Sturm Liouville Theory Orthogonal Functions
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EOF/PC Analysis
ATM 552 Notes: Matrix Methods: EOF, SVD, ETC. D.L.Hartmann Page 68 4. Matrix Methods for Analysis of Structure in Data Sets: Empirical Orthogonal Functions, Principal Component Analysis, Singular Value Decomposition, Maximum Covariance Analysis, Canonical Correlation Analysis, Etc. In this chapter we discuss the use of matrix methods from linear algebra, primarily as a means of searching for structure in data sets. Empirical Orthogonal Function (EOF) analysis seeks structures that explain the maximum amount of variance in a two-dimensional data set. One dimension in the data set represents the dimension in which we are seeking to find structure, and the other dimension represents the dimension in which realizations of this structure are sampled. In seeking characteristic spatial structures that vary with time, for example, we would use space as the structure dimension and time as the sampling dimension. The analysis produces a set of structures in the first dimension, which we call the EOF’s, and which we can think of as being the structures in the spatial dimension. The complementary set of structures in the sampling dimension (e.g. time) we can call the Principal Components (PC’s), and they are related one-to-one to the EOF’s. Both sets of structures are orthogonal in their own dimension. Sometimes it is helpful to sacrifice one or both of these orthogonalities to produce more compact or physically appealing structures, a process called rotation of EOF’s. Singular Value Decomposition (SVD) is a general decomposition of a matrix. It can be used on data matrices to find both the EOF’s and PC’s simultaneously. -
Complete Set of Orthogonal Basis Mathematical Physics
R. I. Badran Complete set of orthogonal basis Mathematical Physics Complete set of orthogonal functions Discrete set of vectors: ˆ ˆ ˆ ˆ ˆ ˆ The two vectors A AX i A y j Az k and B BX i B y j Bz k are 3 orthogonal if A B 0 or Ai Bi 0 . i1 Continuous set of functions on an interval (a, b): a) The two continuous functions A(x) and B (x) are orthogonal b on the interval (a, b) if A(x)B(x)dx 0. a b) The two complex functions A (x) and B (x) are orthogonal on b the interval (a, b) if A (x)B(x)dx 0 , where A*(x) is the a complex conjugate of A (x). c) For a whole set of functions An (x) (where n= 1, 2, 3,) and on the interval (a, b) b 0 if m n An (x)Am (x)dx a , const.t 0 if m n An (x) is called a set of orthogonal functions. Examples: 0 m n sin nx sin mxdx i) , m n 0 where sin nx is a set of orthogonal functions on the interval (-, ). Similarly 0 m n cos nx cos mxdx if if m n 0 R. I. Badran Complete set of orthogonal basis Mathematical Physics ii) sin nx cos mxdx 0 for any n and m 0 (einx ) eimxdx iii) 2 1 vi) P (x)Pm (x)dx 0 unless. m 1 [Try to prove this; also solve problems (2, 5) of section 6]. -
32 FA15 Abstracts
32 FA15 Abstracts IP1 metric simple exclusion process and the KPZ equation. In Vector-Valued Nonsymmetric and Symmetric Jack addition, the experiments of Takeuchi and Sano will be and Macdonald Polynomials briefly discussed. For each partition τ of N there are irreducible representa- Craig A. Tracy tions of the symmetric group SN and the associated Hecke University of California, Davis algebra HN (q) on a real vector space Vτ whose basis is [email protected] indexed by the set of reverse standard Young tableaux of shape τ. The talk concerns orthogonal bases of Vτ -valued polynomials of x ∈ RN . The bases consist of polyno- IP6 mials which are simultaneous eigenfunctions of commuta- Limits of Orthogonal Polynomials and Contrac- tive algebras of differential-difference operators, which are tions of Lie Algebras parametrized by κ and (q, t) respectively. These polynomi- als reduce to the ordinary Jack and Macdonald polynomials In this talk, I will discuss the connection between superin- when the partition has just one part (N). The polynomi- tegrable systems and classical systems of orthogonal poly- als are constructed by means of the Yang-Baxter graph. nomials in particular in the expansion coefficients between There is a natural bilinear form, which is positive-definite separable coordinate systems, related to representations of for certain ranges of parameter values depending on τ,and the (quadratic) symmetry algebras. This connection al- there are integral kernels related to the bilinear form for lows us to extend the Askey scheme of classical orthogonal the group case, of Gaussian and of torus type. The mate- polynomials and the limiting processes within the scheme. -
Orthogonal Functions: the Legendre, Laguerre, and Hermite Polynomials
