Lecture 16: Bessel s Inequality, Parseval s Theorem, Energy convergence
|
|
- Lauren Shelton
- 6 years ago
- Views:
Transcription
1 Introductory lecture notes on Partial Differential Equations - c Anthony Peirce. ot to be copied, used, or revised without explicit written permission from the copyright owner. ecture 6: Bessel s Inequality, Parseval s Theorem, Energy convergence Compiled 4 August 7 In this lecture we consider the counterpart of Pythagoras Theorem for functions whose square is integrable. Square integrable functions are associated with functions describing physical systems having finite energy. For a finite Fourier Series involving terms we derive the so-called Bessel Inequality, in which can be taken to infinity provided the function f is square integrable. The Bessel Inequality is shown to reduce to an equality if and only if the Fourier Series S nx converges to f in the energy norm. The result is known as Parseval s Formula, which has profound consequences for the completeness of the Fourier Basis {, cos, sin }. We see that Parseval s Formula leads to a new class of sums for series of reciprocal powers of n. Key Concepts: Convergence of Fourier Series, Bessel s Inequality, Paresval s Theorem, Plancherel theorem, Pythagoras Theorem, Energy of a function, Convergence in Energy, completeness of the Fourier Basis. 6 Bessel s Inequality and Parseval s Theorem: 6. Bessel s Inequality Definition et fx be a function that is square-integrable on, ] i.e., fx ] dx <, in which case we write f, ]. Consider the Fourier Series associated with fx, namely; et ow fx a S x a + b n sin + b n sin S. fx S x ] f x fxs x + S x
2 Consider the least-square error defined to be ] E f, S fx S x ] dx f x dx fxs x dx + { f, f f, S + S, S } S x dx ow In addition, f, S S, S a a a a + a n ] a a n + b n fxs x dx fx dx + a n + a n + b n. E f, S ] ] + b n sin dx cos nx dx + b n fx cos sin nx dx dx + b n fx sin dx fx S x ] { } a dx f, f a n + b n ow since E f, S ] fx S x ] dx it follows that a a n + b n where Ef] is known as the energy of the -periodic function f. f x dx f, f Ef]
3 Fourier Series 3 Theorem Bessel s Inequality: et f, ] then a a n + b n in particular the series a a n + b n is convergent. f x dx 6. Bessel s Inequality, Components of a Vector and Pythagoras Theorem 6.. D Analogue Consider a D vector f, which is decomposed into components in terms of two orthogonal unit vectors ê and ê, i.e. f a ê + a ê ow f f f a ê + a ê a ê + a ê a + a since ê k are orthogonal unit vectors f a + a which is Pythagoras Theorem D Analogue Suppose we wish to expand a 3-vector f in terms of a set of basis vectors {ê, ê }. Bessel s Inequality assumes the form a + a f Since the subspace span {ê, ê } which represents a plane in R 3 does not include the whole of R 3 the vector a ê + a ê f represents the orthogonal projection of f onto span {ê, ê }. If we include the third basis vector ê 3 in the basis, then the span {ê, ê, ê 3 } R 3. In this case the set {ê, ê, ê 3 } are linearly independent and of full rank and thus span the complete space R 3. {ê, ê, ê 3 } are in this case said to form a complete set. In this case f a ê + a ê + a 3 ê 3 and f a +a +a 3 so that Bessel s Inequality assumes the form of an equality, which in this trivial case reduces to Pythagoras Theorem. For a set of functions, that are complete, the equivalent of Pythagoras Theorem is Parseval s Theorem.
