DOCSLIB.ORG

NOTES on DISTRIBUTIONS MATH 565, FALL 2017 1. Introduction and Examples the Notion of a Distribution (Also Called Generalized Fu

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
NOTES on DISTRIBUTIONS MATH 565, FALL 2017 1. Introduction and Examples the Notion of a Distribution (Also Called Generalized Fu NOTES ON DISTRIBUTIONS MATH 565, FALL 2017 1. Introduction and examples The notion of a distribution (also called generalized functions) was introduced to make sense of operations such as the delta function, which in some sense we very much want to be a function, but we cannot apply the usual rules of integration and differentiation to it. n 1 Given a nonempty open set Ω ⊂ R , consider Cc (Ω) as the collection of smooth compactly supported functions φ :Ω ! C satisfying supp(φ) ⊂ Ω. In the theory of distributions, such functions are also called test functions and very often the notation D(Ω) is used alternatively, so 1 treat it as the same as Cc (Ω). It is not hard to check that D(Ω) is always vector space but it is not a trivial task to endow D(Ω) with a topology for which D(Ω) will be complete in the sense that every Cauchy sequence converges. It turns out this can be done by passing to the theory of topological vector spaces, vector spaces whose topology is not necessarily induced by a norm (or even a metric!). Nonetheless, good sense can be made of convergence in the following way. Definition 1.1. A sequence fφjg in D(Ω) is said to converge to a function φ 2 D(Ω) if there exists a compact set K ⊂ Ω such that supp(φj) ⊂ K for every j 2 N and α α α α (1.1) lim k@ φj − @ φk1 = lim sup j@ φj(x) − @ φ(x)j = 0; for every multi-index α: j!1 j!1 x2Ω Note that since supp(φj) ⊂ K for each j, we must have supp(φ) ⊂ K as well and hence we could replace supx2Ω by supx2K here. Definition 1.2. A distribution on Ω is a linear map u : D(Ω) ! C satisfying the following: for each compact K ⊂ Ω, there exists a constant CK and an integer NK ≥ 0 such that X α (1.2) jhu; φij ≤ CK k@ φk1: jα|≤NK The space of distributions over Ω is denoted by D0(Ω), it is a vector space under the natural addition and scalar multiplication operations 0 (1.3) hc1u1 + c2u2; φi := c1hu1; φi + c2hu2; φi uj 2 D (Ω); cj 2 C; j = 1; 2: Note that our definition of convergence means that u is sequentially continuous in that if φj ! φ in the sense of (1.1), then limj!1hu; φji = hu; φi: Since u is a linear map u : D(Ω) ! C, one might say that the notation for the action of u on a test function φ should be u(φ) rather than hu; φi, but there are advantages to viewing u as a \pairing" with test function. 1 0 0 Definition 1.3. A sequence fukgk=1 ⊂ D (Ω) converges to u in the topology of D (Ω) if lim huk; φi = hu; φi for every φ 2 D(Ω): k!1 Remark 1.4. Note that if φk or uk instead depends on a parameter p lying in a metric space with limit point p0, then we can define limp!p0 φp and limp!p0 up in the natural way by saying that these limits are φ and u respectively if and only if the limit of every sequence pk ! p as k ! 1 satisfies φpk ! φ and upk ! u respectively. 1 2 NOTES ON DISTRIBUTIONS 1.1. Examples. n Example 1.5. Given an open set Ω ⊂ R , define the locally integrable functions on Ω as those which are integrable over any compact set K ⊂ Ω: 1 1 n Lloc(Ω) := ff measurable on Ω: f1K 2 L (R ) for any compact K ⊂ Ωg: 1 R Given any f 2 Lloc(Ω), f defines a distribution via hf; φi := Ω f(x)φ(x)dx. To see that this operation satisfies (1.2), observe that if supp(φ) ⊂ K, then φ = φ1K and hence Z Z Z jhf; φij = f(x)1K (x)φ(x)dx = f(x)φ(x)dx ≤ jf(x)jdx kφk1 Ω K K R by H¨older'sinequality. Hence (1.2) is satisfied with NK = 0 and CK = K jf(x)jdx. The previous example is of fundamental importance in the theory of distributions. Frequently we will conflate the idea of a locally integrable function with the distribution it defines. n Example 1.6. Given any open Ω ⊂ R containing the origin, the Dirac distribution δ is defined by hδ; φi = φ(0). It is nearly trivial that jhδ; φij ≤ kφk1 and