Harmonic Analysis a Comprehensive Course in Analysis, Part 3
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Ladder Operators for Lam\'E Spheroconal Harmonic Polynomials
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 8 (2012), 074, 16 pages Ladder Operators for Lam´eSpheroconal Harmonic Polynomials? Ricardo MENDEZ-FRAGOSO´ yz and Eugenio LEY-KOO z y Facultad de Ciencias, Universidad Nacional Aut´onomade M´exico, M´exico E-mail: [email protected] URL: http://sistemas.fciencias.unam.mx/rich/ z Instituto de F´ısica, Universidad Nacional Aut´onomade M´exico, M´exico E-mail: eleykoo@fisica.unam.mx Received July 31, 2012, in final form October 09, 2012; Published online October 17, 2012 http://dx.doi.org/10.3842/SIGMA.2012.074 Abstract. Three sets of ladder operators in spheroconal coordinates and their respective actions on Lam´espheroconal harmonic polynomials are presented in this article. The poly- nomials are common eigenfunctions of the square of the angular momentum operator and of the asymmetry distribution Hamiltonian for the rotations of asymmetric molecules, in the body-fixed frame with principal axes. The first set of operators for Lam´epolynomials of a given species and a fixed value of the square of the angular momentum raise and lower and lower and raise in complementary ways the quantum numbers n1 and n2 counting the respective nodal elliptical cones. The second set of operators consisting of the cartesian components L^x, L^y, L^z of the angular momentum connect pairs of the four species of poly- nomials of a chosen kind and angular momentum. The third set of operators, the cartesian componentsp ^x,p ^y,p ^z of the linear momentum, connect pairs of the polynomials differing in one unit in their angular momentum and in their parities. -
A Very Short Survey on Boundary Behavior of Harmonic Functions
A VERY SHORT SURVEY ON BOUNDARY BEHAVIOR OF HARMONIC FUNCTIONS BLACK Abstract. This short expository paper aims to use Dirichlet boundary value problem to elab- orate on some of the interactions between complex analysis, potential theory, and harmonic analysis. In particular, I will outline Wiener's solution to the Dirichlet problem for a general planar domain using harmonic measure and prove some elementary results for Hardy spaces. 1. Introduction Definition 1.1. Let Ω Ă R2 be an open set. A function u P C2pΩ; Rq is called harmonic if B2u B2u (1.1) ∆u “ ` “ 0 on Ω: Bx2 By2 The notion of harmonic function can be generalized to any finite dimensional Euclidean space (or on (pseudo)Riemannian manifold), but the theory enjoys a qualitative difference in the planar case due to its relation to the magic properties of functions of one complex variable. Think of Ω Ă C (in this paper I will use R2 and C interchangeably when there is no confusion). Then the Laplace operator takes the form B B B 1 B B B 1 B B (1.2) ∆ “ 4 ; where “ ´ i and “ ` i : Bz¯ Bz Bz 2 Bx By Bz¯ 2 Bx By ˆ ˙ ˆ ˙ Let HolpΩq denote the space of holomorphic functions on Ω. It is easy to see that if f P HolpΩq, then both <f and =f are harmonic functions. Conversely, if u is harmonic on Ω and Ω is simply connected, we can construct a harmonic functionu ~, called the harmonic conjugate of u, via Hilbert transform (or more generally, B¨acklund transform), such that f “ u ` iu~ P HolpΩq. -
Chapter 16 Complex Analysis
Chapter 16 Complex Analysis The term \complex analysis" refers to the calculus of complex-valued functions f(z) depending on a complex variable z. On the surface, it may seem that this subject should merely be a simple reworking of standard real variable theory that you learned in ¯rst year calculus. However, this naijve ¯rst impression could not be further from the truth! Com- plex analysis is the culmination of a deep and far-ranging study of the fundamental notions of complex di®erentiation and complex integration, and has an elegance and beauty not found in the more familiar real arena. For instance, complex functions are always ana- lytic, meaning that they can be represented as convergent power series. As an immediate consequence, a complex function automatically has an in¯nite number of derivatives, and di±culties with degree of smoothness, strange discontinuities, delta functions, and other forms of pathological behavior of real functions never arise in the complex realm. There is a remarkable, profound connection between harmonic functions (solutions of the Laplace equation) of two variables and complex-valued functions. Namely, the real and imaginary parts of a complex analytic function are automatically harmonic. In this manner, complex functions provide a rich lode of new solutions to the two-dimensional Laplace equation to help solve boundary value problems. One of the most useful practical consequences arises from the elementary observation that the composition of two complex functions is also a complex function. We interpret this operation as a complex changes of variables, also known as a conformal mapping since it preserves angles. -