4 4 6.3 Parseval s Theorem Theorem Parseval s Identity et f, ] then the Fourier coefficients a n and b n satisfy Parseval s Formula If and only if a a n + b n lim f x dx Ef] fx S x ] dx. In this case the The east Square Error assumes the form E f, S ] fx S x ] dx a a n + b n n+ a f a x dx a n + b n a n + b n a n + b n Parseval s Theorem for odd functions Theorem 3 Parseval s Identity for odd functions et fx b n sin < x <. Then ] fx dx b n. Proof: fx ] dx m m b m b n sin b m b n δ mn mx sin dx 6. b n. 6.3 Example 6. Recall for x, ], fx x 4 n+ sin. n fx dx x dx 4 4 x n n n 6.4
5 ote: n Also note that n 4 For Fourier Sine Components: Fourier Series 5 evens n m m Example 6. Consider fx x, < x <. odds m + m 4 + m m+ m+ m fx dx b n. 6.6 The Fourier Series Expansion is: x n n cosnx. 6.7 et By Parseval s Formula: n 3 4 cos n x x 4 dx x k k k+ n 4 n n cos n 6.8 k k k. 6.9 n ζ4, 6. n4 where ζ is the Riemann Zeta Function defined by: ζs, s σ + iτ, σ Re{s} > 6. ns
Vectors in Function Spaces
Jim Lambers MAT 66 Spring Semester 15-16 Lecture 18 Notes These notes correspond to Section 6.3 in the text. Vectors in Function Spaces We begin with some necessary terminology. A vector space V, also
More informationExercise 11. Isao Sasano
Exercise Isao Sasano Exercise Calculate the value of the following series by using the Parseval s equality for the Fourier series of f(x) x on the range [, π] following the steps ()-(5). () Calculate the
More informationLecture 19: Heat conduction with distributed sources/sinks
Introductory lecture notes on Partial Differential Equations - c Anthony Peirce. Not to be copied, used, or revised without explicit written permission from the copyright owner. 1 ecture 19: Heat conduction
More informationLecture 11: Fourier Cosine Series
Introductory lecture notes on Partial Differential Equations - c Anthony Peirce Not to be copied, used, or revised without eplicit written permission from the copyright owner ecture : Fourier Cosine Series
More informationLecture 10: Fourier Sine Series
Introductory lecture notes on Partial Differential Equations - c Anthony Peirce. Not to be copied, used, or revised without eplicit written permission from the copyright owner. ecture : Fourier Sine Series
More informationMathematics 324 Riemann Zeta Function August 5, 2005
Mathematics 324 Riemann Zeta Function August 5, 25 In this note we give an introduction to the Riemann zeta function, which connects the ideas of real analysis with the arithmetic of the integers. Define
More informationBernoulli Numbers and their Applications
Bernoulli Numbers and their Applications James B Silva Abstract The Bernoulli numbers are a set of numbers that were discovered by Jacob Bernoulli (654-75). This set of numbers holds a deep relationship
More informationLecture 1: Interpolation and approximation
ecture notes on Variational and Approximate Methods in Applied Mathematics - A Peirce UBC ecture : Interpolation and approximation (Compiled 6 August 207 In this lecture we introduce the concept of approximation
More informationRecall that any inner product space V has an associated norm defined by
Hilbert Spaces Recall that any inner product space V has an associated norm defined by v = v v. Thus an inner product space can be viewed as a special kind of normed vector space. In particular every inner
More informationExamples of the Fourier Theorem (Sect. 10.3). The Fourier Theorem: Continuous case.