ORTHOGONAL FUNCTIONS: THE LEGENDRE, LAGUERRE, AND HERMITE POLYNOMIALS THOMAS COVERSON, SAVARNIK DIXIT, ALYSHA HARBOUR, AND TYLER OTTO Abstract. The Legendre, Laguerre, and Hermite equations are all homogeneous second order Sturm-Liouville equations. Using the Sturm-Liouville Theory we will be able to show that polynomial solutions to these equations are orthogonal. In a more general context, finding that these solutions are orthogonal allows us to write a function as a Fourier series with respect to these solutions. 1. Introduction The Legendre, Laguerre, and Hermite equations have many real world practical uses which we will not discuss here. We will only focus on the methods of solution and use in a mathematical sense. In solving these equations explicit solutions cannot be found. That is solutions in in terms of elementary functions cannot be found. In many cases it is easier to find a numerical or series solution. There is a generalized Fourier series theory which allows one to write a function f(x) as a linear combination of an orthogonal system of functions φ1(x),φ2(x),...,φn(x),... on [a; b]. The series produced is called the Fourier series with respect to the orthogonal system. While the R b a f(x)φn(x)dx coefficients ,which can be determined by the formula cn = R b 2 , a φn(x)dx are called the Fourier coefficients with respect to the orthogonal system. We are concerned only with showing that the Legendre, Laguerre, and Hermite polynomial solutions are orthogonal and can thus be used to form a Fourier series. In order to proceed we must define an inner product and define what it means for a linear operator to be self- adjoint. -
Linear Operators
C H A P T E R 10 Linear Operators Recall that a linear transformation T ∞ L(V) of a vector space into itself is called a (linear) operator. In this chapter we shall elaborate somewhat on the theory of operators. In so doing, we will define several important types of operators, and we will also prove some important diagonalization theorems. Much of this material is directly useful in physics and engineering as well as in mathematics. While some of this chapter overlaps with Chapter 8, we assume that the reader has studied at least Section 8.1. 10.1 LINEAR FUNCTIONALS AND ADJOINTS Recall that in Theorem 9.3 we showed that for a finite-dimensional real inner product space V, the mapping u ’ Lu = Óu, Ô was an isomorphism of V onto V*. This mapping had the property that Lauv = Óau, vÔ = aÓu, vÔ = aLuv, and hence Lau = aLu for all u ∞ V and a ∞ ®. However, if V is a complex space with a Hermitian inner product, then Lauv = Óau, vÔ = a*Óu, vÔ = a*Luv, and hence Lau = a*Lu which is not even linear (this was the definition of an anti- linear (or conjugate linear) transformation given in Section 9.2). Fortunately, there is a closely related result that holds even for complex vector spaces. Let V be finite-dimensional over ç, and assume that V has an inner prod- uct Ó , Ô defined on it (this is just a positive definite Hermitian form on V). Thus for any X, Y ∞ V we have ÓX, YÔ ∞ ç. -
A Complete System of Orthogonal Step Functions
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 132, Number 12, Pages 3491{3502 S 0002-9939(04)07511-2 Article electronically published on July 22, 2004 A COMPLETE SYSTEM OF ORTHOGONAL STEP FUNCTIONS HUAIEN LI AND DAVID C. TORNEY (Communicated by David Sharp) Abstract. We educe an orthonormal system of step functions for the interval [0; 1]. This system contains the Rademacher functions, and it is distinct from the Paley-Walsh system: its step functions use the M¨obius function in their definition. Functions have almost-everywhere convergent Fourier-series expan- sions if and only if they have almost-everywhere convergent step-function-series expansions (in terms of the members of the new orthonormal system). Thus, for instance, the new system and the Fourier system are both complete for Lp(0; 1); 1 <p2 R: 1. Introduction Analytical desiderata and applications, ever and anon, motivate the elaboration of systems of orthogonal step functions|as exemplified by the Haar system, the Paley-Walsh system and the Rademacher system. Our motivation is the example- based classification of digitized images, represented by rational points of the real interval [0,1], the domain of interest in the sequel. It is imperative to establish the \completeness" of any orthogonal system. Much is known, in this regard, for the classical systems, as this has been the subject of numerous investigations, and we use the latter results to establish analogous properties for the orthogonal system developed herein. h i RDefinition 1. Let the inner product f(x);g(x) denote the Lebesgue integral 1 0 f(x)g(x)dx: Definition 2. -
On Some Extensions of Bernstein's Inequality for Self-Adjoint Operators
