Lecture 11. Fourier transform
|
|
- Blanche Newman
- 5 years ago
- Views:
Transcription
1 Lecture. Fourier transform Definition and main resuts Let f L 2 (R). The Fourier transform of a function f is a function f(α) = f(x)t iαx dx () The normaized Fourier transform of f is a function R ˆf = f (2) It is not obvious that the Fourier transform exists for any function f L 2 R. This is proved beow. The main resuts of this ecture are: Theorem (Panchere equaity) f L 2 (R) ˆf L 2 (R), and f = ˆf. Theorem 2 [Inversion Formua] f L 2 (R), f(x) = f(α)e iαx dα R We wi prove these theorems for the case when f is a C 2 function with a finite support, and then extend it to the whoe of L 2 (R). The Fourier transform is a continuous anaog of the Fourier series. The Fourier series are defined on a circe, and the Fourier transform is defined on the rea ine. The atter is obtained from the first one by passing to the imit when the circe ength tends to infinity.
2 2 Decreasing of the Fourier transform for a smooth function Theorem 3 Fourier transform of a C m sower than x m. function with a compact support decreases no The proof is quite the same as for the Fourier series. Proof The Fourier transform brings the derivation to the mutipication by iα: f (α) = iα f(α). (3) This formua is proved by the integration by parts. Denoting the Fourier transform by F, we have: F(f )(α) = f (x)e iαx dx = e iαx df(x) = f(x)de iαx dx = iα f(x)e iαx dx = iαff(α). This impies By induction in M we get: Ff(α) = iα F(f )(α). Ff(α) = (iα) m F(f (m) )(α). But the function F(f (m) ) is bounded because f (m) is continuous with the finite support. Coroary If f C 2,0, then there exists a C such that: f(α) < C + α 2. (4) 3 Fourier series on a ong circe The circe may be represented by a segment with the endpoints identified. Another representation: the functions on a circe of ength T are represented by T -periodic functions on the rea ine. We wi use this atter representation. Consider a -periodic function f : R R and decompose it to the Fourier series with respect to a system that we wi now construct. Consider a -periodic function g(x) = f(x) and decompose it into a cassica Fourier series: g(x) = Σ k Z g k e ikx. (5) 2
3 The factors ik in the exponentia ar caed the wave numbers. For the cassica Fourier series the set of wave numbers coincides with Z. By definition of g, ( x ) f(x) = g = Σg k e ik x = Σ α Z c a e iαx (6) The wave numbers in the atter series range over the set Z. For the arge this set is much thicker than Z. Somewhat oosey we may say that as this set tends to R. Equaity (6) impies that the vectors e iαx, a Z form a basis in L 2 ([ π, π]). Their norms equa. Hence, for α Z, c α = π f(x)e iαx dx π Note that for α Z, c α = F(fχ [ π,π])(α) (7) The Panchere equaity for f has the form The Fourier decomposition of f is given by (6), (7). f 2 = Σ α Z c α 2. (8) 4 Passing to the imit: heuristic proofs of the Panchere equaity and Inversion Formua Fix a function f C 2,0 with the support supp f [ π 0, π 0 ], and for any > 0 consider the restriction of f to the segment [ π, π].formay, these restrictions are different functions, but we wi denote them a by f. Let us prove the Panchere equaity: f = ˆf. It is sufficient to prove that f 2 = f 2 (9) Let us extend the function f -periodicay, end decompose the resuting function to the Fourier series. By (8) and (7), f 2 = Σ α Z f(α) 2 := Σ. (0) 3
4 The expression Σ is an integra sum for the integra f 2 = f(α) 2 dα := I This sum corresponds to the partition of the ine by the segments of ength with the endpoints in the set Z. On one hand, this sum tends to the integra (this shoud be proved!); on the other hand, the sequence Σ is stationary (does not depend on ). This proves (9). In the same way, the Inversion Formua may be proved. By (6), for x <, S is an integra sum for the integra f(x) = Σ α Z I(x) = f(α)e iαx := S. f(α)e iαx dα. Passing to the imit as above (this taking of the imit shoud be justified), we obtain the Inversion Formua: f(x) = f(α)e iαx dα () 5 Forma proof Lemma S R f(α) 2 dα provided that: f C 2,0. Lemma 2 If C 2,0, then Σ (x) I(x) for. R The previous arguments and Lemma impy the Panchere equaity, and Lemma 2 impies the Inversion Formua. Proof [of Lemma ]. If the integra I have been proper, the integra sums woud tend to the integra by the cassica theorem about Riemann integras. We need to overcome an improper integra. This is done by the estimates beow. By Coroary, there exists C > 0 : f(α) < C( + α 2 ). Take ε > 0 and such N that Σ α Z \[ N,N] f(α) 2 < Σ C α Z \[ N,N] ( + α 2 ) 2 < C dα ( + α 2 ) 2 ε 3. α N Then α N f(α) 2 dα < ε 3. 4