s of the Fourier Theorem (Sect. 1.3. The Fourier Theorem: Continuous case. : Using the Fourier Theorem. The Fourier Theorem: Piecewise continuous case. : Using the Fourier Theorem. The Fourier Theorem:
More informationFinite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product
Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a pre-hilbert space, or a unitary space) if there is a mapping (, )
More informationMATH 220: INNER PRODUCT SPACES, SYMMETRIC OPERATORS, ORTHOGONALITY
MATH 22: INNER PRODUCT SPACES, SYMMETRIC OPERATORS, ORTHOGONALITY When discussing separation of variables, we noted that at the last step we need to express the inhomogeneous initial or boundary data as
More informationPHYS 502 Lecture 3: Fourier Series
PHYS 52 Lecture 3: Fourier Series Fourier Series Introduction In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a (possibly infinite) set of simple oscillating
More informationbe the set of complex valued 2π-periodic functions f on R such that
. Fourier series. Definition.. Given a real number P, we say a complex valued function f on R is P -periodic if f(x + P ) f(x) for all x R. We let be the set of complex valued -periodic functions f on
More informationMATH 124B: HOMEWORK 2
MATH 24B: HOMEWORK 2 Suggested due date: August 5th, 26 () Consider the geometric series ( ) n x 2n. (a) Does it converge pointwise in the interval < x
More information2 2 + x =
Lecture 30: Power series A Power Series is a series of the form c n = c 0 + c 1 x + c x + c 3 x 3 +... where x is a variable, the c n s are constants called the coefficients of the series. n = 1 + x +
More information1. Subspaces A subset M of Hilbert space H is a subspace of it is closed under the operation of forming linear combinations;i.e.,
Abstract Hilbert Space Results We have learned a little about the Hilbert spaces L U and and we have at least defined H 1 U and the scale of Hilbert spaces H p U. Now we are going to develop additional
More informationLecture 13: Orthogonal projections and least squares (Section ) Thang Huynh, UC San Diego 2/9/2018
Lecture 13: Orthogonal projections and least squares (Section 3.2-3.3) Thang Huynh, UC San Diego 2/9/2018 Orthogonal projection onto subspaces Theorem. Let W be a subspace of R n. Then, each x in R n can
More informationInner products. Theorem (basic properties): Given vectors u, v, w in an inner product space V, and a scalar k, the following properties hold:
Inner products Definition: An inner product on a real vector space V is an operation (function) that assigns to each pair of vectors ( u, v) in V a scalar u, v satisfying the following axioms: 1. u, v
More informationMath 115 ( ) Yum-Tong Siu 1. Derivation of the Poisson Kernel by Fourier Series and Convolution
Math 5 (006-007 Yum-Tong Siu. Derivation of the Poisson Kernel by Fourier Series and Convolution We are going to give a second derivation of the Poisson kernel by using Fourier series and convolution.
More information7: FOURIER SERIES STEVEN HEILMAN
7: FOURIER SERIES STEVE HEILMA Contents 1. Review 1 2. Introduction 1 3. Periodic Functions 2 4. Inner Products on Periodic Functions 3 5. Trigonometric Polynomials 5 6. Periodic Convolutions 7 7. Fourier
More informationAdvanced Computational Fluid Dynamics AA215A Lecture 2 Approximation Theory. Antony Jameson
Advanced Computational Fluid Dynamics AA5A Lecture Approximation Theory Antony Jameson Winter Quarter, 6, Stanford, CA Last revised on January 7, 6 Contents Approximation Theory. Least Squares Approximation
More informationLecture 4: Frobenius Series about Regular Singular Points
Introductory lecture notes on Partial Differential Equations - c Anthony Peirce. Not to be copied, used, or revised without explicit written permission from the copyright owner. 1 Lecture 4: Frobenius
More information14 Fourier analysis. Read: Boas Ch. 7.
14 Fourier analysis Read: Boas Ch. 7. 14.1 Function spaces A function can be thought of as an element of a kind of vector space. After all, a function f(x) is merely a set of numbers, one for each point
More informationMATH 5640: Fourier Series
MATH 564: Fourier Series Hung Phan, UMass Lowell September, 8 Power Series A power series in the variable x is a series of the form a + a x + a x + = where the coefficients a, a,... are real or complex
More informationThe Fourier series for a 2π-periodic function
The Fourier series for a 2π-periodic function Let f : ( π, π] R be a bounded piecewise continuous function which we continue to be a 2π-periodic function defined on R, i.e. f (x + 2π) = f (x), x R. The
More information3 Orthogonality and Fourier series