On Some Extensions of Bernstein's Inequality for Self-Adjoint Operators Stanislav Minsker∗ e-mail: [email protected] Abstract: We present some extensions of Bernstein's concentration inequality for random matrices. This inequality has become a useful and powerful tool for many problems in statis- tics, signal processing and theoretical computer science. The main feature of our bounds is that, unlike the majority of previous related results, they do not depend on the dimension d of the ambient space. Instead, the dimension factor is replaced by the “effective rank" associated with the underlying distribution that is bounded from above by d. In particular, this makes an extension to the infinite-dimensional setting possible. Our inequalities refine earlier results in this direction obtained by D. Hsu, S. M. Kakade and T. Zhang. Keywords and phrases: Bernstein's inequality, Random Matrix, Effective rank, Concen- tration inequality, Large deviations. 1. Introduction Theoretical analysis of many problems in applied mathematics can often be reduced to problems about random matrices. Examples include numerical linear algebra (randomized matrix decom- positions, approximate matrix multiplication), statistics and machine learning (low-rank matrix recovery, covariance estimation), mathematical signal processing, among others. n P Often, resulting questions can be reduced to estimates for the expectation E Xi − EXi or i=1 n P n probability P Xi − EXi > t , where fXigi=1 is a finite sequence of random matrices and i=1 k · k is the operator norm. Some of the early work on the subject includes the papers on noncom- mutative Khintchine inequalities [LPP91, LP86]; these tools were used by M. -
Generalizations and Specializations of Generating Functions for Jacobi, Gegenbauer, Chebyshev and Legendre Polynomials with Definite Integrals
Journal of Classical Analysis Volume 3, Number 1 (2013), 17–33 doi:10.7153/jca-03-02 GENERALIZATIONS AND SPECIALIZATIONS OF GENERATING FUNCTIONS FOR JACOBI, GEGENBAUER, CHEBYSHEV AND LEGENDRE POLYNOMIALS WITH DEFINITE INTEGRALS HOWARD S. COHL AND CONNOR MACKENZIE Abstract. In this paper we generalize and specialize generating functions for classical orthogo- nal polynomials, namely Jacobi, Gegenbauer, Chebyshev and Legendre polynomials. We derive a generalization of the generating function for Gegenbauer polynomials through extension a two element sequence of generating functions for Jacobi polynomials. Specializations of generat- ing functions are accomplished through the re-expression of Gauss hypergeometric functions in terms of less general functions. Definite integrals which correspond to the presented orthogonal polynomial series expansions are also given. 1. Introduction This paper concerns itself with analysis of generating functions for Jacobi, Gegen- bauer, Chebyshev and Legendre polynomials involving generalization and specializa- tion by re-expression of Gauss hypergeometric generating functions for these orthog- onal polynomials. The generalizations that we present here are for two of the most important generating functions for Jacobi polynomials, namely [4, (4.3.1–2)].1 In fact, these are the first two generating functions which appear in Section 4.3 of [4]. As we will show, these two generating functions, traditionally expressed in terms of Gauss hy- pergeometric functions, can be re-expressed in terms of associated Legendre functions (and also in terms of Ferrers functions, associated Legendre functions on the real seg- ment ( 1,1)). Our Jacobi polynomial generating function generalizations, Theorem 1, Corollary− 1 and Corollary 2, generalize the generating function for Gegenbauer polyno- mials. -
MATRICES WHOSE HERMITIAN PART IS POSITIVE DEFINITE Thesis by Charles Royal Johnson in Partial Fulfillment of the Requirements Fo
MATRICES WHOSE HERMITIAN PART IS POSITIVE DEFINITE Thesis by Charles Royal Johnson In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1972 (Submitted March 31, 1972) ii ACKNOWLEDGMENTS I am most thankful to my adviser Professor Olga Taus sky Todd for the inspiration she gave me during my graduate career as well as the painstaking time and effort she lent to this thesis. I am also particularly grateful to Professor Charles De Prima and Professor Ky Fan for the helpful discussions of my work which I had with them at various times. For their financial support of my graduate tenure I wish to thank the National Science Foundation and Ford Foundation as well as the California Institute of Technology. It has been important to me that Caltech has been a most pleasant place to work. I have enjoyed living with the men of Fleming House for two years, and in the Department of Mathematics the faculty members have always been generous with their time and the secretaries pleasant to work around. iii ABSTRACT We are concerned with the class Iln of n><n complex matrices A for which the Hermitian part H(A) = A2A * is positive definite. Various connections are established with other classes such as the stable, D-stable and dominant diagonal matrices. For instance it is proved that if there exist positive diagonal matrices D, E such that DAE is either row dominant or column dominant and has positive diag onal entries, then there is a positive diagonal F such that FA E Iln. -
ECE 487 Lecture 3 : Foundations of Quantum Mechanics II
