A simple remark on the order ofapproximation by compactly supported radial basis functions, and related networks
|
|
- Owen McDaniel
- 5 years ago
- Views:
Transcription
1 A simple remark on the order ofapproximation by compactly supported radial basis functions, and related networks Xin Li Department of Mathematical Sciences, University of Nevada, Las Abstract We consider simultaneous approximation of multivariate functions and their derivatives by using Wendland's compactly supported radial basis functions <t>s,k' By applying a greedy algorithm, it is presented that, regardless of dimension, an O(ra~*/^) order of approximation can be achieved by a linear combination of m translates of <t>s,k- A similar result on approximation by neural networks is established by using univariate radial functions as the activation functions of the networks. 1. Introduction Multivariate interpolation by radial basis functions has been studied and applied in several areas of mathematics, such as approximation theory (cf. Franke^, Micchelli**, Schaback**), curve and surface fitting (cf. Daehlen, Lyche, and Schumaker*), and numerical solutions of PDE equations (cf. Golberg and Chen^). A function $ is radial if <&(x) = </>( x ), where 0 : R+ > R is a univariate function and x is the usual Euclidean norm of x ft*.for / C(R*) and a set X = {xi,, XAT} C R of distinct points, the radial basis function interpolant a/ is given by N */,x(x) = ]>]a,0(x - x,), (1.1) 3 = 1 where the coefficents c*i,, c%# are determined by 8f,x(*j) = f(xj), l<j<n. (1.2)
2 336 Boundary Element Technology To ensure the solubility of the interpolation problem (1.1) and (1.2), positive definite functions are used. A continuous function $ : R* > R is positive definite, denoted by <& PD, if for any N Z+, any set of pairwise distinct centers X = {x%,,x#} C R*, and any vector a #^\{0}, the quadratic form N N is positive. If 3> is compactly supported, written as $ CS, the coefficients (*i,, OiN in (1.1) are easy to determine by (1.2). A celebrated Theorem of Bochner (cf. Steward^) characterizes all positive definite functions. In the case that $ is compactly supported, the Theorem is interpreted as: <& is positive definite if and only if its Fourier transform is nonnegative and positive on an open subset. Since the Fourier transform of a radial basis function 3>(x) = </>( x ) is given by = (2rr) R* JO, (1.3) where p = w, and J ^ is the Bessel function of thefirstkind, compactly supported radial basis functions are constructed in Wu^ and Wendland^. In this paper, we consider approximation by using Wendland functions, which we now review the definition and their properties. Following Reference [16], let yoc = r sf(s)ds, Jr and, (1.4) where ^ _a/2j+fc+im = (1 --F)+, and [x\ denotes the largest integer < x. It is shown in Reference [16] that <^& is compactly supported in [0,1], and induces a positive definite function on R* in the way that,.m-/? -*(» )> 0<r<l, where r x, and PS^ is a univariate polynomial of degree [s/2j +3&4-1. Moreover, <f>s^ possesses continuous derivatives up to order 2fc, and it is of minimal degree for a given space dimension s and smoothness 2k and is up to a constant factor uniquely determined by this setting. It is also shown in Reference [17] that the Fourier transform of #^^(x) = <^3,k( x ) satisfies w ^)-^^-'^ (1.5)
3 Boundary Element Technology 337 for some constants K\ and K^ and it was derived in the Theorem 2.2 of Reference [17], by applying the results of Wu and Schaback^, that if X = {xi, -,Xn} C fi for some compact subset 0 in R* satisfying uniform interior cone condition, then for any / %(#*), where i = a/2 4- k 4-1/2 and Hi(R*} is the Sobolev space, with /i := sup min llx xji " " being sufficiently small. In this paper, instead of using radial interpolants, we apply a greedy algorithm and discuss the approximation by convexly linear combinations of translates of <f>s,k- By applying the results in Reference [10], we present that, regardless of the dimension, an O(ra~*/^) order of approximation can be achieved by a linear combination of m translates of (^&, which is a well known result in the literature of neural networks. Meanwhile, we derive a similar result on the neural networks by using compactly supported univariate radial functions as the activation functions of the networks. 2. Greedy Algorithm and Order of Approximation by Compactly supported Radial Basis Functions We begin by describing a greedy algorithm. For more information on this subject, readers are referred to Jones^, DeVore and Temlyakov^, Davis, Mallat, and Avellaneda^, and Donahue, Gurvits, Darken, and Sontag^. We state a result by Jones^, and also Maurey in Pisier^. Lemma 1 If f is in the closure of the convex hull of a set G in a Hilbert space, with \\g\\ < b for each g G, then for every n > 1, and every c'> I? (I/IP, there is an fn in the convex hull of n points in G such that An iterative procedure for achieving above approximation order is provided by Jones, which we describe as a greedy algorithm as follows. Let /i G such that Inductively, for k > 1, let