3 Orthogonality and Fourier series We now turn to the concept of orthogonality which is a key concept in inner product spaces and Hilbert spaces. We start with some basic definitions. Definition 3.1. Let
More informationMore Series Convergence
More Series Convergence James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University December 4, 218 Outline Convergence Analysis for Fourier Series Revisited
More informationJim Lambers ENERGY 281 Spring Quarter Lecture 5 Notes
Jim ambers ENERGY 28 Spring Quarter 27-8 ecture 5 Notes These notes are based on Rosalind Archer s PE28 lecture notes, with some revisions by Jim ambers. Fourier Series Recall that in ecture 2, when we
More informationTopics in Harmonic Analysis Lecture 1: The Fourier transform
Topics in Harmonic Analysis Lecture 1: The Fourier transform Po-Lam Yung The Chinese University of Hong Kong Outline Fourier series on T: L 2 theory Convolutions The Dirichlet and Fejer kernels Pointwise
More informationFourier Sin and Cos Series and Least Squares Convergence
Fourier and east Squares Convergence James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University May 7, 28 Outline et s look at the original Fourier sin
More informationFunctional Analysis Exercise Class
Functional Analysis Exercise Class Week: December 4 8 Deadline to hand in the homework: your exercise class on week January 5. Exercises with solutions ) Let H, K be Hilbert spaces, and A : H K be a linear
More informationLinear Algebra. Paul Yiu. Department of Mathematics Florida Atlantic University. Fall A: Inner products
Linear Algebra Paul Yiu Department of Mathematics Florida Atlantic University Fall 2011 6A: Inner products In this chapter, the field F = R or C. We regard F equipped with a conjugation χ : F F. If F =
More informationLecture 23: 6.1 Inner Products
Lecture 23: 6.1 Inner Products Wei-Ta Chu 2008/12/17 Definition An inner product on a real vector space V is a function that associates a real number u, vwith each pair of vectors u and v in V in such
More informationHilbert Spaces. Contents
Hilbert Spaces Contents 1 Introducing Hilbert Spaces 1 1.1 Basic definitions........................... 1 1.2 Results about norms and inner products.............. 3 1.3 Banach and Hilbert spaces......................
More informationMath 489AB A Very Brief Intro to Fourier Series Fall 2008
Math 489AB A Very Brief Intro to Fourier Series Fall 8 Contents Fourier Series. The coefficients........................................ Convergence......................................... 4.3 Convergence
More informationMATH 304 Linear Algebra Lecture 18: Orthogonal projection (continued). Least squares problems. Normed vector spaces.
MATH 304 Linear Algebra Lecture 18: Orthogonal projection (continued). Least squares problems. Normed vector spaces. Orthogonality Definition 1. Vectors x,y R n are said to be orthogonal (denoted x y)
More informationRecall: Dot product on R 2 : u v = (u 1, u 2 ) (v 1, v 2 ) = u 1 v 1 + u 2 v 2, u u = u u 2 2 = u 2. Geometric Meaning:
Recall: Dot product on R 2 : u v = (u 1, u 2 ) (v 1, v 2 ) = u 1 v 1 + u 2 v 2, u u = u 2 1 + u 2 2 = u 2. Geometric Meaning: u v = u v cos θ. u θ v 1 Reason: The opposite side is given by u v. u v 2 =
More informationMA 201: Differentiation and Integration of Fourier Series Applications of Fourier Series Lecture - 10
MA 201: Differentiation and Integration of Fourier Series Applications of Fourier Series ecture - 10 Fourier Series: Orthogonal Sets We begin our treatment with some observations: For m,n = 1,2,3,... cos
More informationLecture 21: The one dimensional Wave Equation: D Alembert s Solution
Introductory lecture notes on Partial Differential Equations - c Anthony Peirce. Not to be copied, used, or revised without explicit written permission from the copyright owner. 1 Lecture 21: The one dimensional
More information23 Elements of analytic ODE theory. Bessel s functions
23 Elements of analytic ODE theory. Bessel s functions Recall I am changing the variables) that we need to solve the so-called Bessel s equation 23. Elements of analytic ODE theory Let x 2 u + xu + x 2
More informationTopics in Fourier analysis - Lecture 2.
Topics in Fourier analysis - Lecture 2. Akos Magyar 1 Infinite Fourier series. In this section we develop the basic theory of Fourier series of periodic functions of one variable, but only to the extent
More informationCOMP 558 lecture 18 Nov. 15, 2010
Least squares We have seen several least squares problems thus far, and we will see more in the upcoming lectures. For this reason it is good to have a more general picture of these problems and how to
More informationζ (s) = s 1 s {u} [u] ζ (s) = s 0 u 1+sdu, {u} Note how the integral runs from 0 and not 1.
