INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS


 Nathaniel Jenkins
 2 years ago
 Views:
Transcription
1 #. INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS Mark A. Pinsky Northwestern University BROOKS/COLE * THOMSON LEARNING Australia Canada Mexico Singapore Spain United Kingdom United States
2 1 FOURIER SERIES ON THE CIRCLE Motivation and Heuristics Motivation from Physics The Vibrating String Heat Flow in Solids Absolutely Convergent Trigonometrie Series *Examples of Factorial and Bessel Functions Poisson Kernel Example *Proof of Laplace's Method *Nonabsolutely Convergent Trigonometrie Series Formulation of Fourier Series Fourier Coefficients and Their Basic Properties Fourier Series of Finite Measures *Rates of Decay of Fourier Coefficients Piecewise Smooth Functions Fourier Characterization of Analytic Functions Sine Integral Öther Proofs That Si(oo) = Pointwise Convergence Criteria * Integration of Fourier Series Convergence of Fourier Series of Measures Riemann Localization Principle GibbsWilbraham Phenomenon The General Case Fourier Series in L Mean Square Approximation Parseval's Theorem * Application to the Isoperimetric Inequality 38 ix
3 * *Rates of Convergence in L Application to AbsolutelyConvergent Fourier Series Norm Convergence and Summability Approximate Identities AlmostEverywhere Convergence of the Abel Means Summability Matrices Fejer Means of a Fourier Series Wiener's Closure Theorem on the Circle *Equidistribution Modulo One *Hardy's Tauberian Theorem Improved Trigonometrie Approximation Rates of Convergence in C(T) Approximation with Fejer Means *Jackson's Theorem *HigherOrder Approximation *Converse Theorems of Bernstein Divergence of Fourier Series The Example of du BoisReymond Analysis via Lebesgue Constants Divergence in the Space L * Appendix: Cqmplements on Laplace's Method First Variation on the ThemeGaussian Approximation Second Variation on the ThemeImproved Error Estimate * Application to Bessel Functions *The Local Limit Theorem of DeMoivreLaplace Appendix: Proof of the Uniform Boundedness Theorem * Appendix: HigherOrder Bessel funetions Appendix: Cantor's Uniqueness Theorem 86 2 FOURIER TRANSFORMS ON THE LINE AND SPACE 2.1 Motivation and Heuristics 2.2 Basic Properties of the Fourier Transform RiemannLebesgue Lemma Approximate Identities and Gaussian Summability Improved Approximate Identities for Pointwise Convergence Application to the Fourier Transform The ndimensional Poisson Kernel
4 fc XI Fourier Transforms of Tempered Distributions *Characterization of the Gaussian Density *Wiener's Density Theorem Fourier Inversion in One Dimension Dirichlet Kernel and Symmetrie Partial Sums Example of the Indicator Function GibbsWilbraham Phenomenon Dini Convergence Theorem Extension to Fourier's Single Integral Smoothing Operations in R'Averaging and Summability Averaging and Weak Convergence Cesäro Summability Approximation Properties of the Fejer Kernel Bernstein's Inequality *OneSided Fourier Integral Representation Fourier Cosine Transform Fourier Sine Transform Generalized /ztransform L 2 Theory in K" Plancherel's Theorem *Bernstein's Theorem for Fourier Transforms The Uncertainty Principle Uncertainty Principle on the Circle Spectral Analysis of the Fourier Transform Hermite Polynomials Eigenfunction of the Fourier Transform Orthogonality Properties Completeness Spherical Fourier Inversion in E" Bochner's Approach Piecewise Smooth Viewpoint Relations with the Wave Equation The Method of Brandolini and Colzani BochnerRiesz Summability A General Theorem on AlmostEverywhere Summability Bessel Functions Fourier Transforms of Radial Functions L 2 Restriction Theorems for the Fourier Transform An Improved Result Limitations on the Range of p The Method of Stationary Phase Statement of the Result Application to Bessel Functions Proof of the Method of Stationary Phase Abel's Lemma 167
