Differential Equations and Linear Algebra

Total Page:16

File Type:pdf, Size:1020Kb

Differential Equations and Linear Algebra Differential Equations and Linear Algebra Jason Underdown December 8, 2014 Contents Chapter 1. First Order Equations1 1. Differential Equations and Modeling1 2. Integrals as General and Particular Solutions5 3. Slope Fields and Solution Curves9 4. Separable Equations and Applications 13 5. Linear First{Order Equations 20 6. Application: Salmon Smolt Migration Model 26 7. Homogeneous Equations 28 Chapter 2. Models and Numerical Methods 31 1. Population Models 31 2. Equilibrium Solutions and Stability 34 3. Acceleration{Velocity Models 39 4. Numerical Solutions 41 Chapter 3. Linear Systems and Matrices 45 1. Linear and Homogeneous Equations 45 2. Introduction to Linear Systems 47 3. Matrices and Gaussian Elimination 50 4. Reduced Row{Echelon Matrices 53 5. Matrix Arithmetic and Matrix Equations 53 6. Matrices are Functions 53 7. Inverses of Matrices 57 8. Determinants 58 Chapter 4. Vector Spaces 61 i ii Contents 1. Basics 61 2. Linear Independence 64 3. Vector Subspaces 65 4. Affine Spaces 65 5. Bases and Dimension 66 6. Abstract Vector Spaces 67 Chapter 5. Higher Order Linear Differential Equations 69 1. Homogeneous Differential Equations 69 2. Linear Equations with Constant Coefficients 70 3. Mechanical Vibrations 74 4. The Method of Undetermined Coefficients 76 5. The Method of Variation of Parameters 78 6. Forced Oscillators and Resonance 80 7. Damped Driven Oscillators 84 Chapter 6. Laplace Transforms 87 1. The Laplace Transform 87 2. The Inverse Laplace Transform 92 3. Laplace Transform Method of Solving IVPs 94 4. Switching 101 5. Convolution 102 Chapter 7. Eigenvalues and Eigenvectors 105 1. Introduction to Eigenvalues and Eigenvectors 105 2. Algorithm for Computing Eigenvalues and Eigenvectors 107 Chapter 8. Systems of Differential Equations 109 1. First Order Systems 109 2. Transforming a Linear DE Into a System of First Order DEs 112 3. Complex Eigenvalues and Eigenvectors 113 4. Second Order Systems 115 Chapter 1 First Order Equations 1. Differential Equations and Modeling A differential equation is simply any equation that involves a function, say y(x) and any of its derivatives. For example, (1) y00 = −y: The above equation uses the prime notation (0) to denote the derivative, which has the benefit of resulting in compact equations. However, the prime notation has the drawback that it does not indicate what the independent variable is. By just looking at equation1 you can't tell if the independent variable is x or t or some other variable. That is, we don't know if we're looking for y(x) or y(t). So sometimes we will write our differential equations using the more verbose, but also more clear Leibniz notation. d2y (1) = −y dx2 In the Leibniz notation, the dependent variable, in this case y, always appears in the numerator of the derivative, and the independent variable always appears in the denominator of the derivative. Definition 1.1. The order of a differential equation is the order of the highest derivative that appears in it. So the order of the previous equation is two. The order of the following equation is also two: (2) x(y00)2 = 36(y + x): Even though y00 is squared in the equation, the highest order derivative is still just a second order derivative. 1 2 1. First Order Equations Our primary goal is to solve differential equations. Solving a differential equa- tion requires us to find a function, that satisfies the equation. This simply means that if you replace every occurence of y in the differential equation with the found function, you get a valid equation. There are some similarities between solving differential equations and solving polynomial equations. For example, given a polynomial equation such as 3x2 − 4x = 4; it is easy to verify that x = 2 is a solution to the equation simply by substituting 2 in for x in the equation and checking whether the resulting statement is true. Analogously, it is easy to verify that y(x) = cos x satisfies, or is a solution to equation1 by simply substituting cos x in for y in the equation and then checking if the resulting statement is true. (cos x)00 =? − cos x (− sin x)0 =? − cos x ? − cos x = − cos x X The biggest difference is that in the case of a polynomial equation our solutions took the form of real numbers, but in the differential equation case, our solutions take the form of functions. Example 1.2. Verify that y(x) = x3 − x is a solution of equation2. y00 = 6x ) x(y00)2 = x(6x)2 = 36x3 = 36(y + x) 4 A basic study of differential equations involves two facets. Creating differential equations which encode the behavior of some real life situation. This is called modeling. The other facet is of course developing systematic solution techniques. We will examine both, but we will focus on developing solution techniques. 