5 Take so arge that Then S f 2 < ε. Σ α Z, α N f(α) 2 α N f(a) 2 dx < ε 3. Lemma 2 is proved in the same way. Concusion. We have proved Theorems and 2: the Panchere equaity and Inversion Formua for smooth functions f with the compact support. The goa: prove the same for f L 2. 6 Operators and their extensions Definition A map A : H H is a inear operator, provided that A(αξ +βη) = αa(ξ)+ βa(η). Definition 2 A map A is an isometry provided that Aξ = ξ ξ H; by defaut, A is inear. Theorem 4 Let E be a dense subset of H, A : E E H be an isometry. Then A may be extended to H up to an isometry A : H H. Proof Let x H, (x r ) E, x n x for n. Lety n = Ax n. Then (x n ) is a Cauchy sequence (y n ) is a Cauchy sequence as we. Let y = im n yn. Set A(x) = y. Exercise The map A is we defined: y depends on x ony, and not on x n x. The map A is an isometry because y n = x n y = x. Therefore, the Fourier transform may be extended to the whoe of L 2 (R) as an isometry; that is, on the whoe os this space the Panchere equaity hods. The Inversion Formua is extended to the space L 2 (R) in the same way. Namey, et, S : f(x) f( x) be an operator of the reversion of the indeterminate. The Inversion Formua is equivaent to the foowing one: F 2 = S. The operators in both sides of this equation are the isometries. Their coincidence on a dense set C 2,0 impies the same on the whoe space L 2 (R). 5
6 7 Case of many variabes In case of many variabes the formua for the Fourier transform has the same form () as for one variabe, with the use of the mutiindex notations. The formua for the normaized Fourier transform is: ˆf = f (2) () n The Panchere equaity for f L 2 (R n ) is quite the same as for n =. The inversion formua obtains a different factor f(x) = f(α)e iαx dα. () n A this is proved in the very same way as for n =. We wi use these facts, but wi not repeat the proofs. R n 6
2M2. Fourier Series Prof Bill Lionheart
M. Fourier Series Prof Bi Lionheart 1. The Fourier series of the periodic function f(x) with period has the form f(x) = a 0 + ( a n cos πnx + b n sin πnx ). Here the rea numbers a n, b n are caed the Fourier
More informationMat 1501 lecture notes, penultimate installment
Mat 1501 ecture notes, penutimate instament 1. bounded variation: functions of a singe variabe optiona) I beieve that we wi not actuay use the materia in this section the point is mainy to motivate the
More informationWeek 6 Lectures, Math 6451, Tanveer
Fourier Series Week 6 Lectures, Math 645, Tanveer In the context of separation of variabe to find soutions of PDEs, we encountered or and in other cases f(x = f(x = a 0 + f(x = a 0 + b n sin nπx { a n
More informationLecture Notes 4: Fourier Series and PDE s
Lecture Notes 4: Fourier Series and PDE s 1. Periodic Functions A function fx defined on R is caed a periodic function if there exists a number T > such that fx + T = fx, x R. 1.1 The smaest number T for
More informationMath 124B January 31, 2012
Math 124B January 31, 212 Viktor Grigoryan 7 Inhomogeneous boundary vaue probems Having studied the theory of Fourier series, with which we successfuy soved boundary vaue probems for the homogeneous heat
More informationMATH 172: MOTIVATION FOR FOURIER SERIES: SEPARATION OF VARIABLES
MATH 172: MOTIVATION FOR FOURIER SERIES: SEPARATION OF VARIABLES Separation of variabes is a method to sove certain PDEs which have a warped product structure. First, on R n, a inear PDE of order m is
More informationThe Group Structure on a Smooth Tropical Cubic
The Group Structure on a Smooth Tropica Cubic Ethan Lake Apri 20, 2015 Abstract Just as in in cassica agebraic geometry, it is possibe to define a group aw on a smooth tropica cubic curve. In this note,
More informationBASIC NOTIONS AND RESULTS IN TOPOLOGY. 1. Metric spaces. Sets with finite diameter are called bounded sets. For x X and r > 0 the set
BASIC NOTIONS AND RESULTS IN TOPOLOGY 1. Metric spaces A metric on a set X is a map d : X X R + with the properties: d(x, y) 0 and d(x, y) = 0 x = y, d(x, y) = d(y, x), d(x, y) d(x, z) + d(z, y), for a