5 v Xll 3 FOURIER ANALYSIS IN Iß SPACES Motivation and Heuristics The M. RieszThorin Interpolation Theorem Generalized Young's Inequality The HausdorffYoung Inequality Stein's Complex Interpolation Theorem The Conjugate Function or Discrete Hilbert Transform LP Theory of the Conjugate Function L 1 Theory of the Conjugate Function Identification as a Singular Integral The Hilbert Transform on K L? Theory of the Hilbert Transform LP Theory of the Hilbert Transform, 1 < p < co Applications to Convergence of Fourier Integrals L 1 Theory of the Hilbert Transform and Extensions Kolmogorov's Inequality for the Hilbert Transform Application to Singular Integrals with Odd Kernels HardyLittlewood Maximal Function Application to the Lebesgue Differentiation Theorem Application to Radial Convolution Operators Maximal Inequalities for Spherical Averages The Marcinkiewicz Interpolation Theorem CalderönZygmund Decomposition A Class of Singular Integrals Properties of Harmonie Functions General Properties Representation Theorems in the Disk Representation Theorems in the Upper HalfPlane Herglotz/Bochner Theorems and Positive Definite Functions 219 POISSON SUMMATION FORMULA AND MULTIPLE FOURIER SERIES Motivation and Heuristics ThePoissonSummationFormulainM Periodization of a Function Statement and Proof Shannon Sampling Multiple Fourier Series Basic L 1 Theory Pointwise Convergence for Smooth Functions Representation of Spherical Partial Sums 233
6 % XÜi Basic L 2 Theory Restriction Theorems for Fourier Coefncients Poisson Summation Formula in R ä *Simultaneous Nonlocalization Application to Lattice Points Kendall's Mean Square Error Landau's Asymptotic Formula Application to Multiple Fourier Series ThreeDimensional Case HigherDimensional Case Schrödinger Equation and Gauss Sums Distributions on the Circle The Schrödinger Equation on the Circle Recurrence of Random Walk APPLICATIONS TO PROBABILITY THEORY Motivation and Heuristics Basic Definitions The Central Limit Theorem Restatement in Terms of Independent Random Variables Extension to Gap Series Extension to Abel Sums Weak Convergence of Measures An Improved Continuity Theorem Another Proof of Bochner's Theorem Convolution Semigroups The BerryEsseen Theorem Extension to Different Distributions The Law of the Iterated Logarifhm INTRODUCTION TO WAVELETS Motivation and Heuristics Heuristic Treatment of the Wavelet Transform Wavelet Transform Wavelet Characterization of Smoothness Haar Wavelet Expansion Haar Functions and Haar Series Haar Sums and Dyadic Projections Completeness of the Haar Functions Haar Series in Co and L p Spaces Pointwise Convergence of Haar Series 298
7 XIV *Construction of Standard Brownian Motion *Haar Function Representation of Brownian Motion *Proof of Continuity *Levy's Modulus of Continuity Multiresolution Analysis Orthonormal Systems and Riesz Systems Scaling Equations and Structure Constants From Scaling Function to MRA Additional Remarks Meyer Wavelets From Scaling Function to Orthonormal Wavelet Direct Proof that V { G V 0 Is Spanned by mtk)} kez Null Integrability of Wavelets Without Scaling Functions Wavelets with Compact Support From Scaling Filter to Scaling Function Explicit Construction of Compact Wavelets Daubechies Recipe HernandezWeiss Recipe Smoothness of Wavelets A Negative Result Cohen's Extension of Theorem Convergence Properties of Wavelet Expansions Wavelet Series in LP Spaces Large Scale Analysis AlmostEverywhere Convergence Convergence at a Preassigned Point Jackson and Bernstein Approximation Theorems Wavelets in Several Variables Two Important Examples Tensor Product of Wavelets General Formulation of MRA and Wavelets in W Notations for Subgroups and Cosets Riesz Systems and Orthonormal Systems in R d Scaling Equation and Structure Constants Existence of the Wavelet Set Proof That the Wavelet Set Spans V, 0 V Cohen's Theorem in R d Examples of Wavelets in W References 365 Notations 369 Index 373
FRACTIONAL INTEGRALS AND DERIVATIVES. Theory and Applications
FRACTIONAL INTEGRALS AND DERIVATIVES Theory and Applications Stefan G. Samko Rostov State University, Russia Anatoly A. Kilbas Belorussian State University, Minsk, Belarus Oleg I. Marichev Belorussian
More informationStatistical Modeling by Wavelets
Statistical Modeling by Wavelets BRANI VIDAKOVIC Duke University A WileyInterscience Publication JOHN WILEY & SONS, INC. New York / Chichester / Weinheim / Brisbane / Singapore / Toronto Contents Preface
More informationPoint Lattices in Computer Graphics and Visualization how signal processing may help computer graphics
Point Lattices in Computer Graphics and Visualization how signal processing may help computer graphics Dimitri Van De Ville Ecole Polytechnique Fédérale de Lausanne Biomedical Imaging Group dimitri.vandeville@epfl.ch
More informationElementary Differential Equations
Elementary Differential Equations EIGHTH EDITION Earl D. Rainville Late Professor of Mathematics University of Michigan Phillip E. Bedient Professor Emeritus of Mathematics Franklin and Marshall College
More informationMean value theorem, Taylors Theorem, Maxima and Minima.