1.1. Mathematical Modeling. Imagine a large population or colony of bacteria in a petri dish. Suppose we wish to model the growth of bacteria in the dish. How could we go about that? Well, we have to start with some educated guesses or assumptions. Assume that the rate of change of this colony in terms of population is directly proportional to the current number of bacteria. That is to say that a larger popu- lation will produce more offspring than a smaller population during the same time interval. This seems reasonable, since we know that a single bacterium reproduces by splitting into two bacteria, and hence more bacteria will result in more offspring. How do we translate this into symbolic language? (3) ∆P = P ∆t 1. Differential Equations and Modeling 3 This says that the change in a population depends on the size of the population and the length of the time interval over which we make our population measure- ments. So if the time interval is short, then the population change will also be small. Similarly it roughly says that more bacteria correspond to more offspring, and vice versa. But if you look closely, the left hand side of equation3 has units of number of bacteria, while the right hand side has units of number of bacteria times time. The equation can't possibly be correct if the units don't match. However to fix this we can multiply the left hand side by some parameter which has units of time, or we can multiply the right hand side by some parameter which has units of 1/time. Let's multiply the right hand side by a parameter k which has units of 1/time. Then our equation becomes: (4) ∆P = kP ∆t Dividing both sides of the equation by ∆t and taking the limit as ∆t goes to zero, we get: ∆P dP lim = = kP ∆t!0 ∆t dt dP (5) = kP dt Here k is a constant of proportionality, a real number which allows us to balance the units on both sides of the equation and it also affords some freedom. In essence it allows us to defer saying how closely P and its derivative are related. If k is a large positive number, then that would imply a large rate of change, and a small positive number greater than zero but less than one would be a small rate of change. If k is negative then that would imply the population is shrinking in number. Example 1.3. If we let P (t) = Cekt, then a simple differentiation reveals that this is a solution to our population model in equation5. Suppose that at time 0, there are 1000 bacteria in the dish. After one hour the population doubles to 2000. This data corresponds to the following two equations which allow us to solve for both C and k: 1000 = P (0) = Ce0 = C =) C = 1000 2000 = P (1) = Cek The second equation implies 2000 = 1000ek which is equivalent to 2 = ek which is equivalent to k = ln 2. Thus we see that with these two bits of data we now know: P (t) = 1000eln(2)·t = 1000(eln(2))t = 1000 · 2t This agrees exactly with our knowledge that bacteria multiply by splitting into two. 4 4 1. First Order Equations 1.2. Linear vs. Nonlinear. As you may have surmised we will not be able to exactly solve every differential equation that you can imagine. So it will be important to recognize which equations we can solve and those which we can't. It turns out that a certain class of equations called linear equations are very amenable to several solution techniques and will always have a solution (under modest assumptions), whereas the complementary set of nonlinear equations are not always solvable. A linear differential equation is any differential equation where solution func- tions can be summed or scaled to get new solutions. Stated precisely, we mean: Definition 1.4. A differential equation is linear is equivalent to saying: If y1(x) and y2(x) are any solutions to the differential equation, and c is any scalar (real) number, then (1) y1(x) + y2(x) will be a solution and, (2) cy1(x) will be a solution. This is a working definition, which we will change later. We will use it for now because it is simple to remember and does capture the essence of linearity, but we will see later on that we can make the definition more inclusive. That is to say that there are linear differential equations which don't satisfy our current definition until after a certain piece of the equation has been removed. Example 1.5. Show that y1(x) + y2(x) is a solution to equation1 when y1(x) = cos x and y2(x) = sin x.