hence δ satisfies the inequality (1.2) with CK = 1 and NK = 0. R −n Moreover, if is integrable with (x)dx = a 2 C, define (x) = (x/). Treating as a distribution as in the previous example, the converge to aδ in the sense of distributions: Z Z Z −n lim h ; φi = lim (x/)φ(x)dx = lim (x)φ(x)dx = φ(0) (x)dx = haδ; φi: !0+ !0+ !0+ where the penultimate identity follows from a standard application of the dominated convergence theorem. In the special case a = 1, this rigorously establishes the common non-rigorous definition of the Dirac distribution as being a limit of functions which are increasingly peaked at the origin. Example 1.7. Let Ω = (0; 1) ⊂ R. Let u be the linear map on D(Ω) defined by 1 X hu; φi = nφ (1=n) : n=1 That u is indeed well-defined is implicit in the ensuing observation establishing (1.2) for this dis- tribution. Suppose K ⊂ (0; 1) is compact. Then d(K; (−∞; 0]) > 0, so there exists N 2 N large such that K ⊂ [1=N; 1) and hence for any φ supported in K, N N N X X X N(N + 1) jhu; φij = nφ (1=n) ≤ njφ (1=n) j ≤ kφk n = kφk : 1 2 1 n=1 n=1 1 N(N+1) Therefore (1.2) is satisfied with CK = 2 and NK = 0. Note that u does not define a distribution on R. n Example 1.8. Let Ω ⊂ R and let S be any parameterized k-surface with surface measure dS. Then Z hu; φi := φdS; S is easily verified to define a distribution. In particular, if supp(φ) ⊂ K, then the constants in (1.2) R 1 n−1 can be taken as NK = 0 and CK = K\S 1 dS. Cases of special interest will be when S = S is the unit sphere or more generally, S is quadratic surface. P1 (n) Exercise 1.9. Check that hu; φi := n=1 nφ (n) defines a distribution on R satisfying (1.2) for (n) any compact K ⊂ R (where φ is the n-th derivative of φ). 1 R There is a slight technicality here in that its not obvious that K\S 1 dS is always well-defined, but it is possible to enlarge K so that the integral is indeed well-defined. NOTES ON DISTRIBUTIONS 3 2. Operations on distributions 1 2.1. Multiplication by a smooth function. As we saw, any function f 2 Lloc(Ω) defines a distribution via Z hf; φi = f(x)φ(x) dx: If 2 C1(Ω), we of course define the product of and f as simply the function ( f)(x) = (x)f(x), its corresponding action on D(Ω) is then Z Z h f; φi = (x)f(x)φ(x) dx = f(x) (x)φ(x) dx = hf; φi: Indeed, the product φ defines a test function and supp( φ) ⊂ supp(φ), so the right hand side here is meaningful. Definition 2.1. Given u 2 D0(Ω) and 2 C1(Ω), the product u is defined to be the distribution h u; φi := hu; φi: Note that this does indeed define a distribution in that it satisfies (1.2): if K ⊂ Ω is compact, and C , N are the constants satisfied by u in (1.2), then sup sup j@α (x)j is finite, and K K jα|≤NK x2K hence for supp(φ) ⊂ K, the Liebniz rule implies the existence of a constant M depending on this quantity such that X α X α jh u; φij = jhu; φij ≤ CK k@ ( φ)k1 ≤ MCK k@ φk1: jα|≤NK jα|≤NK Hence u also satisfies (1.2), with the constant MCK replacing CK . k β 1 2.2. Differentiation of a distribution. Now suppose f 2 C (Ω) is such that @ f 2 Lloc(Ω) for every multi-index jβj ≤ k. Then we would have that for any fixed jβj ≤ k, @βf defines a distribution via Z h@βf; φi = @βf(x)φ(x) dx: However, an integration by parts shows that Z Z h@βf; φi = @βf(x)φ(x) dx = (−1)jβj f(x)@βφ(x) dx = (−1)jβjhf; @βφi: Indeed, @βφ defines a test function and hence we are led to make the following definition: Definition 2.2. Given u 2 D0(Ω) and a multi-index β, the partial derivative @βu is defined as h@βu; φi := (−1)jβjhu; @βφi: Once again there is the matter of checking that this operation actually defines a distribution on Ω, so suppose K ⊂ Ω is any compact subset and that CK ;NK are the constants satisfied by u in (1.2). Then β β X α β X α jh@ u; φij = jhu; @ φij ≤ CK k@ (@ φ)k1 ≤ CK k@ φk1: jα|≤NK jα|≤NK +jβj Example 2.3. Let H(x) be the Heaviside function on R ( 1; if x ≥ 0 H(x) = 0; if x < 0: 4 NOTES ON DISTRIBUTIONS R 1 R 1 Since H is locally integrable, hH; φi = −∞ H(x)φ(x)dx = 0 φ(x)dx defines a distribution on R.