Problem Sheet 3. From Theorem 3. we have ζ (s) = + s s {u} u+sdu, (45) valid for Res > 0. i) Deduce that for Res >. [u] ζ (s) = s u +sdu ote the integral contains [u] in place of {u}. ii) Deduce that for
More information3 rd class Mech. Eng. Dept. hamdiahmed.weebly.com Fourier Series
Definition 1 Fourier Series A function f is said to be piecewise continuous on [a, b] if there exists finitely many points a = x 1 < x 2
More informationHilbert Spaces. Hilbert space is a vector space with some extra structure. We start with formal (axiomatic) definition of a vector space.
Hilbert Spaces Hilbert space is a vector space with some extra structure. We start with formal (axiomatic) definition of a vector space. Vector Space. Vector space, ν, over the field of complex numbers,
More informationFunctional Analysis (2006) Homework assignment 2
Functional Analysis (26) Homework assignment 2 All students should solve the following problems: 1. Define T : C[, 1] C[, 1] by (T x)(t) = t x(s) ds. Prove that this is a bounded linear operator, and compute
More informationCommunication Signals (Haykin Sec. 2.4 and Ziemer Sec Sec. 2.4) KECE321 Communication Systems I
Communication Signals (Haykin Sec..4 and iemer Sec...4-Sec..4) KECE3 Communication Systems I Lecture #3, March, 0 Prof. Young-Chai Ko 년 3 월 일일요일 Review Signal classification Phasor signal and spectra Representation
More informationVector Spaces. Vector space, ν, over the field of complex numbers, C, is a set of elements a, b,..., satisfying the following axioms.
Vector Spaces Vector space, ν, over the field of complex numbers, C, is a set of elements a, b,..., satisfying the following axioms. For each two vectors a, b ν there exists a summation procedure: a +
More informationFourier Series. Department of Mathematical and Statistical Sciences University of Alberta
1 Lecture Notes on Partial Differential Equations Chapter IV Fourier Series Ilyasse Aksikas Department of Mathematical and Statistical Sciences University of Alberta aksikas@ualberta.ca DEFINITIONS 2 Before
More informationCompletion Date: Monday February 11, 2008
MATH 4 (R) Winter 8 Intermediate Calculus I Solutions to Problem Set #4 Completion Date: Monday February, 8 Department of Mathematical and Statistical Sciences University of Alberta Question. [Sec..9,
More informationMa 221 Eigenvalues and Fourier Series
Ma Eigenvalues and Fourier Series Eigenvalue and Eigenfunction Examples Example Find the eigenvalues and eigenfunctions for y y 47 y y y5 Solution: The characteristic equation is r r 47 so r 44 447 6 Thus
More informationx s 1 e x dx, for σ > 1. If we replace x by nx in the integral then we obtain x s 1 e nx dx. x s 1
Recall 9. The Riemann Zeta function II Γ(s) = x s e x dx, for σ >. If we replace x by nx in the integral then we obtain Now sum over n to get n s Γ(s) = x s e nx dx. x s ζ(s)γ(s) = e x dx. Note that as
More informationMath 345: Applied Mathematics
Math 345: Applied Mathematics Introduction to Fourier Series, I Tones, Harmonics, and Fourier s Theorem Marcus Pendergrass Hampden-Sydney College Fall 2012 1 Sounds as waveforms I ve got a bad feeling
More informationINTRODUCTION TO REAL ANALYSIS II MATH 4332 BLECHER NOTES
INTRODUCTION TO REAL ANALYSIS II MATH 433 BLECHER NOTES. As in earlier classnotes. As in earlier classnotes (Fourier series) 3. Fourier series (continued) (NOTE: UNDERGRADS IN THE CLASS ARE NOT RESPONSIBLE
More informationMODULE 8 Topics: Null space, range, column space, row space and rank of a matrix
MODULE 8 Topics: Null space, range, column space, row space and rank of a matrix Definition: Let L : V 1 V 2 be a linear operator. The null space N (L) of L is the subspace of V 1 defined by N (L) = {x
More informationGeneral Inner Product and The Fourier Series