More informationLecture Notes for Math 251: ODE and PDE. Lecture 32: 10.2 Fourier Series
Lecture Notes for Math 251: ODE and PDE. Lecture 32: 1.2 Fourier Series Shawn D. Ryan Spring 212 Last Time: We studied the heat equation and the method of Separation of Variabes. We then used Separation
More informationA Two-Parameter Trigonometric Series
A Two-Parameter Trigonometric Series Xiang-Qian Chang Xiang.Chang@bos.mcphs.edu Massachusetts Coege of Pharmacy and Heath Sciences Boston MA 25 Let us consider the foowing dua questions. A. Givenafunction
More informationCS229 Lecture notes. Andrew Ng
CS229 Lecture notes Andrew Ng Part IX The EM agorithm In the previous set of notes, we taked about the EM agorithm as appied to fitting a mixture of Gaussians. In this set of notes, we give a broader view
More informationLECTURE NOTES 8 THE TRACELESS SYMMETRIC TENSOR EXPANSION AND STANDARD SPHERICAL HARMONICS
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.07: Eectromagnetism II October, 202 Prof. Aan Guth LECTURE NOTES 8 THE TRACELESS SYMMETRIC TENSOR EXPANSION AND STANDARD SPHERICAL HARMONICS
More information221B Lecture Notes Notes on Spherical Bessel Functions
Definitions B Lecture Notes Notes on Spherica Besse Functions We woud ike to sove the free Schrödinger equation [ h d r R(r) = h k R(r). () m r dr r m R(r) is the radia wave function ψ( x) = R(r)Y m (θ,
More informationMath 124B January 17, 2012
Math 124B January 17, 212 Viktor Grigoryan 3 Fu Fourier series We saw in previous ectures how the Dirichet and Neumann boundary conditions ead to respectivey sine and cosine Fourier series of the initia
More information1D Heat Propagation Problems
Chapter 1 1D Heat Propagation Probems If the ambient space of the heat conduction has ony one dimension, the Fourier equation reduces to the foowing for an homogeneous body cρ T t = T λ 2 + Q, 1.1) x2
More information14 Separation of Variables Method
14 Separation of Variabes Method Consider, for exampe, the Dirichet probem u t = Du xx < x u(x, ) = f(x) < x < u(, t) = = u(, t) t > Let u(x, t) = T (t)φ(x); now substitute into the equation: dt
More informationFourier Series. 10 (D3.9) Find the Cesàro sum of the series. 11 (D3.9) Let a and b be real numbers. Under what conditions does a series of the form
Exercises Fourier Anaysis MMG70, Autumn 007 The exercises are taken from: Version: Monday October, 007 DXY Section XY of H F Davis, Fourier Series and orthogona functions EÖ Some exercises from earier
More informationMARKOV CHAINS AND MARKOV DECISION THEORY. Contents
MARKOV CHAINS AND MARKOV DECISION THEORY ARINDRIMA DATTA Abstract. In this paper, we begin with a forma introduction to probabiity and expain the concept of random variabes and stochastic processes. After
More informationQuantum Mechanical Models of Vibration and Rotation of Molecules Chapter 18
Quantum Mechanica Modes of Vibration and Rotation of Moecues Chapter 18 Moecuar Energy Transationa Vibrationa Rotationa Eectronic Moecuar Motions Vibrations of Moecues: Mode approximates moecues to atoms
More informationAssignment 7 Due Tuessday, March 29, 2016
Math 45 / AMCS 55 Dr. DeTurck Assignment 7 Due Tuessday, March 9, 6 Topics for this week Convergence of Fourier series; Lapace s equation and harmonic functions: basic properties, compuations on rectanges
More informationarxiv: v1 [math.fa] 23 Aug 2018
An Exact Upper Bound on the L p Lebesgue Constant and The -Rényi Entropy Power Inequaity for Integer Vaued Random Variabes arxiv:808.0773v [math.fa] 3 Aug 08 Peng Xu, Mokshay Madiman, James Mebourne Abstract
More informationAn Extension of Almost Sure Central Limit Theorem for Order Statistics
An Extension of Amost Sure Centra Limit Theorem for Order Statistics T. Bin, P. Zuoxiang & S. Nadarajah First version: 6 December 2007 Research Report No. 9, 2007, Probabiity Statistics Group Schoo of
More informationHomework 5 Solutions
Stat 310B/Math 230B Theory of Probabiity Homework 5 Soutions Andrea Montanari Due on 2/19/2014 Exercise [5.3.20] 1. We caim that n 2 [ E[h F n ] = 2 n i=1 A i,n h(u)du ] I Ai,n (t). (1) Indeed, integrabiity
More information6 Wave Equation on an Interval: Separation of Variables
6 Wave Equation on an Interva: Separation of Variabes 6.1 Dirichet Boundary Conditions Ref: Strauss, Chapter 4 We now use the separation of variabes technique to study the wave equation on a finite interva.