Load more

Recommended publications
  • A Corrective View of Neural Networks: Representation, Memorization and Learning
    Proceedings of Machine Learning Research vol 125:1{54, 2020 33rd Annual Conference on Learning Theory A Corrective View of Neural Networks: Representation, Memorization and Learning Guy Bresler [email protected] Department of EECS, MIT Cambridge, MA, USA. Dheeraj Nagaraj [email protected] Department of EECS, MIT Cambridge, MA, USA. Editors: Jacob Abernethy and Shivani Agarwal Abstract We develop a corrective mechanism for neural network approximation: the total avail- able non-linear units are divided into multiple groups and the first group approx- imates the function under consideration, the second approximates the error in ap- proximation produced by the first group and corrects it, the third group approxi- mates the error produced by the first and second groups together and so on. This technique yields several new representation and learning results for neural networks: 1. Two-layer neural networks in the random features regime (RF) can memorize arbi- Rd ~ n trary labels for n arbitrary points in with O( θ4 ) ReLUs, where θ is the minimum distance between two different points. This bound can be shown to be optimal in n up to logarithmic factors. 2. Two-layer neural networks with ReLUs and smoothed ReLUs can represent functions with an error of at most with O(C(a; d)−1=(a+1)) units for a 2 N [ f0g when the function has Θ(ad) bounded derivatives. In certain cases d can be replaced with effective dimension q d. Our results indicate that neural networks with only a single nonlinear layer are surprisingly powerful with regards to representation, and show that in contrast to what is suggested in recent work, depth is not needed in order to represent highly smooth functions.
    [Show full text]
  • Smooth Bumps, a Borel Theorem and Partitions of Smooth Functions on P.C.F. Fractals
    SMOOTH BUMPS, A BOREL THEOREM AND PARTITIONS OF SMOOTH FUNCTIONS ON P.C.F. FRACTALS. LUKE G. ROGERS, ROBERT S. STRICHARTZ, AND ALEXANDER TEPLYAEV Abstract. We provide two methods for constructing smooth bump functions and for smoothly cutting off smooth functions on fractals, one using a prob- abilistic approach and sub-Gaussian estimates for the heat operator, and the other using the analytic theory for p.c.f. fractals and a fixed point argument. The heat semigroup (probabilistic) method is applicable to a more general class of metric measure spaces with Laplacian, including certain infinitely ramified fractals, however the cut off technique involves some loss in smoothness. From the analytic approach we establish a Borel theorem for p.c.f. fractals, showing that to any prescribed jet at a junction point there is a smooth function with that jet. As a consequence we prove that on p.c.f. fractals smooth functions may be cut off with no loss of smoothness, and thus can be smoothly decom- posed subordinate to an open cover. The latter result provides a replacement for classical partition of unity arguments in the p.c.f. fractal setting. 1. Introduction Recent years have seen considerable developments in the theory of analysis on certain fractal sets from both probabilistic and analytic viewpoints [1, 14, 26]. In this theory, either a Dirichlet energy form or a diffusion on the fractal is used to construct a weak Laplacian with respect to an appropriate measure, and thereby to define smooth functions. As a result the Laplacian eigenfunctions are well un- derstood, but we have little knowledge of other basic smooth functions except in the case where the fractal is the Sierpinski Gasket [19, 5, 20].