MA 001 Preparatory Mathematics I. Complex numbers as ordered pairs. Argand s diagram. Triangle inequality. De Moivre s Theorem. Algebra: Quadratic equations and expressions. Permutations and Combinations.
More informationMATHEMATICAL METHODS OF STATISTICS
MATHEMATICAL METHODS OF STATISTICS By HARALD CRAMER TROFESSOK IN THE UNIVERSITY OF STOCKHOLM Princeton PRINCETON UNIVERSITY PRESS 1946 TABLE OF CONTENTS. First Part. MATHEMATICAL INTRODUCTION. CHAPTERS
More informationAN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS
AN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS Revised Edition James Epperson Mathematical Reviews BICENTENNIAL 0, 1 8 0 7 z ewiley wu 2007 r71 BICENTENNIAL WILEYINTERSCIENCE A John Wiley & Sons, Inc.,
More informationWAVES AND FIELDS IN INHOMOGENEOUS MEDIA
WAVES AND FIELDS IN INHOMOGENEOUS MEDIA WENG CHO CHEW UNIVERSITY OF ILLINOIS URBANACHAMPAIGN IEEE PRESS Series on Electromagnetic Waves Donald G. Dudley, Series Editor IEEE Antennas and Propagation Society,
More informationWavelet analysis. Wavelet requirements. Example signals. Stationary signal 2 Hz + 10 Hz + 20Hz. Zero mean, oscillatory (wave) Fast decay (let)
Wavelet analysis In the case of Fourier series, the orthonormal basis is generated by integral dilation of a single function e jx Every 2πperiodic squareintegrable function is generated by a superposition
More informationSCHWEITZER ENGINEERING LABORATORIES, COMERCIAL LTDA.
Pocket book of Electrical Engineering Formulas Content 1. Elementary Algebra and Geometry 1. Fundamental Properties (real numbers) 1 2. Exponents 2 3. Fractional Exponents 2 4. Irrational Exponents 2 5.
More informationApplied Linear Algebra I Review page 1
Applied Linear Algebra Review 1 I. Determinants A. Definition of a determinant 1. Using sum a. Permutations i. Sign of a permutation ii. Cycle 2. Uniqueness of the determinant function in terms of properties
More informationNumerical Analysis An Introduction
Walter Gautschi Numerical Analysis An Introduction 1997 Birkhauser Boston Basel Berlin CONTENTS PREFACE xi CHAPTER 0. PROLOGUE 1 0.1. Overview 1 0.2. Numerical analysis software 3 0.3. Textbooks and monographs
More informationMATH MathematicsNursing. MATH Remedial Mathematics IBusiness & Economics. MATH Remedial Mathematics IIBusiness and Economics
MATH 090  MathematicsNursing MATH 091  Remedial Mathematics IBusiness & Economics MATH 094  Remedial Mathematics IIBusiness and Economics MATH 095  Remedial Mathematics IScience (3 CH) MATH 096
More informationTheory of Sobolev Multipliers
Vladimir G. Maz'ya Tatyana O. Shaposhnikova Theory of Sobolev Multipliers With Applications to Differential and Integral Operators ^ Springer Introduction Part I Description and Properties of Multipliers
More informationNEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS
NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document
More informationColumbia University in the City of New York New York, N.Y. 10027
Columbia University in the City of New York New York, N.Y. 10027 DEPARTMENT OF MATHEMATICS 508 Mathematics Building 2990 Broadway Fall Semester 2005 Professor Ioannis Karatzas W4061: MODERN ANALYSIS Description
More informationThe Fourier Series of a Periodic Function
1 Chapter 1 he Fourier Series of a Periodic Function 1.1 Introduction Notation 1.1. We use the letter with a double meaning: a) [, 1) b) In the notations L p (), C(), C n () and C () we use the letter
More informationThinkwell s Homeschool Algebra 2 Course Lesson Plan: 34 weeks
Thinkwell s Homeschool Algebra 2 Course Lesson Plan: 34 weeks Welcome to Thinkwell s Homeschool Algebra 2! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More informationof Functional Analysis and Applications Group
FIRST ANNUAL WORKSHOP TALKS: May 15, 2010, Room Sousa Pinto, 08:15 AM Luís Castro Anabela Silva Alberto Simões Ana Paula Nolasco Saburou Saitoh Anabela Ramos António Caetano Alexandre Almeida Sofia Lopes