More informationCourse 2BA1, Section 11: Periodic Functions and Fourier Series
Course BA, 8 9 Section : Periodic Functions and Fourier Series David R. Wikins Copyright c David R. Wikins 9 Contents Periodic Functions and Fourier Series 74. Fourier Series of Even and Odd Functions...........
More informationFourier Transform & Sobolev Spaces
Fourier Transform & Sobolev Spaces Michael Reiter, Arthur Schuster Summer Term 2008 Abstract We introduce the concept of weak derivative that allows us to define new interesting Hilbert spaces the Sobolev
More informationWave Equation Dirichlet Boundary Conditions
Wave Equation Dirichet Boundary Conditions u tt x, t = c u xx x, t, < x 1 u, t =, u, t = ux, = fx u t x, = gx Look for simpe soutions in the form ux, t = XxT t Substituting into 13 and dividing
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 informationBourgain s Theorem. Computational and Metric Geometry. Instructor: Yury Makarychev. d(s 1, s 2 ).
Bourgain s Theorem Computationa and Metric Geometry Instructor: Yury Makarychev 1 Notation Given a metric space (X, d) and S X, the distance from x X to S equas d(x, S) = inf d(x, s). s S The distance
More information4 Separation of Variables
4 Separation of Variabes In this chapter we describe a cassica technique for constructing forma soutions to inear boundary vaue probems. The soution of three cassica (paraboic, hyperboic and eiptic) PDE
More informationCompletion. is dense in H. If V is complete, then U(V) = H.
Competion Theorem 1 (Competion) If ( V V ) is any inner product space then there exists a Hibert space ( H H ) and a map U : V H such that (i) U is 1 1 (ii) U is inear (iii) UxUy H xy V for a xy V (iv)
More informationSome Measures for Asymmetry of Distributions
Some Measures for Asymmetry of Distributions Georgi N. Boshnakov First version: 31 January 2006 Research Report No. 5, 2006, Probabiity and Statistics Group Schoo of Mathematics, The University of Manchester
More informationHaar Decomposition and Reconstruction Algorithms
Jim Lambers MAT 773 Fa Semester 018-19 Lecture 15 and 16 Notes These notes correspond to Sections 4.3 and 4.4 in the text. Haar Decomposition and Reconstruction Agorithms Decomposition Suppose we approximate
More informationTHE REACHABILITY CONES OF ESSENTIALLY NONNEGATIVE MATRICES
THE REACHABILITY CONES OF ESSENTIALLY NONNEGATIVE MATRICES by Michae Neumann Department of Mathematics, University of Connecticut, Storrs, CT 06269 3009 and Ronad J. Stern Department of Mathematics, Concordia
More informationMore Scattering: the Partial Wave Expansion
More Scattering: the Partia Wave Expansion Michae Fower /7/8 Pane Waves and Partia Waves We are considering the soution to Schrödinger s equation for scattering of an incoming pane wave in the z-direction
More informationHomogeneity properties of subadditive functions
Annaes Mathematicae et Informaticae 32 2005 pp. 89 20. Homogeneity properties of subadditive functions Pá Burai and Árpád Száz Institute of Mathematics, University of Debrecen e-mai: buraip@math.kte.hu
More informationPartial permutation decoding for MacDonald codes
Partia permutation decoding for MacDonad codes J.D. Key Department of Mathematics and Appied Mathematics University of the Western Cape 7535 Bevie, South Africa P. Seneviratne Department of Mathematics
More informationADELIC ANALYSIS AND FUNCTIONAL ANALYSIS ON THE FINITE ADELE RING. Ilwoo Cho
Opuscua Math. 38, no. 2 208, 39 85 https://doi.org/0.7494/opmath.208.38.2.39 Opuscua Mathematica ADELIC ANALYSIS AND FUNCTIONAL ANALYSIS ON THE FINITE ADELE RING Iwoo Cho Communicated by.a. Cojuhari Abstract.