    [Show full text]
  • Robinson Showed That It Is Possible to Rep- Resent Each Schwartz Distribution T As an Integral T(4) = Sf4, Where F Is Some Nonstandard Smooth Function
    Indag. Mathem., N.S., 11 (1) 129-138 March 30,200O Constructive nonstandard representations of generalized functions by Erik Palmgren* Department of Mathematics,Vppsala University, P. 0. Box 480, S-75106 Uppa& &v&n, e-mail:[email protected] Communicated by Prof. A.S. Troelstra at the meeting of May 31, 1999 ABSTRACT Using techniques of nonstandard analysis Abraham Robinson showed that it is possible to rep- resent each Schwartz distribution T as an integral T(4) = sf4, where f is some nonstandard smooth function. We show that the theory based on this representation can be developed within a constructive setting. I. INTRODUCTION Robinson (1966) demonstrated that Schwartz’ theory of distributions could be given a natural formulation using techniques of nonstandard analysis, so that distributions become certain nonstandard smooth functions. In particular, Dirac’s delta-function may then be taken to be the rational function where E is a positive infinitesimal. As is wellknown, classical nonstandard analysis is based on strongly nonconstructive assumptions. In this paper we present a constructive version of Robinson’s theory using the framework of constructive nonstandard analysis developed in Palmgren (1997, 1998), which was based on Moerdijk’s (1995) sheaf-theoretic nonstandard models. The main *The author gratefully acknowledges support from the Swedish Research Council for Natural Sciences (NFR). 129 points of the present paper are the elimination of the standard part map from Robinson’s theory and a constructive proof of the representation of distribu- tions as nonstandard smooth functions (Theorem 2.6). We note that Schmieden and Laugwitz already in 1958 introduced their ver- sion of nonstandard analysis to give an interpretation of generalized functions which are closer to those of the Mikusinski-Sikorski calculus.
    [Show full text]
  • Delta Functions and Distributions
    When functions have no value(s): Delta functions and distributions Steven G. Johnson, MIT course 18.303 notes Created October 2010, updated March 8, 2017. Abstract x = 0. That is, one would like the function δ(x) = 0 for all x 6= 0, but with R δ(x)dx = 1 for any in- These notes give a brief introduction to the mo- tegration region that includes x = 0; this concept tivations, concepts, and properties of distributions, is called a “Dirac delta function” or simply a “delta which generalize the notion of functions f(x) to al- function.” δ(x) is usually the simplest right-hand- low derivatives of discontinuities, “delta” functions, side for which to solve differential equations, yielding and other nice things. This generalization is in- a Green’s function. It is also the simplest way to creasingly important the more you work with linear consider physical effects that are concentrated within PDEs, as we do in 18.303. For example, Green’s func- very small volumes or times, for which you don’t ac- tions are extremely cumbersome if one does not al- tually want to worry about the microscopic details low delta functions. Moreover, solving PDEs with in this volume—for example, think of the concepts of functions that are not classically differentiable is of a “point charge,” a “point mass,” a force plucking a great practical importance (e.g. a plucked string with string at “one point,” a “kick” that “suddenly” imparts a triangle shape is not twice differentiable, making some momentum to an object, and so on.
    [Show full text]
  • [Math.FA] 1 Apr 1998 E Words Key 1991 Abstract *)Tersac Fscn N H Hr Uhrhsbe Part Been Has Author 93-0452
    RENORMINGS OF Lp(Lq) by R. Deville∗, R. Gonzalo∗∗ and J. A. Jaramillo∗∗ (*) Laboratoire de Math´ematiques, Universit´eBordeaux I, 351, cours de la Lib´eration, 33400 Talence, FRANCE. email : [email protected] (**) Departamento de An´alisis Matem´atico, Universidad Complutense de Madrid, 28040 Madrid, SPAIN. email : [email protected] and [email protected] Abstract : We investigate the best order of smoothness of Lp(Lq). We prove in particular ∞ that there exists a C -smooth bump function on Lp(Lq) if and only if p and q are even integers and p is a multiple of q. arXiv:math/9804002v1 [math.FA] 1 Apr 1998 1991 Mathematics Subject Classification : 46B03, 46B20, 46B25, 46B30. Key words : Higher order smoothness, Renormings, Geometry of Banach spaces, Lp spaces. (**) The research of second and the third author has been partially supported by DGICYT grant PB 93-0452. 1 Introduction The first results on high order differentiability of the norm in a Banach space were given by Kurzweil [10] in 1954, and Bonic and Frampton [2] in 1965. In [2] the best order of differentiability of an equivalent norm on classical Lp-spaces, 1 < p < ∞, is given. The best order of smoothness of equivalent norms and of bump functions in Orlicz spaces has been investigated by Maleev and Troyanski (see [13]). The work of Leonard and Sundaresan [11] contains a systematic investigation of the high order differentiability of the norm function in the Lebesgue-Bochner function spaces Lp(X). Their results relate the continuously n-times differentiability of the norm on X and of the norm on Lp(X).