More informationMatrices and Polynomials
APPENDIX 9 Matrices and Polynomials he Multiplication of Polynomials Let α(z) =α 0 +α 1 z+α 2 z 2 + α p z p and y(z) =y 0 +y 1 z+y 2 z 2 + y n z n be two polynomials of degrees p and n respectively. hen,
More informationThe Quantum Harmonic Oscillator Stephen Webb
The Quantum Harmonic Oscillator Stephen Webb The Importance of the Harmonic Oscillator The quantum harmonic oscillator holds a unique importance in quantum mechanics, as it is both one of the few problems
More informationMICROLOCAL ANALYSIS OF THE BOCHNERMARTINELLI INTEGRAL
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 00029939(XX)00000 MICROLOCAL ANALYSIS OF THE BOCHNERMARTINELLI INTEGRAL NIKOLAI TARKHANOV AND NIKOLAI VASILEVSKI
More informationPuzzling features of data from asteroseismology space missions
Puzzling features of data from asteroseismology space missions Javier Pascual Granado Rafael Garrido Haba XI CoRoT Week La Laguna, Tenerife, Spain 1922 March 2013 Motivation Old problems like the search
More informationComplex Function Theory. Second Edition. Donald Sarason >AMS AMERICAN MATHEMATICAL SOCIETY
Complex Function Theory Second Edition Donald Sarason >AMS AMERICAN MATHEMATICAL SOCIETY Contents Preface to the Second Edition Preface to the First Edition ix xi Chapter I. Complex Numbers 1 1.1. Definition
More informationA RIGOROUS AND COMPLETED STATEMENT ON HELMHOLTZ THEOREM
Progress In Electromagnetics Research, PIER 69, 287 304, 2007 A RIGOROU AND COMPLETED TATEMENT ON HELMHOLTZ THEOREM Y. F. Gui and W. B. Dou tate Key Lab of Millimeter Waves outheast University Nanjing,
More informationMathematics (MAT) MAT 061 Basic Euclidean Geometry 3 Hours. MAT 051 PreAlgebra 4 Hours
MAT 051 PreAlgebra Mathematics (MAT) MAT 051 is designed as a review of the basic operations of arithmetic and an introduction to algebra. The student must earn a grade of C or in order to enroll in MAT
More informationMoving Least Squares Approximation
Chapter 7 Moving Least Squares Approimation An alternative to radial basis function interpolation and approimation is the socalled moving least squares method. As we will see below, in this method the
More informationNonlinear Iterative Partial Least Squares Method
Numerical Methods for Determining Principal Component Analysis Abstract Factors Béchu, S., RichardPlouet, M., Fernandez, V., Walton, J., and Fairley, N. (2016) Developments in numerical treatments for
More informationElementary Differential Equations and Boundary Value Problems. 10th Edition International Student Version
Brochure More information from http://www.researchandmarkets.com/reports/3148843/ Elementary Differential Equations and Boundary Value Problems. 10th Edition International Student Version Description:
More information2 Fourier Analysis and Analytic Functions
2 Fourier Analysis and Analytic Functions 2.1 Trigonometric Series One of the most important tools for the investigation of linear systems is Fourier analysis. Let f L 1 be a complexvalued Lebesgueintegrable
More informationThe Dirichlet Unit Theorem
Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if
More informationSIGNAL PROCESSING & SIMULATION NEWSLETTER
1 of 10 1/25/2008 3:38 AM SIGNAL PROCESSING & SIMULATION NEWSLETTER Note: This is not a particularly interesting topic for anyone other than those who ar e involved in simulation. So if you have difficulty
More informationCalculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum
Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum UNIT I: The Hyperbolic Functions basic calculus concepts, including techniques for curve sketching, exponential and logarithmic
More informationBANACH AND HILBERT SPACE REVIEW
BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but
More informationAn Introduction to Partial Differential Equations in the Undergraduate Curriculum