Recommended publications
  • FUNCTIONAL ANALYSIS 1. Banach and Hilbert Spaces in What
    FUNCTIONAL ANALYSIS PIOTR HAJLASZ 1. Banach and Hilbert spaces In what follows K will denote R of C. Definition. A normed space is a pair (X, k · k), where X is a linear space over K and k · k : X → [0, ∞) is a function, called a norm, such that (1) kx + yk ≤ kxk + kyk for all x, y ∈ X; (2) kαxk = |α|kxk for all x ∈ X and α ∈ K; (3) kxk = 0 if and only if x = 0. Since kx − yk ≤ kx − zk + kz − yk for all x, y, z ∈ X, d(x, y) = kx − yk defines a metric in a normed space. In what follows normed paces will always be regarded as metric spaces with respect to the metric d. A normed space is called a Banach space if it is complete with respect to the metric d. Definition. Let X be a linear space over K (=R or C). The inner product (scalar product) is a function h·, ·i : X × X → K such that (1) hx, xi ≥ 0; (2) hx, xi = 0 if and only if x = 0; (3) hαx, yi = αhx, yi; (4) hx1 + x2, yi = hx1, yi + hx2, yi; (5) hx, yi = hy, xi, for all x, x1, x2, y ∈ X and all α ∈ K. As an obvious corollary we obtain hx, y1 + y2i = hx, y1i + hx, y2i, hx, αyi = αhx, yi , Date: February 12, 2009. 1 2 PIOTR HAJLASZ for all x, y1, y2 ∈ X and α ∈ K. For a space with an inner product we define kxk = phx, xi . Lemma 1.1 (Schwarz inequality).
    [Show full text]
  • Bessel's Differential Equation Mathematical Physics
    R. I. Badran Bessel's Differential Equation Mathematical Physics Bessel's Differential Equation: 2 Recalling the DE y n y 0 which has a sinusoidal solution (i.e. sin nx and cos nx) and knowing that these solutions can be treated as power series, we can find a solution to the Bessel's DE which is written as: 2 2 2 x y xy (x p )y 0 . The solution is represented by a series. This series very much look like a damped sine or cosine. It is called a Bessel function. To solve the Bessel' DE, we apply the Frobenius method by mn assuming a series solution of the form y an x . n0 Substitute this solution into the DE equation and after some mathematical steps you may find that the indicial equation is 2 2 m p 0 where m= p. Also you may get a1=0 and hence all odd a's are zero as well. The recursion relation can be found as a a n n2 (n m 2)2 p 2 with (n=0, 1, 2, 3, ……etc.). Case (1): m= p (seeking the first solution of Bessel DE) We get the solution 2 4 p x x y a x 1 2(2p 2) 2 4(2p 2)(2p 4) This can be rewritten as: x p2n J (x) y a (1)n p 2n n0 2 n!( p n)! 1 Put a 2 p p! The Bessel function has the factorial form x p2n J (x) (1)n p 2n p , n0 n!( p n)!2 R.