    [Show full text]
  • Be a Smooth Manifold, and F : M → R a Function
    LECTURE 3: SMOOTH FUNCTIONS 1. Smooth Functions Definition 1.1. Let (M; A) be a smooth manifold, and f : M ! R a function. (1) We say f is smooth at p 2 M if there exists a chart f'α;Uα;Vαg 2 A with −1 p 2 Uα, such that the function f ◦ 'α : Vα ! R is smooth at 'α(p). (2) We say f is a smooth function on M if it is smooth at every x 2 M. −1 Remark. Suppose f ◦ 'α is smooth at '(p). Let f'β;Uβ;Vβg be another chart in A with p 2 Uβ. Then by the compatibility of charts, the function −1 −1 −1 f ◦ 'β = (f ◦ 'α ) ◦ ('α ◦ 'β ) must be smooth at 'β(p). So the smoothness of a function is independent of the choice of charts in the given atlas. Remark. According to the chain rule, it is easy to see that if f : M ! R is smooth at p 2 M, and h : R ! R is smooth at f(p), then h ◦ f is smooth at p. We will denote the set of all smooth functions on M by C1(M). Note that this is a (commutative) algebra, i.e. it is a vector space equipped with a (commutative) bilinear \multiplication operation": If f; g are smooth, so are af + bg and fg; moreover, the multiplication is commutative, associative and satisfies the usual distributive laws. 1 n+1 i n Example. Each coordinate function fi(x ; ··· ; x ) = x is a smooth function on S , since ( 2yi −1 1 n 1+jyj2 ; 1 ≤ i ≤ n fi ◦ '± (y ; ··· ; y ) = 1−|yj2 ± 1+jyj2 ; i = n + 1 are smooth functions on Rn.
    [Show full text]
  • Generalized Functions: Homework 1
    Generalized Functions: Homework 1 Exercise 1. Prove that there exists a function f Cc∞(R) which isn’t the zero function. ∈ Solution: Consider the function: 1 e x x > 0 g = − 0 x 0 ( ≤ 1 We claim that g C∞(R). Indeed, all of the derivatives of e− x are vanishing when x 0: ∈ n → d 1 1 lim e− x = lim q(x) e− x = 0 x 0 n x 0 → dx → ∙ where q(x) is a rational function of x. The function f = g(x) g(1 x) is smooth, as a multiplication of two smooth functions, has compact support∙ − (the interval 1 1 [0, 1]), and is non-zero (f 2 = e ). Exercise 2. Find a sequence of functions fn n Z (when fn Cc∞ (R) n) that weakly converges to Dirac Delta function.{ } ∈ ∈ ∀ Solution: Consider the sequence of function: ˜ 1 1 fn = g x g + x C∞(R) n − ∙ n ∈ c where g(x) Cc∞(R) is the function defined in the first question. Notice that ∈ 1 1 the support of fn is the interval n , n . Normalizing these functions so that the integral will be equal to 1 we get:− 1 ∞ − f = d(n)f˜ , d(n) = f˜ (x)dx n n n Z −∞ For every test function F (x) C∞(R), we can write F (x) = F (0) + x G(x), ∈ c ∙ where G(x) C∞(R). Therefore we have: ∈ c 1/n ∞ F (x) f (x)dx = F (x) f (x)dx ∙ n ∙ n Z 1Z/n −∞ − 1/n 1/n = F (0) f (x)dx + xG(x) f (x)dx ∙ n ∙ n 1Z/n 1Z/n − − 1/n = F (0) + xG(x) f (x)dx ∙ n 1Z/n − 1 1/n ∞ ∞ So F (x) fn(x)dx F (x) δ(x)dx = 1/n xG(x) fn(x)dx.