A Linear Algebra Approach Department of Mathematics University of Puget Sound 4-20-14 / Spring Semester Outline 1 2 Inner Product The inner product is an algebraic operation that takes two vectors and
More informationI teach myself... Hilbert spaces
I teach myself... Hilbert spaces by F.J.Sayas, for MATH 806 November 4, 2015 This document will be growing with the semester. Every in red is for you to justify. Even if we start with the basic definition
More information1 A complete Fourier series solution
Math 128 Notes 13 In this last set of notes I will try to tie up some loose ends. 1 A complete Fourier series solution First here is an example of the full solution of a pde by Fourier series. Consider
More information1 Math 241A-B Homework Problem List for F2015 and W2016
1 Math 241A-B Homework Problem List for F2015 W2016 1.1 Homework 1. Due Wednesday, October 7, 2015 Notation 1.1 Let U be any set, g be a positive function on U, Y be a normed space. For any f : U Y let
More information4 Hilbert spaces. The proof of the Hilbert basis theorem is not mathematics, it is theology. Camille Jordan
The proof of the Hilbert basis theorem is not mathematics, it is theology. Camille Jordan Wir müssen wissen, wir werden wissen. David Hilbert We now continue to study a special class of Banach spaces,
More informationLinear Algebra Massoud Malek
CSUEB Linear Algebra Massoud Malek Inner Product and Normed Space In all that follows, the n n identity matrix is denoted by I n, the n n zero matrix by Z n, and the zero vector by θ n An inner product
More informationAMS 212A Applied Mathematical Methods I Appendices of Lecture 06 Copyright by Hongyun Wang, UCSC. ( ) cos2
AMS 22A Applied Mathematical Methods I Appendices of Lecture 06 Copyright by Hongyun Wang UCSC Appendix A: Proof of Lemma Lemma : Let (x ) be the solution of x ( r( x)+ q( x) )sin 2 + ( a) 0 < cos2 where
More informationFourier Series. 1. Review of Linear Algebra
Fourier Series In this section we give a short introduction to Fourier Analysis. If you are interested in Fourier analysis and would like to know more detail, I highly recommend the following book: Fourier
More information6 Lecture 6b: the Euler Maclaurin formula
Queens College, CUNY, Department of Computer Science Numerical Methods CSCI 361 / 761 Fall 217 Instructor: Dr. Sateesh Mane c Sateesh R. Mane 217 March 26, 218 6 Lecture 6b: the Euler Maclaurin formula
More informationMore on Fourier Series
More on Fourier Series R. C. Trinity University Partial Differential Equations Lecture 6.1 New Fourier series from old Recall: Given a function f (x, we can dilate/translate its graph via multiplication/addition,
More informationExercises * on Linear Algebra
Exercises * on Linear Algebra Laurenz Wiskott Institut für Neuroinformatik Ruhr-Universität Bochum, Germany, EU 4 February 7 Contents Vector spaces 4. Definition...............................................
More informationA Proof of the Riemann Hypothesis and Determination of the Relationship Between Non- Trivial Zeros of Zeta Functions and Prime Numbers
A Proof of the Riemann Hypothesis and Determination of the Relationship Between on- Trivial Zeros of Zeta Functions and Prime umbers ABSTRACT Murad Ahmed Abu Amr MSc Degree in Physics, Mutah University
More informationLecture 1: Review of methods to solve Ordinary Differential Equations
Introductory lecture notes on Partial Differential Equations - c Anthony Peirce Not to be copied, used, or revised without explicit written permission from the copyright owner 1 Lecture 1: Review of methods
More informationMath 350 Fall 2011 Notes about inner product spaces. In this notes we state and prove some important properties of inner product spaces.