    [Show full text]
  • The Standard Form of a Differential Equation
    Fall 06 The Standard Form of a Differential Equation Format required to solve a differential equation or a system of differential equations using one of the command-line differential equation solvers such as rkfixed, Rkadapt, Radau, Stiffb, Stiffr or Bulstoer. For a numerical routine to solve a differential equation (DE), we must somehow pass the differential equation as an argument to the solver routine. A standard form for all DEs will allow us to do this. Basic idea: get rid of any second, third, fourth, etc. derivatives that appear, leaving only first derivatives. Example 1 2 d d 5 yx()+3 ⋅ yx()−5yx ⋅ () 4x⋅ 2 dx dx This DE contains a second derivative. How do we write a second derivative as a first derivative? A second derivative is a first derivative of a first derivative. d2 d d yx() yx() 2 dx dxxd Step 1: STANDARDIZATION d Let's define two functions y0(x) and y1(x) as y0()x yx() and y1()x y0()x dx Then this differential equation can be written as d 5 y1()x +3y ⋅ 1()x −5y ⋅ 0()x 4x dx This DE contains 2 functions instead of one, but there is a strong relationship between these two functions d y1()x y0()x dx So, the original DE is now a system of two DEs, d d 5 y1()x y0()x and y1()x +3y ⋅ 1()x −5y ⋅ 0()x 4x⋅ dx dx The convention is to write these equations with the derivatives alone on the left-hand side. d y0()x y1()x dx This is the first step in the standardization process.
    [Show full text]
  • Semi-Indefinite-Inner-Product and Generalized Minkowski Spaces
    Semi-indefinite-inner-product and generalized Minkowski spaces A.G.Horv´ath´ Department of Geometry, Budapest University of Technology and Economics, H-1521 Budapest, Hungary Nov. 3, 2008 Abstract In this paper we parallelly build up the theories of normed linear spaces and of linear spaces with indefinite metric, called also Minkowski spaces for finite dimensions in the literature. In the first part of this paper we collect the common properties of the semi- and indefinite-inner-products and define the semi-indefinite- inner-product and the corresponding structure, the semi-indefinite-inner- product space. We give a generalized concept of Minkowski space embed- ded in a semi-indefinite-inner-product space using the concept of a new product, that contains the classical cases as special ones. In the second part of this paper we investigate the real, finite dimen- sional generalized Minkowski space and its sphere of radius i. We prove that it can be regarded as a so-called Minkowski-Finsler space and if it is homogeneous one with respect to linear isometries, then the Minkowski- Finsler distance its points can be determined by the Minkowski-product. MSC(2000):46C50, 46C20, 53B40 Keywords: normed linear space, indefinite and semi-definite inner product, arXiv:0901.4872v1 [math.MG] 30 Jan 2009 orthogonality, Finsler space, group of isometries 1 Introduction 1.1 Notation and Terminology concepts without definition: real and complex vector spaces, basis, dimen- sion, direct sum of subspaces, linear and bilinear mapping, quadratic forms, inner (scalar) product, hyperboloid, ellipsoid, hyperbolic space and hyperbolic metric, kernel and rank of a linear mapping.
    [Show full text]
  • CORSO ESTIVO DI MATEMATICA Differential Equations Of
    CORSO ESTIVO DI MATEMATICA Differential Equations of Mathematical Physics G. Sweers http://go.to/sweers or http://fa.its.tudelft.nl/ sweers ∼ Perugia, July 28 - August 29, 2003 It is better to have failed and tried, To kick the groom and kiss the bride, Than not to try and stand aside, Sparing the coal as well as the guide. John O’Mill ii Contents 1Frommodelstodifferential equations 1 1.1Laundryonaline........................... 1 1.1.1 Alinearmodel........................ 1 1.1.2 Anonlinearmodel...................... 3 1.1.3 Comparingbothmodels................... 4 1.2Flowthroughareaandmore2d................... 5 1.3Problemsinvolvingtime....................... 11 1.3.1 Waveequation........................ 11 1.3.2 Heatequation......................... 12 1.4 Differentialequationsfromcalculusofvariations......... 15 1.5 Mathematical solutions are not always physically relevant . 19 2 Spaces, Traces and Imbeddings 23 2.1Functionspaces............................ 23 2.1.1 Hölderspaces......................... 23 2.1.2 Sobolevspaces........................ 24 2.2Restrictingandextending...................... 29 2.3Traces................................. 34 1,p 2.4 Zero trace and W0 (Ω) ....................... 36 2.5Gagliardo,Nirenberg,SobolevandMorrey............. 38 3 Some new and old solution methods I 43 3.1Directmethodsinthecalculusofvariations............ 43 3.2 Solutions in flavours......................... 48 3.3PreliminariesforCauchy-Kowalevski................ 52 3.3.1 Ordinary differentialequations............... 52 3.3.2 Partial differentialequations...............