    [Show full text]
  • On the Size of the Sets of Gradients of Bump Functions and Starlike Bodies on the Hilbert Space
    ON THE SIZE OF THE SETS OF GRADIENTS OF BUMP FUNCTIONS AND STARLIKE BODIES ON THE HILBERT SPACE DANIEL AZAGRA AND MAR JIMENEZ-SEVILLA´ Abstract. We study the size of the sets of gradients of bump functions on the Hilbert space `2, and the related question as to how small the set of tangent hyperplanes to a smooth bounded starlike body in `2 can be. We find that those sets can be quite small. On the one hand, the usual norm of the Hilbert space 1 `2 can be uniformly approximated by C smooth Lipschitz functions ψ so that 0 the cones generated by the ranges of its derivatives ψ (`2) have empty interior. 1 This implies that there are C smooth Lipschitz bumps in `2 so that the cones generated by their sets of gradients have empty interior. On the other hand, we 1 construct C -smooth bounded starlike bodies A ⊂ `2, which approximate the unit ball, so that the cones generated by the hyperplanes which are tangent to A have empty interior as well. We also explain why this is the best answer to the above questions that one can expect. RESUM´ E.´ On ´etudie la taille de l’ensemble de valeurs du gradient d’une fonction lisse, non identiquement nulle et `asupport born´edans l’espace de Hilbert `2, et aussi celle de l’ensemble des hyperplans tangents `aun corps ´etoil´edans `2. On montre que ces ensembles peuvent ˆetreassez petits. D’un cˆot´e,la norme de l’espace de Hilbert est approxim´eeuniformement par des fonctions de classe C1 et Lipschitziennes ψ telles que les cˆonesengendr´es par les 0 ensembles de valeurs des d´eriv´ees ψ (`2) sont d’int´erieur vide.
    [Show full text]
  • 1.6 Smooth Functions and Partitions of Unity
    1.6 Smooth functions and partitions of unity The set C1(M; R) of smooth functions on M inherits much of the structure of R by composition. R is a ring, having addition + : R × R −! R and multiplication × : R × R −! R which are both smooth. As a result, C1(M; R) is as well: One way of seeing why is to use the smooth diagonal map ∆ : M −! M × M, i.e. ∆(p) = (p; p). Then, given functions f ; g 2 C1(M; R) we have the sum f + g, defined by the composition ∆ f ×g + M / M × M / R × R / R : We also have the product f g, defined by the composition ∆ f ×g × M / M × M / R × R / R : Given a smooth map ' : M −! N of manifolds, we obtain a natural operation '∗ : C1(N; R) −! C1(M; R), given by f 7! f ◦ '. This is called the pullback of functions, and defines a homomorphism of rings since ∆ ◦ ' = (' × ') ◦ ∆. The association M 7! C1(M; R) and ' 7! '∗ takes objects and arrows of C1-Man to objects and arrows of the category of rings, respectively, in such a way which respects identities and composition of morphisms. Such a map is called a functor. In this case, it has the peculiar property that it switches the source and target of morphisms. It is therefore a contravariant functor from the category of manifolds to the category of rings, and is the basis for algebraic geometry, the algebraic representation of geometrical objects. It is easy to see from this that any diffeomorphism ' : M −! M defines an automorphism '∗ of C1(M; R), but actually all automorphisms are of this form (Why?).