ECE 487 Lecture 12 : Tools to Understand Nanotechnology III Class Outline: •Inverse Operator •Unitary Operators •Hermitian Operators Things you should know when you leave… Key Questions • What is an inverse operator? • What is the definition of a unitary operator? • What is a Hermetian operator and what do we use them for? M. J. Gilbert ECE 487 – Lecture 1 2 02/24/1 1 Tools to Understand Nanotechnology- III Let’s begin with a problem… Prove that the sum of the modulus squared of the matrix elements of a linear operator  is independent of the complete orthonormal basis used to represent the operator. M. J. Gilbert ECE 487 – Lecture 1 2 02/24/1 1 Tools to Understand Nanotechnology- III Last time we considered operators and how to represent them… •Most of the time we spent trying to understand the most simple of the operator classes we consider…the identity operator. •But he was only one of four operator classes to consider. 1. The identity operator…important for operator algebra. 2. Inverse operators…finding these often solves a physical problem mathematically and the are also important in operator algebra. 3. Unitary operators…these are very useful for changing the basis for representing the vectors and describing the evolution of quantum mechanical systems. 4. Hermitian operators…these are used to represent measurable quantities in quantum mechanics and they have some very powerful mathematical properties. M. J. Gilbert ECE 487 – Lecture 1 2 02/24/1 1 Tools to Understand Nanotechnology- III So, let’s move on to start discussing the inverse operator… Consider an operator  operating on some arbitrary function |f>. -
Arxiv:2008.08079V2 [Math.FA] 29 Dec 2020 Hypergroups Is Not Required)
HARMONIC ANALYSIS OF LITTLE q-LEGENDRE POLYNOMIALS STEFAN KAHLER Abstract. Many classes of orthogonal polynomials satisfy a specific linearization prop- erty giving rise to a polynomial hypergroup structure, which offers an elegant and fruitful link to harmonic and functional analysis. From the opposite point of view, this allows regarding certain Banach algebras as L1-algebras, associated with underlying orthogonal polynomials or with the corresponding orthogonalization measures. The individual be- havior strongly depends on these underlying polynomials. We study the little q-Legendre polynomials, which are orthogonal with respect to a discrete measure. Their L1-algebras have been known to be not amenable but to satisfy some weaker properties like right character amenability. We will show that the L1-algebras associated with the little q- Legendre polynomials share the property that every element can be approximated by linear combinations of idempotents. This particularly implies that these L1-algebras are weakly amenable (i. e., every bounded derivation into the dual module is an inner deriva- tion), which is known to be shared by any L1-algebra of a locally compact group. As a crucial tool, we establish certain uniform boundedness properties of the characters. Our strategy relies on continued fractions, character estimations and asymptotic behavior. 1. Introduction 1.1. Motivation. One of the most famous results of mathematics, the ‘Banach–Tarski paradox’, states that any ball in d ≥ 3 dimensions can be split into a finite number of pieces in such a way that these pieces can be reassembled into two balls of the original size. It is also well-known that there is no analogue for d 2 f1; 2g, and the Banach–Tarski paradox heavily relies on the axiom of choice [37]. -
Shifted Jacobi Polynomials and Delannoy Numbers
SHIFTED JACOBI POLYNOMIALS AND DELANNOY NUMBERS GABOR´ HETYEI A` la m´emoire de Pierre Leroux Abstract. We express a weigthed generalization of the Delannoy numbers in terms of shifted Jacobi polynomials. A specialization of our formulas extends a relation between the central Delannoy numbers and Legendre polynomials, observed over 50 years ago [8], [13], [14], to all Delannoy numbers and certain Jacobi polynomials. Another specializa- tion provides a weighted lattice path enumeration model for shifted Jacobi polynomials and allows the presentation of a combinatorial, non-inductive proof of the orthogonality of Jacobi polynomials with natural number parameters. The proof relates the orthogo- nality of these polynomials to the orthogonality of (generalized) Laguerre polynomials, as they arise in the theory of rook polynomials. We also find finite orthogonal polynomial sequences of Jacobi polynomials with negative integer parameters and expressions for a weighted generalization of the Schr¨odernumbers in terms of the Jacobi polynomials. Introduction It has been noted more than fifty years ago [8], [13], [14] that the diagonal entries of the Delannoy array (dm,n), introduced by Henri Delannoy [5], and the Legendre polynomials Pn(x) satisfy the equality (1) dn,n = Pn(3), but this relation was mostly considered a “coincidence”. An important observation of our present work is that (1) can be extended to (α,0) (2) dn+α,n = Pn (3) for all α ∈ Z such that α ≥ −n, (α,0) where Pn (x) is the Jacobi polynomial with parameters (α, 0). This observation in itself is a strong indication that the interaction between Jacobi polynomials (generalizing Le- gendre polynomials) and the Delannoy numbers is more than a mere coincidence.