Mean Growth of the Derivative of Analytic Functions, Bounded Mean Oscillation and Normal Functions
MEAN GROWTH OF THE DERIVATIVE OF ANALYTIC FUNCTIONS, BOUNDED MEAN OSCILLATION AND NORMAL FUNCTIONS Oscar Blasco, Daniel Girela and M. Auxiliadora Marquez´ Abstract. For a given positive function φ defined in [0, 1) and 1 ≤ p<∞, we consider L the space (p, φ) which consists of all functions f analytic in the unit disc ∆ for which 1/p 1 π iθ p → 2π −π f (re ) dθ =O(φ(r)), as r 1.A result of Bourdon, Shapiro and Sledd 1 −1 implies that such a space is contained in BMOA for φ(r)=(1− r) p .Among other results, in this paper we prove that this result is sharp in a very strong sense, showing that, 1 −1 for a large class of weight functions φ, the function φ(r)=(1− r) p is the best one to 1− 1 get L(p, φ) ⊂ BMOA.Actually, if φ(r)(1 − r) p ↑∞,asr ↑ 1, we construct a function f ∈L(p, φ) which is not a normal function.These results improve other obtained recently by the second author.We also characterize the functions φ, among a certain class of weight functions, to be able to embedd L(p, φ)intoHq for q>por into the space B of Bloch functions. 1. Introduction and statement of results. Let ∆ denote the unit disc {z ∈ C : |z| < 1} and T the unit circle {ξ ∈ C : |ξ| =1}.If 0 <r<1 and g is a function which is analytic in ∆, we set π 1/p 1 iθ p Mp(r, g)= g(re ) dθ , 0 <p<∞, 2π −π M∞(r, g) = max |g(z)|. -
The Halphen Family of Distributions for Modeling Maximum
Transactions on Ecology and the Environment vol 6, © 1994 WIT Press, www.witpress.com, ISSN 1743-3541 The Halphen family of distributions for modeling maximum annual flood series: New developments L. Perreault, V. Fortin & B. Bobee Chair on Statistical Hydrology, 2800 rue Einstein, Sainte-Foy, Quebec, Canada, G1V 4C7 Abstract Design of hydraulic structures, optimal operation of reservoirs, and flood forecasting involve estimation of flood flows x^,, with given specified return period T. This estimation is generally achieved by fitting a probability distribution to observed data. In this paper, we briefly present some recent theoretical results of a study on Halphen distributions which were developed in the early forties but have been little used due to the computational complexity. This family of three distributions was motivated by Halphen's desire to construct probability models which met specific requirements of fitting hydrological variables. Halphen probability models have great potential applicability in statistical flood analysis but are not yet familiar to hydrologists. 1 Introduction Engineering activities such as the design of hydraulic structures, the management of water resources systems, and the prevention of flood damage, all require an estimation of flood characteristics. The objective of flood frequency analysis FFA is to estimate the flood x^ exceeded in average once every T year, which may be used to design, for example, spillways or as prior information in real- time flood forcasting models. At gauged locations, x? may be estimated by fitting a parametric distribution F(x; 0) to observed maximum annual floods, where x refer to an observation and 0 to a vector of parameters. -
Harmonic Measurement and Reduction in Power Systems. Rutisurhata Kurniawan Hartana Louisiana State University and Agricultural & Mechanical College
Louisiana State University LSU Digital Commons LSU Historical Dissertations and Theses Graduate School 1989 Harmonic Measurement and Reduction in Power Systems. Rutisurhata Kurniawan Hartana Louisiana State University and Agricultural & Mechanical College Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_disstheses Recommended Citation Hartana, Rutisurhata Kurniawan, "Harmonic Measurement and Reduction in Power Systems." (1989). LSU Historical Dissertations and Theses. 4849. https://digitalcommons.lsu.edu/gradschool_disstheses/4849 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Historical Dissertations and Theses by an authorized administrator of LSU Digital Commons. For more information, please contact [email protected]. INFORMATION TO USERS The most advanced technology has been used to photograph and reproduce this manuscript from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely afreet reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand corner and continuing from left to right in equal sections with small overlaps. -