    [Show full text]
  • Examples Discussion 05
    (December 18, 2017) Examples discussion 05 Paul Garrett [email protected] http:=/www.math.umn.edu/egarrett/ [This document is http:=/www.math.umn.edu/egarrett/m/real/examples 2017-18/real-disc-05.pdf] [05.1] Give a persuasive proof that the function 0 (for x ≤ 0) f(x) = e−1=x (for x > 0) is infinitely differentiable at 0. Use this kind of construction to make a smooth step function: 0 for x ≤ 0 and 1 for x ≥ 1, and goes monotonically from 0 to 1 in the interval [0; 1]. Use this to construct a family of smooth cut-off functions ffn : n = 1; 2; 3;:::g: for each n, fn(x) = 1 for x 2 [−n; n], fn(x) = 0 for x 62 [−(n + 1); n + 1], and fn goes monotonically from 0 to 1 in [−(n + 1); −n] and monotonically from 1 to 0 in [n; n + 1]. Discussion: In x > 0, by induction, the derivatives are finite linear linear combinations of functions of the −n −1=x −n −1=x n −x form x e . It suffices to show that limx!0+ x e = 0. Equivalently, that limx!+1 x e = 0, which follows from e−x = 1=ex, and xn xn xn x−ne−1=x = = ≤ −! 0 (as x ! +1) ex P xm xn+1 m≥0 m! (n+1)! (This is perhaps a little better than appeals to L'Hospital's Rule.) Thus, f is smooth at 0, with all derivatives 0 there. === Next, we make a smooth bump function by 8 0 (for x ≤ −1) > <> 1 b(x) = e 1−x2 (for −1 < x < 1) > > : 0 (for x ≥ 1) A similar argument to the previous shows that this is smooth.
    [Show full text]
  • Fourier Series Windowed by a Bump Function
    FOURIER SERIES WINDOWED BY A BUMP FUNCTION PAUL BERGOLD AND CAROLINE LASSER Abstract. We study the Fourier transform windowed by a bump function. We transfer Jackson's classical results on the convergence of the Fourier series of a periodic function to windowed series of a not necessarily periodic function. Numerical experiments illustrate the obtained theoretical results. 1. Introduction The theory of Fourier series plays an essential role in numerous applications of contemporary mathematics. It allows us to represent a periodic function in terms of complex exponentials. Indeed, any square integrable function f : R C of period 2π has a norm-convergent Fourier series such that (see e.g. [BN71, Prop.! 4.2.3.]) 1 f(x) = f(k)eikx almost everywhere, k= X−∞ where the Fourier coefficients areb defined according to π 1 ikx f(k) := f(x)e− dx; k Z: 2π π 2 Z− By the classical resultsb of Jackson in 1930, see [Jac94], the decay rate of the Fourier coefficients and therefore the convergence speed of the Fourier series depend on the regularity of the function. If f has a jump discontinuity, then the order of magnitude of the coefficients is (1= k ), as k . Moreover, if f is a smooth function of O j j j j ! 1 period 2π, say f Cs+1(R) for some s 1, then the order improves to (1= k s+1). 2 ≥ O j j In the present paper we focus on the reconstruction of a not necessarily periodic function with respect to a finite interval ( λ, λ). For this purpose let us think of − a smooth, non-periodic real function : R R, which we want to represent by a Fourier series in ( λ, λ).
    [Show full text]
  • Partitions of Unity
    Partitions of Unity Rich Schwartz April 1, 2015 1 The Result Let M be a smooth manifold. This means that • M is a metric space. • M is a countable union of compact subsets. • M is locally homeomorphic to Rn. These local homeomorphisms are the coordinate charts. • M has a maximal covering by coordinate charts, such that all overlap functions are smooth. Let {Θα} be an open cover of M. The goal of these notes is to prove that M has a partition of unity subordinate to {Θα}. This means that there is a countable collection {fi} of smooth functions on M such that: • fi(p) ∈ [0, 1] for all p ∈ M. • The support of fi is a compact subset of some Θα from the cover. • For any compact subset K ⊂ M, we have fi = 0 on K except for finitely many indices i. • P fi(p)=1 for all p ∈ M. The support of fi is the closure of the set p ∈ M such that fi(p) > 0. These notes will assume that you already know how to construct bump functions in Rn. Note: I deliberately picked a weird letter for the cover, so that it doesn’t interfere with the rest of the construction. 1 2 The Compact Case As a warm-up, let’s consider the case when M is compact. For every p ∈ M there is some open set Vp such that • p ∈ Vp. • Vp ⊂ Θα for some Θα from our cover. • Vp is contained in a coordinate chart. Using the fact that we are entirely inside a coordinate chart, we can construct a bump function f : M → [0, 1] such that f(p) > 0 and the support of f is contained in a compact subset of Vp.
    [Show full text]