More informationSmoothers for ecient multigrid methods in IGA
Smoothers for ecient mutigrid methods in IGA Cemens Hofreither, Stefan Takacs, Water Zuehner DD23, Juy 2015 supported by The work was funded by the Austrian Science Fund (FWF): NFN S117 (rst and third
More informationOn Some Basic Properties of Geometric Real Sequences
On Some Basic Properties of eometric Rea Sequences Khirod Boruah Research Schoar, Department of Mathematics, Rajiv andhi University Rono His, Doimukh-791112, Arunacha Pradesh, India Abstract Objective
More information4 1-D Boundary Value Problems Heat Equation
4 -D Boundary Vaue Probems Heat Equation The main purpose of this chapter is to study boundary vaue probems for the heat equation on a finite rod a x b. u t (x, t = ku xx (x, t, a < x < b, t > u(x, = ϕ(x
More informationOn the Number of Limit Cycles for Discontinuous Generalized Liénard Polynomial Differential Systems
Internationa Journa of Bifurcation and Chaos Vo. 25 No. 10 2015 1550131 10 pages c Word Scientific Pubishing Company DOI: 10.112/S02181271550131X On the Number of Limit Cyces for Discontinuous Generaized
More informationThe Arzelà-Ascoli Theorem
John Nachbar Washington University March 27, 2016 The Arzelà-Ascoli Theorem The Arzelà-Ascoli Theorem gives sufficient conditions for compactness in certain function spaces. Among other things, it helps
More informationSingular Integrals. 1 Calderon-Zygmund decomposition
Singular Integrals Analysis III Calderon-Zygmund decomposition Let f be an integrable function f dx 0, f = g + b with g Cα almost everywhere, with b
More informationXSAT of linear CNF formulas
XSAT of inear CN formuas Bernd R. Schuh Dr. Bernd Schuh, D-50968 Kön, Germany; bernd.schuh@netcoogne.de eywords: compexity, XSAT, exact inear formua, -reguarity, -uniformity, NPcompeteness Abstract. Open
More informationORTHOGONAL MULTI-WAVELETS FROM MATRIX FACTORIZATION
J. Korean Math. Soc. 46 2009, No. 2, pp. 281 294 ORHOGONAL MLI-WAVELES FROM MARIX FACORIZAION Hongying Xiao Abstract. Accuracy of the scaing function is very crucia in waveet theory, or correspondingy,
More informationDavid Eigen. MA112 Final Paper. May 10, 2002
David Eigen MA112 Fina Paper May 1, 22 The Schrodinger equation describes the position of an eectron as a wave. The wave function Ψ(t, x is interpreted as a probabiity density for the position of the eectron.
More informationNOISE-INDUCED STABILIZATION OF STOCHASTIC DIFFERENTIAL EQUATIONS
NOISE-INDUCED STABILIZATION OF STOCHASTIC DIFFERENTIAL EQUATIONS TONY ALLEN, EMILY GEBHARDT, AND ADAM KLUBALL 3 ADVISOR: DR. TIFFANY KOLBA 4 Abstract. The phenomenon of noise-induced stabiization occurs
More informationLecture Notes for Math 251: ODE and PDE. Lecture 34: 10.7 Wave Equation and Vibrations of an Elastic String
ecture Notes for Math 251: ODE and PDE. ecture 3: 1.7 Wave Equation and Vibrations of an Eastic String Shawn D. Ryan Spring 212 ast Time: We studied other Heat Equation probems with various other boundary
More informationSTATISTICS ON HILBERT S SIXTEENTH PROBLEM
STATISTICS ON HILBERT S SIXTEENTH PROBLEM ANTONIO LERARIO AND ERIK LUNDBERG Abstract. We study the statistics of the number of connected components and the voume of a random rea agebraic hypersurface in
More informationFormulas for Angular-Momentum Barrier Factors Version II
BNL PREPRINT BNL-QGS-06-101 brfactor1.tex Formuas for Anguar-Momentum Barrier Factors Version II S. U. Chung Physics Department, Brookhaven Nationa Laboratory, Upton, NY 11973 March 19, 2015 abstract A
More informationV.B The Cluster Expansion
V.B The Custer Expansion For short range interactions, speciay with a hard core, it is much better to repace the expansion parameter V( q ) by f(q ) = exp ( βv( q )) 1, which is obtained by summing over
More informationFRIEZE GROUPS IN R 2
FRIEZE GROUPS IN R 2 MAXWELL STOLARSKI Abstract. Focusing on the Eucidean pane under the Pythagorean Metric, our goa is to cassify the frieze groups, discrete subgroups of the set of isometries of the
More informationMost Continuous Functions are Nowhere Differentiable
Most Continuous Functions are Nowhere Differentiable Spring 2004 The Space of Continuous Functions Let K = [0, 1] and let C(K) be the set of all continuous functions f : K R. Definition 1 For f C(K) we
More informationPHYS 110B - HW #1 Fall 2005, Solutions by David Pace Equations referenced as Eq. # are from Griffiths Problem statements are paraphrased
PHYS 110B - HW #1 Fa 2005, Soutions by David Pace Equations referenced as Eq. # are from Griffiths Probem statements are paraphrased [1.] Probem 6.8 from Griffiths A ong cyinder has radius R and a magnetization
More informationMath 124B January 24, 2012
Mth 24B Jnury 24, 22 Viktor Grigoryn 5 Convergence of Fourier series Strting from the method of seprtion of vribes for the homogeneous Dirichet nd Neumnn boundry vue probems, we studied the eigenvue probem
More information1 Fourier Integrals on L 2 (R) and L 1 (R).