An Introduction to Partial Differential Equations in the Undergraduate Curriculum J. Tolosa & M. Vajiac LECTURE 11 Laplace s Equation in a Disk 11.1. Outline of Lecture The Laplacian in Polar Coordinates
More informationINTEGRAL METHODS IN LOWFREQUENCY ELECTROMAGNETICS
INTEGRAL METHODS IN LOWFREQUENCY ELECTROMAGNETICS I. Dolezel Czech Technical University, Praha, Czech Republic P. Karban University of West Bohemia, Plzeft, Czech Republic P. Solin University of Nevada,
More informationOverview of Math Standards
Algebra 2 Welcome to math curriculum design maps for Manhattan Ogden USD 383, striving to produce learners who are: Effective Communicators who clearly express ideas and effectively communicate with diverse
More informationIdeal Class Group and Units
Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals
More informationGeneralized Inverse of Matrices and its Applications
Generalized Inverse of Matrices and its Applications C. RADHAKRISHNA RAO, Sc.D., F.N.A., F.R.S. Director, Research and Training School Indian Statistical Institute SUJIT KUMAR MITRA, Ph.D. Professor of
More informationISU Department of Mathematics. Graduate Examination Policies and Procedures
ISU Department of Mathematics Graduate Examination Policies and Procedures There are four primary criteria to be used in evaluating competence on written or oral exams. 1. Knowledge Has the student demonstrated
More informationCITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION
No: CITY UNIVERSITY LONDON BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2005 Date: August
More informationTheta Functions. Lukas Lewark. Seminar on Modular Forms, 31. Januar 2007
Theta Functions Lukas Lewark Seminar on Modular Forms, 31. Januar 007 Abstract Theta functions are introduced, associated to lattices or quadratic forms. Their transformation property is proven and the
More informationComputational Optical Imaging  Optique Numerique.  Deconvolution 
Computational Optical Imaging  Optique Numerique  Deconvolution  Winter 2014 Ivo Ihrke Deconvolution Ivo Ihrke Outline Deconvolution Theory example 1D deconvolution Fourier method Algebraic method
More informationThnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More informationAnalytically Tractable Stochastic Stock Price Models
Archil Gulisashvili Analytically Tractable Stochastic Stock Price Models 4Q Springer Contents 1 Volatility Processes 1 1.1 Brownian Motion 1 1.2 s Geometric Brownian Motion 6 1.3 LongTime Behavior of
More informationAdvanced Signal Processing and Digital Noise Reduction
Advanced Signal Processing and Digital Noise Reduction Saeed V. Vaseghi Queen's University of Belfast UK WILEY HTEUBNER A Partnership between John Wiley & Sons and B. G. Teubner Publishers Chichester New
More informationChebyshev Expansions
Chapter 3 Chebyshev Expansions The best is the cheapest. Benjamin Franklin 3.1 Introduction In Chapter, approximations were considered consisting of expansions around a specific value of the variable (finite
More informationThe Geometry of Graphs
The Geometry of Graphs Paul Horn Department of Mathematics University of Denver May 21, 2016 Graphs Ultimately, I want to understand graphs: Collections of vertices and edges. Graphs Ultimately, I want
More informationBOOK REVIEWS. [f(y)/(*y)]dy.
BOOK REVIEWS Orthogonal Polynomials. By Gabor Szegö. (American Mathematical Society Colloquium Publications, vol. 23.) New York, American Mathematical Society, 1939. 10+401 pp. The general concept of orthogonal
More informationVery Preliminary Program
Very Preliminary Program Two of the participants will give Colloquium talks before the meeting. The workshop it self starts on Friday morning. All talks will take place in Lockett, 277. The room is equipped
More informationReference: Introduction to Partial Differential Equations by G. Folland, 1995, Chap. 3.