Lecture Note for Math 220A Complex Analysis of One Variable
Lecture Note for Math 220A Complex Analysis of One Variable Song-Ying Li University of California, Irvine Contents 1 Complex numbers and geometry 2 1.1 Complex number field . 2 1.2 Geometry of the complex numbers . 3 1.2.1 Euler's Formula . 3 1.3 Holomorphic linear factional maps . 6 1.3.1 Self-maps of unit circle and the unit disc. 6 1.3.2 Maps from line to circle and upper half plane to disc. 7 2 Smooth functions on domains in C 8 2.1 Notation and definitions . 8 2.2 Polynomial of degree n ...................... 9 2.3 Rules of differentiations . 11 3 Holomorphic, harmonic functions 14 3.1 Holomorphic functions and C-R equations . 14 3.2 Harmonic functions . 15 3.3 Translation formula for Laplacian . 17 4 Line integral and cohomology group 18 4.1 Line integrals . 18 4.2 Cohomology group . 19 4.3 Harmonic conjugate . 21 1 5 Complex line integrals 23 5.1 Definition and examples . 23 5.2 Green's theorem for complex line integral . 25 6 Complex differentiation 26 6.1 Definition of complex differentiation . 26 6.2 Properties of complex derivatives . 26 6.3 Complex anti-derivative . 27 7 Cauchy's theorem and Morera's theorem 31 7.1 Cauchy's theorems . 31 7.2 Morera's theorem . 33 8 Cauchy integral formula 34 8.1 Integral formula for C1 and holomorphic functions . 34 8.2 Examples of evaluating line integrals . 35 8.3 Cauchy integral for kth derivative f (k)(z) . 36 9 Application of the Cauchy integral formula 36 9.1 Mean value properties . -
An Algebraic Approach to Harmonic Polynomials on S3
AN ALGEBRAIC APPROACH TO HARMONIC POLYNOMIALS ON S3 Kevin Mandira Limanta A thesis in fulfillment of the requirements for the degree of Master of Science (by Research) SCHOOL OF MATHEMATICS AND STATISTICS FACULTY OF SCIENCE UNIVERSITY OF NEW SOUTH WALES June 2017 ORIGINALITY STATEMENT 'I hereby declare that this submission is my own work and to the best of my knowledge it contains no materials previously published or written by another person, or substantial proportions of material which have been accepted for the award of any other degree or diploma at UNSW or any other educational institution, except where due acknowledgement is made in the thesis. Any contribution made to the research by others, with whom I have worked at UNSW or elsewhere, is explicitly acknowledged in the thesis. I also declare that the intellectual content of this thesis is the product of my own work, except to the extent that assistance from others in the project's design and conception or in style, presentation and linguistic expression is acknowledged.' Signed .... Date Show me your ways, Lord, teach me your paths. Guide me in your truth and teach me, for you are God my Savior, and my hope is in you all day long. { Psalm 25:4-5 { i This is dedicated for you, Papa. ii Acknowledgement This thesis is the the result of my two years research in the School of Mathematics and Statistics, University of New South Wales. I learned quite a number of life lessons throughout my entire study here, for which I am very grateful of. -
Spherical Harmonics and Homogeneous Har- Monic Polynomials
SPHERICAL HARMONICS AND HOMOGENEOUS HAR- MONIC POLYNOMIALS 1. The spherical Laplacean. Denote by S ½ R3 the unit sphere. For a function f(!) de¯ned on S, let f~ denote its extension to an open neighborhood N of S, constant along normals to S (i.e., constant along rays from the origin). We say f 2 C2(S) if f~ is a C2 function in N , and for such functions de¯ne a di®erential operator ¢S by: ¢Sf := ¢f;~ where ¢ on the right-hand side is the usual Laplace operator in R3. With a little work (omitted here) one may derive the expression for ¢ in polar coordinates (r; !) in R2 (r > 0;! 2 S): 2 1 ¢u = u + u + ¢ u: rr r r r2 S (Here ¢Su(r; !) is the operator ¢S acting on the function u(r; :) in S, for each ¯xed r.) A homogeneous polynomial of degree n ¸ 0 in three variables (x; y; z) is a linear combination of `monomials of degree n': d1 d2 d3 x y z ; di ¸ 0; d1 + d2 + d3 = n: This de¯nes a vector space (over R) denoted Pn. A simple combinatorial argument (involving balls and separators, as most of them do), seen in class, yields the dimension: 1 d := dim(P ) = (n + 1)(n + 2): n n 2 Writing a polynomial p 2 Pn in polar coordinates, we necessarily have: n p(r; !) = r f(!); f = pjS; where f is the restriction of p to S. This is an injective linear map p 7! f, but the functions on S so obtained are rather special (a dn-dimensional subspace of the in¯nite-dimensional space C(S) of continuous functions-let's call it Pn(S)) We are interested in the subspace Hn ½ Pn of homogeneous harmonic polynomials of degree n (¢p = 0). -