18.103 Fall 2013 1 Fourier Integrals on L 2 () and L 1 (). The first part of these notes cover 3.5 of AG, without proofs. When we get to things not covered in the book, we will start giving proofs. The
More informationNotes on Complex Analysis
Michael Papadimitrakis Notes on Complex Analysis Department of Mathematics University of Crete Contents The complex plane.. The complex plane...................................2 Argument and polar representation.........................
More informationV.B The Cluster Expansion
V.B The Custer Expansion For short range interactions, speciay with a hard core, it is much better to repace the expansion parameter V( q ) by f( q ) = exp ( βv( q )), which is obtained by summing over
More informationLecture Note 3: Stationary Iterative Methods
MATH 5330: Computationa Methods of Linear Agebra Lecture Note 3: Stationary Iterative Methods Xianyi Zeng Department of Mathematica Sciences, UTEP Stationary Iterative Methods The Gaussian eimination (or
More informationMath 205b Homework 2 Solutions
Math 5b Homework Solutions January 5, 5 Problem (R-S, II.) () For the R case, we just expand the right hand side and use the symmetry of the inner product: ( x y x y ) = = ((x, x) (y, y) (x, y) (y, x)
More information1/12/05: sec 3.1 and my article: How good is the Lebesgue measure?, Math. Intelligencer 11(2) (1989),
Real Analysis 2, Math 651, Spring 2005 April 26, 2005 1 Real Analysis 2, Math 651, Spring 2005 Krzysztof Chris Ciesielski 1/12/05: sec 3.1 and my article: How good is the Lebesgue measure?, Math. Intelligencer
More informationMAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9
MAT 570 REAL ANALYSIS LECTURE NOTES PROFESSOR: JOHN QUIGG SEMESTER: FALL 204 Contents. Sets 2 2. Functions 5 3. Countability 7 4. Axiom of choice 8 5. Equivalence relations 9 6. Real numbers 9 7. Extended
More informationb n n=1 a n cos nx (3) n=1
Fourier Anaysis The Fourier series First some terminoogy: a function f(x) is periodic if f(x ) = f(x) for a x for some, if is the smaest such number, it is caed the period of f(x). It is even if f( x)
More informationAnalysis Finite and Infinite Sets The Real Numbers The Cantor Set
Analysis Finite and Infinite Sets Definition. An initial segment is {n N n n 0 }. Definition. A finite set can be put into one-to-one correspondence with an initial segment. The empty set is also considered
More informationx y More precisely, this equation means that given any ε > 0, there exists some δ > 0 such that
Chapter 2 Limits and continuity 21 The definition of a it Definition 21 (ε-δ definition) Let f be a function and y R a fixed number Take x to be a point which approaches y without being equal to y If there
More informationIndeed, the family is still orthogonal if we consider a complex valued inner product ( or an inner product on complex vector space)
Fourier series of complex valued functions Suppose now f is a piecewise continuous complex valued function on [, π], that is f(x) = u(x)+iv(x) such that both u and v are real valued piecewise continuous
More informationLECTURE 5: THE METHOD OF STATIONARY PHASE
LECTURE 5: THE METHOD OF STATIONARY PHASE Some notions.. A crash course on Fourier transform For j =,, n, j = x j. D j = i j. For any multi-index α = (α,, α n ) N n. α = α + + α n. α! = α! α n!. x α =
More informationLecture 17 - The Secrets we have Swept Under the Rug
Lecture 17 - The Secrets we have Swept Under the Rug Today s ectures examines some of the uirky features of eectrostatics that we have negected up unti this point A Puzze... Let s go back to the basics
More informationf(x)e ikx dx. (19.1) ˆf(k)e ikx dk. (19.2)
9 Fourier transform 9 A first look at the Fourier transform In Math 66 you all studied the Laplace transform, which was used to turn an ordinary differential equation into an algebraic one There are a
More informationPoisson summation and the discrete Fourier transform John Kerl University of Arizona, Math 523B April 14, 2006
Poisson summation and the discrete Fourier transform John Kerl University of Arizona, Math 523B April 4, 2006 In Euclidean space, given a vector,... 2 ... we can put down a coordinate frame (say an orthonormal
More informationReichenbachian Common Cause Systems
Reichenbachian Common Cause Systems G. Hofer-Szabó Department of Phiosophy Technica University of Budapest e-mai: gszabo@hps.ete.hu Mikós Rédei Department of History and Phiosophy of Science Eötvös University,
More informationPricing Multiple Products with the Multinomial Logit and Nested Logit Models: Concavity and Implications
Pricing Mutipe Products with the Mutinomia Logit and Nested Logit Modes: Concavity and Impications Hongmin Li Woonghee Tim Huh WP Carey Schoo of Business Arizona State University Tempe Arizona 85287 USA
More informationSTABLE GRAPHS BENJAMIN OYE
STABLE GRAPHS BENJAMIN OYE Abstract. In Reguarity Lemmas for Stabe Graphs [1] Maiaris and Sheah appy toos from mode theory to obtain stronger forms of Ramsey's theorem and Szemeredi's reguarity emma for
More informationRelated Topics Maxwell s equations, electrical eddy field, magnetic field of coils, coil, magnetic flux, induced voltage
Magnetic induction TEP Reated Topics Maxwe s equations, eectrica eddy fied, magnetic fied of cois, coi, magnetic fux, induced votage Principe A magnetic fied of variabe frequency and varying strength is
More informationWe denote the space of distributions on Ω by D ( Ω) 2.