    [Show full text]
  • Differential Equations and Linear Algebra
    Chapter 1 First Order Equations 1.1 Four Examples : Linear versus Nonlinear A first order differential equation connects a function y.t/ to its derivative dy=dt. That rate of change in y is decided by y itself (and possibly also by the time t). Here are four examples. Example 1 is the most important differential equation of all. dy dy dy dy 1/ y 2/ y 3/ 2ty 4/ y2 dt D dt D dt D dt D Those examples illustrate three linear differential equations (1, 2, and 3) and a nonlinear differential equation. The unknown function y.t/ is squared in Example 4. The derivative y or y or 2ty is proportional to the function y in Examples 1, 2, 3. The graph of dy=dt versus y becomes a parabola in Example 4, because of y2. It is true that t multiplies y in Example 3. That equation is still linear in y and dy=dt. It has a variable coefficient 2t, changing with time. Examples 1 and 2 have constant coefficient (the coefficients of y are 1 and 1). Solutions to the Four Examples We can write down a solution to each example. This will be one solution but it is not the complete solution, because each equation has a family of solutions. Eventually there will be a constant C in the complete solution. This number C is decided by the starting value of y at t 0, exactly as in ordinary integration. The integral of f.t/ solves the simplest differentialD equation of all, with y.0/ C : D dy t 5/ f.t/ The complete solution is y.t/ f.s/ds C : dt D D C Z0 1 2 Chapter 1.
    [Show full text]
  • The Calculus of Trigonometric Functions the Calculus of Trigonometric Functions – a Guide for Teachers (Years 11–12)
    1 Supporting Australian Mathematics Project 2 3 4 5 6 12 11 7 10 8 9 A guide for teachers – Years 11 and 12 Calculus: Module 15 The calculus of trigonometric functions The calculus of trigonometric functions – A guide for teachers (Years 11–12) Principal author: Peter Brown, University of NSW Dr Michael Evans, AMSI Associate Professor David Hunt, University of NSW Dr Daniel Mathews, Monash University Editor: Dr Jane Pitkethly, La Trobe University Illustrations and web design: Catherine Tan, Michael Shaw Full bibliographic details are available from Education Services Australia. Published by Education Services Australia PO Box 177 Carlton South Vic 3053 Australia Tel: (03) 9207 9600 Fax: (03) 9910 9800 Email: [email protected] Website: www.esa.edu.au © 2013 Education Services Australia Ltd, except where indicated otherwise. You may copy, distribute and adapt this material free of charge for non-commercial educational purposes, provided you retain all copyright notices and acknowledgements. This publication is funded by the Australian Government Department of Education, Employment and Workplace Relations. Supporting Australian Mathematics Project Australian Mathematical Sciences Institute Building 161 The University of Melbourne VIC 3010 Email: [email protected] Website: www.amsi.org.au Assumed knowledge ..................................... 4 Motivation ........................................... 4 Content ............................................. 5 Review of radian measure ................................. 5 An important limit ....................................