Research.Pdf (1.262Mb)
Lossless Data Compression of Monitored Power Signals Using the PLHaar Transform A Thesis Presented to the Faculty of the Graduate School at the University of Missouri{Columbia In Partial Fulfillment of the Requirements for the Degree Master of Science By Nicholas J. Hudson Dr. Michael J. Devaney, Thesis Supervisor July 2015 c Nicholas Hudson 2015, All rights reserved. The undersigned have examined the thesis entitled Lossless Data Compression of Monitored Power Signals Using the PLHaar Transform presented by Nicholas Hudson and determined it worthy of acceptance. Professor Michael J. Devaney Professor Zhihai "Henry" He Professor Jeffrey Uhlmann for the old man Acknowledgements I've unmeasurable gratitude to my advisor Dr. Michael Devaney. Without his boundless patience, breadth of knowledge, excellent advice, and boundless patience not only would this work have been not just impossible but unimaginable to me. I'm both deeply honored and fantastically fortunate to be his student. I'd also like to thank Dr. James Keller, who not only introduced me to the non-Euclidean spaces fundamental to this work, but also extended my education with a philosophical ele- ment of value nigh-impossible to overstate. And I'm deeply indebted to my mother and brother, whose faith in me served as bedrock to this research. ii Contents Acknowledgements ii Contents iii List of Figures v List of Tables vi 1 Introduction 1 1.1 Motivation . .1 1.2 Relevance . .2 1.3 Proposed Approach . .4 1.4 Organization . .4 2 Background 5 2.1 History . .5 2.1.1 Fourier . .5 2.1.2 Windowed Transforms . -
The Schur Lemma
Appendix A The Schur Lemma Schur’s lemma provides sufficient conditions for linear operators to be bounded on Lp. Moreover, for positive operators it provides necessary and sufficient such condi- tions. We discuss these situations. A.1 The Classical Schur Lemma We begin with an easy situation. Suppose that K(x,y) is a locally integrable function on a product of two σ-finite measure spaces (X, μ) × (Y,ν), and let T be a linear operator given by T( f )(x)= K(x,y) f (y)dν(y) Y when f is bounded and compactly supported. It is a simple consequence of Fubini’s theorem that for almost all x ∈ X the integral defining T converges absolutely. The following lemma provides a sufficient criterion for the Lp boundedness of T. Lemma. Suppose that a locally integrable function K(x,y) satisfies sup |K(x,y)|dν(y)=A < ∞, x∈X Y sup |K(x,y)|dμ(x)=B < ∞. y∈Y X Then the operator T extends to a bounded operator from Lp(Y) to Lp(X) with norm − 1 1 A1 p B p for 1 ≤ p ≤ ∞. Proof. The second condition gives that T maps L1 to L1 with bound B, while the first condition gives that T maps L∞ to L∞ with bound A. It follows by the Riesz– − 1 1 Thorin interpolation theorem that T maps Lp to Lp with bound A1 p B p . This lemma can be improved significantly when the operators are assumed to be positive. A.2 Schur’s Lemma for Positive Operators We have the following necessary and sufficient condition for the Lp boundedness of positive operators. -
On the Equivalence Between Weak BMO and the Space of Derivatives of the Zygmund Class
On the equivalence between Weak BMO and the space of derivatives of the Zygmund class Eddy Kwessi* Abstract In this paper, we will discuss the space of functions of weak bounded mean oscillation. In particular, we will show that this space is the dual space of the special atom space, whose dual space was already known to be the space of derivative of functions (in the sense of distribution) belonging to the Zygmund class of functions. We show in particular that this proves that the Hardy space H1 strictly contains the space special atom space. Keywords: BMO, Derivative, Distributions, Zygmund Class AMS Subject Classification: 42B05, 42B30, 30B50, 30E20 1 Introduction The space of functions of bounded means oscillation has taken central stage in the mathematical litera- ture after the work of Charles Fefferman [4] where he showed that it is the dual space of the real Hardy 1 space H , a long sought after result. Right after, Ronald Coifman [2] showed this result using a different 1 method. Essentially, he showed that H has an atomic decomposition. De Souza [3] showed there is a 1 1 1 1 subset B of H , formed by special atoms that is contained in H . This space B has the particularity that it contains some functions whose Fourier series diverge. The question of whether B1 is equivalent 1 1 1 to H was never truly answered explicitly but it was always suspected that the inclusion B ⊂ H was 1 1 strict, that is, there must be at least one function in H that is not in B .