Sep. 1 0, 008 Distributions Distributions are generalized functions. Some familiarity with the theory of distributions helps understanding of various function spaces which play important roles in the study
More informationThus f is continuous at x 0. Matthew Straughn Math 402 Homework 6
Matthew Straughn Math 402 Homework 6 Homework 6 (p. 452) 14.3.3, 14.3.4, 14.3.5, 14.3.8 (p. 455) 14.4.3* (p. 458) 14.5.3 (p. 460) 14.6.1 (p. 472) 14.7.2* Lemma 1. If (f (n) ) converges uniformly to some
More informationGeneralized Bell polynomials and the combinatorics of Poisson central moments
Generaized Be poynomias and the combinatorics of Poisson centra moments Nicoas Privaut Division of Mathematica Sciences Schoo of Physica and Mathematica Sciences Nanyang Technoogica University SPMS-MAS-05-43,
More information$, (2.1) n="# #. (2.2)
Chapter. Eectrostatic II Notes: Most of the materia presented in this chapter is taken from Jackson, Chap.,, and 4, and Di Bartoo, Chap... Mathematica Considerations.. The Fourier series and the Fourier
More informationBohr s atomic model. 1 Ze 2 = mv2. n 2 Z
Bohr s atomic mode Another interesting success of the so-caed od quantum theory is expaining atomic spectra of hydrogen and hydrogen-ike atoms. The eectromagnetic radiation emitted by free atoms is concentrated
More informationAn explicit Jordan Decomposition of Companion matrices
An expicit Jordan Decomposition of Companion matrices Fermín S V Bazán Departamento de Matemática CFM UFSC 88040-900 Forianópois SC E-mai: fermin@mtmufscbr S Gratton CERFACS 42 Av Gaspard Coriois 31057
More informationThe Binary Space Partitioning-Tree Process Supplementary Material
The inary Space Partitioning-Tree Process Suppementary Materia Xuhui Fan in Li Scott. Sisson Schoo of omputer Science Fudan University ibin@fudan.edu.cn Schoo of Mathematics and Statistics University of
More informationHow the backpropagation algorithm works Srikumar Ramalingam School of Computing University of Utah
How the backpropagation agorithm works Srikumar Ramaingam Schoo of Computing University of Utah Reference Most of the sides are taken from the second chapter of the onine book by Michae Nieson: neuranetworksanddeepearning.com
More information6.434J/16.391J Statistics for Engineers and Scientists May 4 MIT, Spring 2006 Handout #17. Solution 7
6.434J/16.391J Statistics for Engineers and Scientists May 4 MIT, Spring 2006 Handout #17 Soution 7 Probem 1: Generating Random Variabes Each part of this probem requires impementation in MATLAB. For the
More information(f) is called a nearly holomorphic modular form of weight k + 2r as in [5].