5 Potential Theory Reference: Introduction to Partial Differential Equations by G. Folland, 995, Chap. 3. 5. Problems of Interest. In what follows, we consider Ω an open, bounded subset of R n with C 2
More informationMean Value Coordinates
Mean Value Coordinates Michael S. Floater Abstract: We derive a generalization of barycentric coordinates which allows a vertex in a planar triangulation to be expressed as a convex combination of its
More informationOn Chebyshev interpolation of analytic functions
On Chebyshev interpolation of analytic functions Laurent Demanet Department of Mathematics Massachusetts Institute of Technology Lexing Ying Department of Mathematics University of Texas at Austin March
More informationDirichlet forms methods for error calculus and sensitivity analysis
Dirichlet forms methods for error calculus and sensitivity analysis Nicolas BOULEAU, Osaka university, november 2004 These lectures propose tools for studying sensitivity of models to scalar or functional
More informationarxiv:math/0010057v1 [math.dg] 5 Oct 2000
arxiv:math/0010057v1 [math.dg] 5 Oct 2000 HEAT KERNEL ASYMPTOTICS FOR LAPLACE TYPE OPERATORS AND MATRIX KDV HIERARCHY IOSIF POLTEROVICH Preliminary version Abstract. We study the heat kernel asymptotics
More informationStephane Crepey. Financial Modeling. A Backward Stochastic Differential Equations Perspective. 4y Springer
Stephane Crepey Financial Modeling A Backward Stochastic Differential Equations Perspective 4y Springer Part I An Introductory Course in Stochastic Processes 1 Some Classes of DiscreteTime Stochastic
More informationIllPosed Problems in Probability and Stability of Random Sums. Lev B. Klebanov, Tomasz J. Kozubowski, and Svetlozar T. Rachev
IllPosed Problems in Probability and Stability of Random Sums By Lev B. Klebanov, Tomasz J. Kozubowski, and Svetlozar T. Rachev Preface This is the first of two volumes concerned with the illposed problems
More informationPOLYNOMIAL HISTOPOLATION, SUPERCONVERGENT DEGREES OF FREEDOM, AND PSEUDOSPECTRAL DISCRETE HODGE OPERATORS
POLYNOMIAL HISTOPOLATION, SUPERCONVERGENT DEGREES OF FREEDOM, AND PSEUDOSPECTRAL DISCRETE HODGE OPERATORS N. ROBIDOUX Abstract. We show that, given a histogram with n bins possibly noncontiguous or consisting
More informationWavelet Analysis Based Estimation of Probability Density function of Wind Data
, pp.2334 http://dx.doi.org/10.14257/ijeic.2014.5.3.03 Wavelet Analysis Based Estimation of Probability Density function of Wind Data Debanshee Datta Department of Mechanical Engineering Indian Institute
More informationNumerical Recipes in C++
Numerical Recipes in C++ The Art of Scientific Computing Second Edition William H. Press Los Alamos National Laboratory Saul A. Teukolsky Department of Physics, Cornell University William T. Vetterling
More informationAPPLIED MATHEMATICS ADVANCED LEVEL
APPLIED MATHEMATICS ADVANCED LEVEL INTRODUCTION This syllabus serves to examine candidates knowledge and skills in introductory mathematical and statistical methods, and their applications. For applications
More informationMATH 590: Meshfree Methods
MATH 590: Meshfree Methods Chapter 7: Conditionally Positive Definite Functions Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2010 fasshauer@iit.edu MATH 590 Chapter
More informationTHE SQUARE PARTIAL SUMS OF THE FOURIER TRANSFORM OF RADIAL FUNCTIONS IN THREE DIMENSIONS
Scientiae Mathematicae Japonicae Online, Vol. 5,, 9 9 9 THE SQUARE PARTIAL SUMS OF THE FOURIER TRANSFORM OF RADIAL FUNCTIONS IN THREE DIMENSIONS CHIKAKO HARADA AND EIICHI NAKAI Received May 4, ; revised
More informationOption Pricing Formulae using Fourier Transform: Theory and Application
Option Pricing Formulae using Fourier Transform: Theory and Application Martin Schmelzle * April 2010 Abstract Fourier transform techniques are playing an increasingly important role in Mathematical Finance.
More informationChapter 3: Mathematical Models and Numerical Methods Involving FirstOrder Differential Equations
Massasoit Community College Instructor: Office: Email: Phone: Office Hours: Course: Differential Equations Course Number: MATH230XX Semester: Classroom: Day and Time: Course Description: This course is
More informationSine and Cosine Series; Odd and Even Functions
Sine and Cosine Series; Odd and Even Functions A sine series on the interval [, ] is a trigonometric series of the form k = 1 b k sin πkx. All of the terms in a series of this type have values vanishing
More informationWHICH LINEARFRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE?