Math 350 Fall 2011 Notes about inner product spaces In this notes we state and prove some important properties of inner product spaces. First, recall the dot product on R n : if x, y R n, say x = (x 1,...,
More informationA proof for the full Fourier series on [ π, π] is given here.
niform convergence of Fourier series A smooth function on an interval [a, b] may be represented by a full, sine, or cosine Fourier series, and pointwise convergence can be achieved, except possibly at
More informationFourier and Partial Differential Equations
Chapter 5 Fourier and Partial Differential Equations 5.1 Fourier MATH 294 SPRING 1982 FINAL # 5 5.1.1 Consider the function 2x, 0 x 1. a) Sketch the odd extension of this function on 1 x 1. b) Expand the
More informationLINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008. This chapter is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,
More informationDS-GA 1002 Lecture notes 10 November 23, Linear models
DS-GA 2 Lecture notes November 23, 2 Linear functions Linear models A linear model encodes the assumption that two quantities are linearly related. Mathematically, this is characterized using linear functions.
More informationf(s) e -i n π s/l d s
Pointwise convergence of complex Fourier series Let f(x) be a periodic function with period l defined on the interval [,l]. The complex Fourier coefficients of f( x) are This leads to a Fourier series
More informationLECTURE 7. k=1 (, v k)u k. Moreover r
LECTURE 7 Finite rank operators Definition. T is said to be of rank r (r < ) if dim T(H) = r. The class of operators of rank r is denoted by K r and K := r K r. Theorem 1. T K r iff T K r. Proof. Let T
More informationSEISMIC WAVE PROPAGATION. Lecture 2: Fourier Analysis
SEISMIC WAVE PROPAGATION Lecture 2: Fourier Analysis Fourier Series & Fourier Transforms Fourier Series Review of trigonometric identities Analysing the square wave Fourier Transform Transforms of some
More informationMathematical Methods wk 1: Vectors
Mathematical Methods wk : Vectors John Magorrian, magog@thphysoxacuk These are work-in-progress notes for the second-year course on mathematical methods The most up-to-date version is available from http://www-thphysphysicsoxacuk/people/johnmagorrian/mm
More informationMathematical Methods wk 1: Vectors
Mathematical Methods wk : Vectors John Magorrian, magog@thphysoxacuk These are work-in-progress notes for the second-year course on mathematical methods The most up-to-date version is available from http://www-thphysphysicsoxacuk/people/johnmagorrian/mm
More informationMATH2210 Notebook 3 Spring 2018
MATH2210 Notebook 3 Spring 2018 prepared by Professor Jenny Baglivo c Copyright 2009 2018 by Jenny A. Baglivo. All Rights Reserved. 3 MATH2210 Notebook 3 3 3.1 Vector Spaces and Subspaces.................................
More informationmultiply both sides of eq. by a and projection overlap
Fourier Series n x n x f xa ancos bncos n n periodic with period x consider n, sin x x x March. 3, 7 Any function with period can be represented with a Fourier series Examples (sawtooth) (square wave)
More informationMath 4263 Homework Set 1
Homework Set 1 1. Solve the following PDE/BVP 2. Solve the following PDE/BVP 2u t + 3u x = 0 u (x, 0) = sin (x) u x + e x u y = 0 u (0, y) = y 2 3. (a) Find the curves γ : t (x (t), y (t)) such that that
More informationGeneral Inner Product & Fourier Series
General Inner Products 1 General Inner Product & Fourier Series Advanced Topics in Linear Algebra, Spring 2014 Cameron Braithwaite 1 General Inner Product The inner product is an algebraic operation that
More informationRank & nullity. Defn. Let T : V W be linear. We define the rank of T to be rank T = dim im T & the nullity of T to be nullt = dim ker T.