PRODUCTS OF NEARLY HOLOMORPHIC EIGENFORMS JEFFREY BEYERL, KEVIN JAMES, CATHERINE TRENTACOSTE, AND HUI XUE Abstract. We prove that the product of two neary hoomorphic Hece eigenforms is again a Hece eigenform
More informationSample Problems for Third Midterm March 18, 2013
Mat 30. Treibergs Sampe Probems for Tird Midterm Name: Marc 8, 03 Questions 4 appeared in my Fa 000 and Fa 00 Mat 30 exams (.)Let f : R n R n be differentiabe at a R n. (a.) Let g : R n R be defined by
More informationUNIFORM CONVERGENCE OF MULTIPLIER CONVERGENT SERIES
royecciones Vo. 26, N o 1, pp. 27-35, May 2007. Universidad Catóica de Norte Antofagasta - Chie UNIFORM CONVERGENCE OF MULTILIER CONVERGENT SERIES CHARLES SWARTZ NEW MEXICO STATE UNIVERSITY Received :
More informationHYDROGEN ATOM SELECTION RULES TRANSITION RATES
DOING PHYSICS WITH MATLAB QUANTUM PHYSICS Ian Cooper Schoo of Physics, University of Sydney ian.cooper@sydney.edu.au HYDROGEN ATOM SELECTION RULES TRANSITION RATES DOWNLOAD DIRECTORY FOR MATLAB SCRIPTS
More informationC. Fourier Sine Series Overview
12 PHILIP D. LOEWEN C. Fourier Sine Series Overview Let some constant > be given. The symboic form of the FSS Eigenvaue probem combines an ordinary differentia equation (ODE) on the interva (, ) with a
More informationEstablishment of Weak Conditions for Darboux- Goursat-Beudon Theorem
Georgia Southern University Digita Commons@Georgia Southern Mathematica Sciences Facuty Pubications Department of Mathematica Sciences 2009 Estabishment of Weak Conditions for Darboux- Goursat-Beudon Theorem
More informationMAS 315 Waves 1 of 8 Answers to Examples Sheet 1. To solve the three problems, we use the methods of 1.3 (with necessary changes in notation).
MAS 35 Waves of 8 Answers to Exampes Sheet. From.) and.5), the genera soution of φ xx = c φ yy is φ = fx cy) + gx + cy). Put c = : the genera soution of φ xx = φ yy is therefore φ = fx y) + gx + y) ) To
More informationMath 172 Problem Set 8 Solutions
Math 72 Problem Set 8 Solutions Problem. (i We have (Fχ [ a,a] (ξ = χ [ a,a] e ixξ dx = a a e ixξ dx = iξ (e iax e iax = 2 sin aξ. ξ (ii We have (Fχ [, e ax (ξ = e ax e ixξ dx = e x(a+iξ dx = a + iξ where
More informationLet R be the line parameterized by x. Let f be a complex function on R that is integrable. The Fourier transform ˆf = F f is. e ikx f(x) dx. (1.
Chapter 1 Fourier transforms 1.1 Introduction Let R be the line parameterized by x. Let f be a complex function on R that is integrable. The Fourier transform ˆf = F f is ˆf(k) = e ikx f(x) dx. (1.1) It
More informationTHE THREE POINT STEINER PROBLEM ON THE FLAT TORUS: THE MINIMAL LUNE CASE
THE THREE POINT STEINER PROBLEM ON THE FLAT TORUS: THE MINIMAL LUNE CASE KATIE L. MAY AND MELISSA A. MITCHELL Abstract. We show how to identify the minima path network connecting three fixed points on
More information(Refer Slide Time: 2:34) L C V
Microwave Integrated Circuits Professor Jayanta Mukherjee Department of Eectrica Engineering Indian Intitute of Technoogy Bombay Modue 1 Lecture No 2 Refection Coefficient, SWR, Smith Chart. Heo wecome
More informationSummer Jump-Start Program for Analysis, 2012 Song-Ying Li
Summer Jump-Start Program for Analysis, 01 Song-Ying Li 1 Lecture 6: Uniformly continuity and sequence of functions 1.1 Uniform Continuity Definition 1.1 Let (X, d 1 ) and (Y, d ) are metric spaces and
More informationDo Schools Matter for High Math Achievement? Evidence from the American Mathematics Competitions Glenn Ellison and Ashley Swanson Online Appendix
VOL. NO. DO SCHOOLS MATTER FOR HIGH MATH ACHIEVEMENT? 43 Do Schoos Matter for High Math Achievement? Evidence from the American Mathematics Competitions Genn Eison and Ashey Swanson Onine Appendix Appendix
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 informationSobolev Spaces. Chapter 10
Chapter 1 Sobolev Spaces We now define spaces H 1,p (R n ), known as Sobolev spaces. For u to belong to H 1,p (R n ), we require that u L p (R n ) and that u have weak derivatives of first order in L p
More informationA. Distribution of the test statistic
A. Distribution of the test statistic In the sequentia test, we first compute the test statistic from a mini-batch of size m. If a decision cannot be made with this statistic, we keep increasing the mini-batch
More information