WHICH LINEARFRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE? JOEL H. SHAPIRO Abstract. These notes supplement the discussion of linear fractional mappings presented in a beginning graduate course
More informationUnivariate and Multivariate Methods PEARSON. Addison Wesley
Time Series Analysis Univariate and Multivariate Methods SECOND EDITION William W. S. Wei Department of Statistics The Fox School of Business and Management Temple University PEARSON Addison Wesley Boston
More informationSouth Carolina College and CareerReady (SCCCR) PreCalculus
South Carolina College and CareerReady (SCCCR) PreCalculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More informationTHE KADISONSINGER PROBLEM IN MATHEMATICS AND ENGINEERING: A DETAILED ACCOUNT
THE ADISONSINGER PROBLEM IN MATHEMATICS AND ENGINEERING: A DETAILED ACCOUNT PETER G. CASAZZA, MATTHEW FICUS, JANET C. TREMAIN, ERIC WEBER Abstract. We will show that the famous, intractible 1959 adisonsinger
More informationIntroduction to the Finite Element Method
Introduction to the Finite Element Method 09.06.2009 Outline Motivation Partial Differential Equations (PDEs) Finite Difference Method (FDM) Finite Element Method (FEM) References Motivation Figure: cross
More informationSummary Nonstationary Time Series Multitude of Representations Possibilities from Applied Computational Harmonic Analysis Tests of Stationarity
Nonstationary Time Series, Priestley s Evolutionary Spectra and Wavelets Guy Nason, School of Mathematics, University of Bristol Summary Nonstationary Time Series Multitude of Representations Possibilities
More informationAlgebra 1 Course Title
Algebra 1 Course Title Course wide 1. What patterns and methods are being used? Course wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept
More informationInner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
More informationLecture notes: harmonic analysis. Russell Brown Department of mathematics University of Kentucky Lexington, KY 405060027
Lecture notes: harmonic analysis Russell Brown Department of mathematics University of Kentucky Lexington, KY 405060027 12 June 2001 ii Contents Preface v 1 The Fourier transform on L 1 1 1.1 Definition
More informationCollege of Arts and Sciences. Mathematics
108R INTERMEDIATE ALGEBRA. (3) This course is remedial in nature and covers material commonly found in second year high school algebra. Specific topics to be discussed include numbers, fractions, algebraic
More informationFUNDAMENTAL FINITE ELEMENT ANALYSIS AND APPLICATIONS
FUNDAMENTAL FINITE ELEMENT ANALYSIS AND APPLICATIONS With Mathematica and MATLAB Computations M. ASGHAR BHATTI WILEY JOHN WILEY & SONS, INC. CONTENTS OF THE BOOK WEB SITE PREFACE xi xiii 1 FINITE ELEMENT
More informationTaylor Series and Asymptotic Expansions
Taylor Series and Asymptotic Epansions The importance of power series as a convenient representation, as an approimation tool, as a tool for solving differential equations and so on, is pretty obvious.
More informationEstimated Pre Calculus Pacing Timeline
Estimated Pre Calculus Pacing Timeline 20102011 School Year The timeframes listed on this calendar are estimates based on a fiftyminute class period. You may need to adjust some of them from time to
More information1D 3D 1D 3D. is called eigenstate or state function. When an operator act on a state, it can be written as
Chapter 3 (Lecture 45) Postulates of Quantum Mechanics Now we turn to an application of the preceding material, and move into the foundations of quantum mechanics. Quantum mechanics is based on a series
More information1. Let P be the space of all polynomials (of one real variable and with real coefficients) with the norm
Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: F3B, F4Sy, NVP 0050615 Skrivtid: 9 14 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok
More informationLearning Objectives for Math 165
Learning Objectives for Math 165 Chapter 2 Limits Section 2.1: Average Rate of Change. State the definition of average rate of change Describe what the rate of change does and does not tell us in a given
More informationSampling 50 Years After Shannon
Sampling 50 Years After Shannon MICHAEL UNSER, FELLOW, IEEE This paper presents an account of the current state of sampling, 50 years after Shannon s formulation of the sampling theorem. The emphasis is
More informationMATH BOOK OF PROBLEMS SERIES. New from Pearson Custom Publishing!