Rank & nullity Aim lecture: We further study vector space complements, which is a tool which allows us to decompose linear problems into smaller ones. We give an algorithm for finding complements & an
More informationFourier Series. Spectral Analysis of Periodic Signals
Fourier Series. Spectral Analysis of Periodic Signals he response of continuous-time linear invariant systems to the complex exponential with unitary magnitude response of a continuous-time LI system at
More informationFurther Mathematical Methods (Linear Algebra) 2002
Further Mathematical Methods (Linear Algebra) Solutions For Problem Sheet 9 In this problem sheet, we derived a new result about orthogonal projections and used them to find least squares approximations
More informationMath Computer Lab 4 : Fourier Series
Math 227 - Computer Lab 4 : Fourier Series Dylan Zwick Fall 212 This lab should be a pretty quick lab. It s goal is to introduce you to one of the coolest ideas in mathematics, the Fourier series, and
More informationMath Real Analysis II
Math 4 - Real Analysis II Solutions to Homework due May Recall that a function f is called even if f( x) = f(x) and called odd if f( x) = f(x) for all x. We saw that these classes of functions had a particularly
More informationMathematics of Information Spring semester 2018
Communication Technology Laboratory Prof. Dr. H. Bölcskei Sternwartstrasse 7 CH-809 Zürich Mathematics of Information Spring semester 08 Solution to Homework Problem Overcomplete expansion in R a) Consider
More informationOrthonormal Systems. Fourier Series
Yuliya Gorb Orthonormal Systems. Fourier Series October 31 November 3, 2017 Yuliya Gorb Orthonormal Systems (cont.) Let {e α} α A be an orthonormal set of points in an inner product space X. Then {e α}
More informationFunctional Analysis MATH and MATH M6202
Functional Analysis MATH 36202 and MATH M6202 1 Inner Product Spaces and Normed Spaces Inner Product Spaces Functional analysis involves studying vector spaces where we additionally have the notion of
More informationwhere the bar indicates complex conjugation. Note that this implies that, from Property 2, x,αy = α x,y, x,y X, α C.
Lecture 4 Inner product spaces Of course, you are familiar with the idea of inner product spaces at least finite-dimensional ones. Let X be an abstract vector space with an inner product, denoted as,,
More informationMath 259: Introduction to Analytic Number Theory More about the Gamma function
Math 59: Introduction to Analytic Number Theory More about the Gamma function We collect some more facts about Γs as a function of a complex variable that will figure in our treatment of ζs and Ls, χ.
More informationReal Variables # 10 : Hilbert Spaces II
randon ehring Real Variables # 0 : Hilbert Spaces II Exercise 20 For any sequence {f n } in H with f n = for all n, there exists f H and a subsequence {f nk } such that for all g H, one has lim (f n k,
More informationDivergent Series: why = 1/12. Bryden Cais
Divergent Series: why + + 3 + = /. Bryden Cais Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever.. H. Abel. Introduction The notion of convergence
More informationMTH 2032 SemesterII
MTH 202 SemesterII 2010-11 Linear Algebra Worked Examples Dr. Tony Yee Department of Mathematics and Information Technology The Hong Kong Institute of Education December 28, 2011 ii Contents Table of Contents
More informationwhich arises when we compute the orthogonal projection of a vector y in a subspace with an orthogonal basis. Hence assume that P y = A ij = x j, x i
MODULE 6 Topics: Gram-Schmidt orthogonalization process We begin by observing that if the vectors {x j } N are mutually orthogonal in an inner product space V then they are necessarily linearly independent.
More informationMATH 20F: LINEAR ALGEBRA LECTURE B00 (T. KEMP)
MATH 20F: LINEAR ALGEBRA LECTURE B00 (T KEMP) Definition 01 If T (x) = Ax is a linear transformation from R n to R m then Nul (T ) = {x R n : T (x) = 0} = Nul (A) Ran (T ) = {Ax R m : x R n } = {b R m
More informationChapter 1. Preliminaries. The purpose of this chapter is to provide some basic background information. Linear Space. Hilbert Space.
Chapter 1 Preliminaries The purpose of this chapter is to provide some basic background information. Linear Space Hilbert Space Basic Principles 1 2 Preliminaries Linear Space The notion of linear space
More informationOrthogonal Complements
Orthogonal Complements Definition Let W be a subspace of R n. If a vector z is orthogonal to every vector in W, then z is said to be orthogonal to W. The set of all such vectors z is called the orthogonal
More information