheodory Functions,'' JCAM 54 (1994) 371-376. Strong and Weak Convergence of rational functions orthogonal on Unit Circle,'' J of London Math. On the orthogonal rational functions with arbitrary poles and interpolation properties'' JCAM 60 (1995) 347-355. euclid.barry.edu /~pan/pub.html   (467 words)

 Math 2000 An important formula was derived to demonstrate that the sampling error is relatively insensitive to the exact shapes of empirical orthogonal functions. Two examples are described: the global average of the annual surface air temperature using 63 stations and the regional average of the monthly tropical Pacific sea surface temperature. A systematic theory for the interpolation is described and it uses the emperical orthogonal functions computed from the recent and more accurate observed data. camel.math.ca /CMS/Events/math2000/abs/is.html   (719 words)

 Manuel Alfaro García Alfaro, M. Cantero, L. Moral y L. Velázquez, Hermitian modifications of Toeplitz linear functionals and orthogonal polynomials, enviado para su publicación. Alfaro, M.J. Cantero y L. Moral, Semiorthogonal functions and orthogonal polynomials on the unit circle, J. Alfaro y F. Marcellán, Carathéodory functions and orthogonal polynomials on the unit circle, Complex Methods in Approximation Theory, A. Martínez-Finkelshtein et al. www.unizar.es /analisis_matematico/alfaro/alfaro.html   (400 words)

 Modeled and observed empirical orthogonal functions of currents in the Yucatan Channel, Gulf of Mexico   (Site not responding. Last check: 2007-11-01) Modeled and observed empirical orthogonal functions of currents in the Yucatan Channel, Gulf of Mexico have reported empirical orthogonal function (EOF) analyses based on 23-month current-meter and acoustic Doppler current profiler measurements in the Yucatan Channel. Citation: Oey, L.-Y. er, and W. Sturges (2004), Modeled and observed empirical orthogonal functions of currents in the Yucatan Channel, Gulf of Mexico, J. www.agu.org /pubs/crossref/2004/2004JC002345.shtml   (348 words)

 Orthogonal Functions, Gaussian Quadrature and Fourier Analysis with SCILAB: Infoclearinghouse.com Orthogonal Functions, Gaussian Quadrature and Fourier Analysis with SCILAB: Infoclearinghouse.com Orthogonal Functions, Gaussian Quadrature and Fourier Analysis with SCILAB An alternative for calculating the coefficients of the Fourier series is to use function filonquad, defined earlier for calculating integrals of the form 52 www.infoclearinghouse.com /Merchant2/merchant.mv?Screen=PROD&Product_Code=scilab010&Category_Code=S   (326 words)

 [No title]   (Site not responding. Last check: 2007-11-01) N. Ahmed a.o.: On notation and definition of terms related to a class of complete orthogonal functions. Ze-lin He: On representation of quasi-convex functions by L(R+)-Walsh transforms. Walsh: A closed set of normal orthogonal functions. felix.unife.it /Root/d-Mathematics/d-Analysis/b-Walsh-functions-theory   (395 words)

 Orthogonal functions   (Site not responding. Last check: 2007-11-01) In mathematics, two functions f andg are orthogonal if their innerproduct Whether or nottwo particular functions are orthogonal depends on how their innerproduct has been defined. Solutions of linear differential equations withboundary conditions can often be written as a weighted sum of orthogonal solution functions (a.k.a. www.therfcc.org /orthogonal-functions-214771.html   (73 words)

 [No title]   (Site not responding. Last check: 2007-11-01) The SIAM Activity Group on Orthogonal Polynomials and Special Functions has solicited reactions via the elctronic Newsletter OP-SF Net (freely available for all interested people) and the printed Newsletter of the Activity Group (only for members). On the basis of the reactions we received I have made a comprehensive proposal for changes of those parts of the 1991 Classification which deal with Orthogonal Plynomials and Special Functions (in particular part 33). If basic spherical harmonics mean the elements in irreducible subspaces of the algebra of polynomials on a quantum sphere then these are also covered by 33D80. www.math.yorku.ca /Who/Faculty/Muldoon/siamopsf/misc/mr4.html   (206 words)

 Mailgate: sci.math.num-analysis: Re: Orthogonal polynomials   (Site not responding. Last check: 2007-11-01) Orthogonal functions are a list of functions w0(x), w1(x), w2(x),...wN(x) with the property that an integration over the product of any two of them wi(x)*wj(x) is non-zero only when i=j. Orthogonal functions are useful because it is easy to invert a linear sum of them in a series using this property. Lucas wrote: >Hi, >what are orthogonal polynomials and how they can be used? mailgate.supereva.it /sci/sci.math.num-analysis/msg13593.html   (77 words)

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