MATH BOOK OF PROBLEMS SERIES New from Pearson Custom Publishing! The Math Book of Problems Series is a database of math problems for the following courses: Prealgebra Algebra Precalculus Calculus Statistics
More informationDepartment of Mathematics
250 Department of Mathematics Department of Mathematics Chairperson: Raji, Wissam V. Professors Emeriti: Muwafi, Amin; Yff, Peter Professors: AbiKhuzam, Faruk F; AbuKhuzam, Hazar M.; KhuriMakdisi, Kamal
More informationLinear Algebra: Vectors
A Linear Algebra: Vectors A Appendix A: LINEAR ALGEBRA: VECTORS TABLE OF CONTENTS Page A Motivation A 3 A2 Vectors A 3 A2 Notational Conventions A 4 A22 Visualization A 5 A23 Special Vectors A 5 A3 Vector
More informationRESONANCES AND BALLS IN OBSTACLE SCATTERING WITH NEUMANN BOUNDARY CONDITIONS
RESONANCES AND BALLS IN OBSTACLE SCATTERING WITH NEUMANN BOUNDARY CONDITIONS T. J. CHRISTIANSEN Abstract. We consider scattering by an obstacle in R d, d 3 odd. We show that for the Neumann Laplacian if
More informationPreCalculus Semester 1 Course Syllabus
PreCalculus Semester 1 Course Syllabus The Plano ISD eschool Mission is to create a borderless classroom based on a positive studentteacher relationship that fosters independent, innovative critical
More informationSome remarks on PhragménLindelöf theorems for weak solutions of the stationary Schrödinger operator
Wan Boundary Value Problems (2015) 2015:239 DOI 10.1186/s1366101505080 R E S E A R C H Open Access Some remarks on PhragménLindelöf theorems for weak solutions of the stationary Schrödinger operator
More informationGeorgia Department of Education Kathy Cox, State Superintendent of Schools 7/19/2005 All Rights Reserved 1
Accelerated Mathematics 3 This is a course in precalculus and statistics, designed to prepare students to take AB or BC Advanced Placement Calculus. It includes rational, circular trigonometric, and inverse
More information2. Introduction to quantum mechanics
2. Introduction to quantum mechanics 2.1 Linear algebra Dirac notation Complex conjugate Vector/ket Dual vector/bra Inner product/bracket Tensor product Complex conj. matrix Transpose of matrix Hermitian
More informationAsymptotic Analysis of Fields in MultiStructures
Asymptotic Analysis of Fields in MultiStructures VLADIMIR KOZLOV Department of Mathematics, Linkoeping University, Sweden VLADIMIR MAZ'YA Department of Mathematics, Linkoeping University, Sweden ALEXANDER
More informationand s n (x) f(x) for all x and s.t. s n is measurable if f is. REAL ANALYSIS Measures. A (positive) measure on a measurable space
RAL ANALYSIS A survey of MA 641643, UAB 19992000 M. Griesemer Throughout these notes m denotes Lebesgue measure. 1. Abstract Integration σalgebras. A σalgebra in X is a nonempty collection of subsets
More informationPrinciples of Digital Communication
Principles of Digital Communication Robert G. Gallager January 5, 2008 ii Preface: introduction and objectives The digital communication industry is an enormous and rapidly growing industry, roughly comparable
More informationALGEBRAIC EIGENVALUE PROBLEM
ALGEBRAIC EIGENVALUE PROBLEM BY J. H. WILKINSON, M.A. (Cantab.), Sc.D. Technische Universes! Dsrmstedt FACHBEREICH (NFORMATiK BIBL1OTHEK Sachgebieto:. Standort: CLARENDON PRESS OXFORD 1965 Contents 1.
More informationFEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL
FEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL STATIsTICs 4 IV. RANDOm VECTORs 1. JOINTLY DIsTRIBUTED RANDOm VARIABLEs If are two rom variables defined on the same sample space we define the joint
More informationNMR Measurement of T1T2 Spectra with Partial Measurements using Compressive Sensing
NMR Measurement of T1T2 Spectra with Partial Measurements using Compressive Sensing Alex Cloninger Norbert Wiener Center Department of Mathematics University of Maryland, College Park http://www.norbertwiener.umd.edu
More informationRegression With Gaussian Measures
Regression With Gaussian Measures Michael J. Meyer Copyright c April 11, 2004 ii PREFACE We treat the basics of Gaussian processes, Gaussian measures, kernel reproducing Hilbert spaces and related topics.
More informationMathematics (MS) www.utrgv.edu/grad. Admissio n Requirements Apply to the UTRGV Graduate College:
Mathematics (MS) The Master of Science (MS) in Mathematics program is designed to provide a graduate level education for students who intend to teach at various levels, students who will continue or seek
More informationMATHEMATICS (CLASSES XI XII)
MATHEMATICS (CLASSES XI XII) General Guidelines (i) All concepts/identities must be illustrated by situational examples. (ii) The language of word problems must be clear, simple and unambiguous. (iii)
More information