    [Show full text]
  • MTHE / MATH 237 Differential Equations for Engineering Science
    i Queen’s University Mathematics and Engineering and Mathematics and Statistics MTHE / MATH 237 Differential Equations for Engineering Science Supplemental Course Notes Serdar Y¨uksel November 26, 2015 ii This document is a collection of supplemental lecture notes used for MTHE / MATH 237: Differential Equations for Engineering Science. Serdar Y¨uksel Contents 1 Introduction to Differential Equations ................................................... .......... 1 1.1 Introduction.................................... ............................................ 1 1.2 Classification of DifferentialEquations ............ ............................................. 1 1.2.1 OrdinaryDifferentialEquations ................. ........................................ 2 1.2.2 PartialDifferentialEquations .................. ......................................... 2 1.2.3 HomogeneousDifferentialEquations .............. ...................................... 2 1.2.4 N-thorderDifferentialEquations................ ........................................ 2 1.2.5 LinearDifferentialEquations ................... ........................................ 2 1.3 SolutionsofDifferentialequations ................ ............................................. 3 1.4 DirectionFields................................. ............................................ 3 1.5 Fundamental Questions on First-Order Differential Equations ...................................... 4 2 First-Order Ordinary Differential Equations ..................................................
    [Show full text]
  • Linear Differential Equations Math 130 Linear Algebra
    where A and B are arbitrary constants. Both of these equations are homogeneous linear differential equations with constant coefficients. An nth-order linear differential equation is of the form Linear Differential Equations n 00 0 Math 130 Linear Algebra anf (t) + ··· + a2f (t) + a1f (t) + a0f(t) = b: D Joyce, Fall 2013 that is, some linear combination of the derivatives It's important to know some of the applications up through the nth derivative is equal to b. In gen- of linear algebra, and one of those applications is eral, the coefficients an; : : : ; a1; a0, and b can be any to homogeneous linear differential equations with functions of t, but when they're all scalars (either constant coefficients. real or complex numbers), then it's a linear differ- ential equation with constant coefficients. When b What is a differential equation? It's an equa- is 0, then it's homogeneous. tion where the unknown is a function and the equa- For the rest of this discussion, we'll only consider tion is a statement about how the derivatives of the at homogeneous linear differential equations with function are related. constant coefficients. The two example equations 0 Perhaps the most important differential equation are such; the first being f −kf = 0, and the second 00 is the exponential differential equation. That's the being f + f = 0. one that says a quantity is proportional to its rate of change. If we let t be the independent variable, f(t) What do they have to do with linear alge- the quantity that depends on t, and k the constant bra? Their solutions form vector spaces, and the of proportionality, then the differential equation is dimension of the vector space is the order n of the equation.
    [Show full text]
  • On the Continuity of Fourier Multipliers on the Homogeneous Sobolev Spaces
    R AN IE N R A U L E O S F D T E U L T I ’ I T N S ANNALES DE L’INSTITUT FOURIER Krystian KAZANIECKI & Michał WOJCIECHOWSKI On the continuity of Fourier multipliers on the homogeneous Sobolev spaces . 1 d W1 (R ) Tome 66, no 3 (2016), p. 1247-1260. <http://aif.cedram.org/item?id=AIF_2016__66_3_1247_0> © Association des Annales de l’institut Fourier, 2016, Certains droits réservés. Cet article est mis à disposition selon les termes de la licence CREATIVE COMMONS ATTRIBUTION – PAS DE MODIFICATION 3.0 FRANCE. http://creativecommons.org/licenses/by-nd/3.0/fr/ L’accès aux articles de la revue « Annales de l’institut Fourier » (http://aif.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://aif.cedram.org/legal/). cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Ann. Inst. Fourier, Grenoble 66, 3 (2016) 1247-1260 ON THE CONTINUITY OF FOURIER MULTIPLIERS . 1 d ON THE HOMOGENEOUS SOBOLEV SPACES W1 (R ) by Krystian KAZANIECKI & Michał WOJCIECHOWSKI (*) Abstract. — In this paper we prove that every Fourier multiplier on the ˙ 1 d homogeneous Sobolev space W1 (R ) is a continuous function. This theorem is a generalization of the result of A. Bonami and S. Poornima for Fourier multipliers, which are homogeneous functions of degree zero. Résumé. — Dans cet article, nous prouvons que chaque multiplicateur de Fou- ˙ 1 d rier sur l’espace homogène W1 (R ) de Sobolev est une fonction continue. Notre théorème est une généralisation du résultat de A.
    [Show full text]
  • Weak Convergence of Asymptotically Homogeneous Functions of Measures
    h’on,inec,r ~ndysrr, Theory. Methods & App/;cmonr. Vol. IS. No. I. pp. I-16. 1990. 0362-546X/90 13.M)+ .OO Printed in Great Britain. 0 1990 Pergamon Press plc WEAK CONVERGENCE OF ASYMPTOTICALLY HOMOGENEOUS FUNCTIONS OF MEASURES FRANGOISE DEMENGEL Laboratoire d’Analyse Numerique, Universite Paris-Sud, 91405 Orsay, France and JEFFREY RAUCH* University of Michigan, Ann Arbor, Michigan 48109, U.S.A. (Received 28 April 1989; received for publication 15 September 1989) Key words and phruses: Functions of measures, weak convergence, measures. INTRODUCTION FUNCTIONS of measures have already been introduced and studied by Goffman and Serrin [8], Reschetnyak [lo] and Demengel and Temam [5]: especially for the convex case, some lower continuity results for the weak tolopology (see [5]) permit us to clarify and solve, from a mathematical viewpoint, some mechanical problems, namely in the theory of elastic plastic materials. More recently, in order to study weak convergence of solutions of semilinear hyperbolic systems, which arise from mechanical fluids, we have been led to answer the following ques- tion: on what conditions on a sequence pn of bounded measures have we f(~,) --*f(p) where P is a bounded measure and f is any “function of a measure” (not necessarily convex)? We answer this question in Section 1 when the functions f are homogeneous, and in Section 2 for sublinear functions; the general case of asymptotically homogeneous function is then a direct consequence of the two previous cases! An extension to x dependent functions is described in Section 4. In the one dimensional case, we give in proposition 3.3 a criteria which is very useful in practice, namely for the weak continuity of solutions measures of hyperbolic semilinear systems with respect to weakly convergent Cauchy data.
    [Show full text]
  • G = U Ytv2
    PROCEEDINGS of the AMERICAN MATHEMATICAL SOCIETY Volume 81, Number 1, January 1981 A CHARACTERIZATION OF MINIMAL HOMOGENEOUS BANACH SPACES HANS G. FEICHTINGER Abstract. Let G be a locally compact group. It is shown that for a homogeneous Banach space B on G satisfying a slight additional condition there exists a minimal space fimm in the family of all homogeneous Banach spaces which contain all elements of B with compact support. Two characterizations of Bm¡1¡aie given, the first one in terms of "atomic" representations. The equivalence of these two characterizations is derived by means of certain (bounded) partitions of unity which are of interest for themselves. Notations. In the sequel G denotes a locally compact group. \M\ denotes the (Haar) measure of a measurable subset M G G, or the cardinality of a finite set. ®(G) denotes the space of continuous, complex-valued functions on G with compact support (supp). For y G G the (teft) translation operator ly is given by Lyf(x) = f(y ~ xx). A translation invariant Banach space B is called a homogeneous Banach space (in the sense of Katznelson [9]) if it is continuously embedded into the topological vector space /^(G) of all locally integrable functions on G, satisfies \\Lyf\\B= 11/11,for all y G G, and lim^iy-- f\\B - 0 for all / G B (hence, as usual, two measurable functions coinciding l.a.e. are identified). If, furthermore, B is a dense subspace of LX(G) it is called a Segal algebra (in the sense of Reiter [15], [16]). Any homogeneous Banach space is a left L'((7)-Banach- module with respect to convolution, i.e./ G B, h G LX(G) implies h */ G B, and \\h */\\b < ll*Uill/IJi» m particular any Segal algebra is a Banach ideal of LX(G).
    [Show full text]