4 338 Boundary Element Technology such that Then, _«_^ (l-o)a + c%7-/. This algorithm is greedy in the sense that we choose optimal approximation in each step. As understood, to ensure that an algorithm converges, certain conditions are needed to be imposed on the function /. Lemma 1 is first applied by Barron* to establish a well known result on approximating a multivariate function in the order of O(ra~*/^) by a network withraneurons (cf. Section 3). It is then applied by the author^ to derive a similar result on simultanuous approximation by translated radial networks, which we especially apply to compactly supported radial basis functions in this section. For a region ft G A*, denote by %%(ft) the Sobolev space consisting of all distributions / on ft with D*f L^(ft) for any k Z+, k < n, where n > 0 is an integer, with 1/2 When ft = jr*, an equivalent norm of / G Hn(R*) is given by a ^1/2 7(w)»(l + w Vdw) l- / In this paper, we consider approximation in Hn := "Hn([ K,K]*)' For a compactly supported 0a,fc> we define a function G by G(x)= ^^( x-27rk ), (2.1) which is 2?r-periodic coordinately. (The function G introduced in (2.1) is slightly different from the one in Reference [10], where ^] 0( 27rk ^) is used to ensure the study of simultaneous approximation.) For / [ 7T,7r]^), its Fourier series expansion is /(%) = where, k
5 Boundary Element Technology 339 are the Fourier coefficients of /. Notice = (27r)-/ 0..fc( x )e-'< Jfl» = *.,fc(k) (2.2) Let oo}, q > 0. Then, the following lemma follows from (1.5), (2.2), and Lemma 4.3 of Reference [10] and its proof, which is similar in spirit to Lemma 5 proved in the next section. Lemma 2 Let G be given in (2.1) with respect to 0s, A;- Then for any / %%+^ where 26 > m, 6% Wmf #, = ^ke^ W/)A,k(k), / 6, in the closure of the convex hull of ME, (G) = {cg(x t) : t G [ TT, TT]^, c e R with\c\ <Ef}. In the above Lemma and following results, we request 2k > n to ensure <&a,& E "Hn since (j)g^ G C?*. As an obvious conseqence of Lemmas 1 and 2, the following result arrives. Proposition 3 Let f W*~*"^+*. Let 0^ denote the compactly supported radial basis functions of minimal degree, positive definite on R* and in where 2k >n. Then for any integer m > I, there is a Junction where Cj G R, tj G [ 37T,37r]% for I <j < 2*m, such that where C/ is a constant independent of m. Proof By Lemmas 1 and 2, there is a function where Cj G #, tj G [ 7T,7r]^, for 1 < j < m, such that
6 340 Boundary Element Technology Notice that for each t [ 7T,7r]*, by (2.1), G(- t) is a sum of at most 2* function </>«,&( t 2?rk), k Z% which do not vanish in [ 7T,7r]*, since <&s,&(") = </>s,fc( * ) is supported in [ 1,1]*. The conclusion then follows. 3. Neural Networks by Compactly Supported Functions A neural network with one hidden layer is mathematically expressed as fc=i &,x) 4-0%), (3.1) where a is the activation function of the network, c& R, w& E #*, 0& #, for 1 < k < n, and n is the number of neurons. Approximation by the networks in (3.1) has been extensively studied in the literature, with various results established by many authors under more or less conditions (cf. [1,9], and references therein). In this section we use a compactly supported univariate function <t>i,k(%) to construct neural networks, and establish the following result. Proposition 4 Let f H^*. Let 4%% denote the compactly supported univariate radial basis function of minimal degree, positive definite on R and in C^, where 2k > n. Then for any m > 1, there is a network in the form such that 6=1 where Cf is independent of m. Let For a constant M > 0, set a(x) = V* <t>i,k(x 4-2mr), x E [-TT, TT]. (3.2) c,0e^,w ^, with c < M, w < 1, 0 <?r}. (3.3) Then we have the following lemma. For the sake of convenience to the reader, we present its proof by using a similar argument as in Reference [10]. Lemma 5 Let a be given in (3.2) with respect to 4>i^. Then for any f *H%*~^, where 2k > n, /(x) <%(/) is in the closure of the convex hull oftom,(<r) inhn norm, where M/ = M/)/$i,jb(U ).
7 Boundary Element Technology 341 Proof Without loss of generality, assume <%(/) = 0. Denote the closure of the convex hull of fi^/ (&) by Co(QMf (<?)) Then Co( lmf (#)) is a bounded subset in Hn- Suppose that / is not in Co(fi^(cr)), then by a standard argument on the dual space and applying the Hahn-Banach Theorem (cf. [13], for instance), there exist g& L?[ TT, TT]*, a < n, such that sup Dhgadx < r < \a\<n (3.4) for some constant r. Observe that for any e > 0, one can easily construct <?a with all order partial derivatives, compactly supported in [ 7r,7r]*, such that \\9a~ 9<*\\L*[-ir,ir]' < Therefore, without loss of generality, we assume that the functions g&, in (3.4) have all order partial derivatives and are compactly supported in [ 7r,7r]*. Evaluating the integrals in (3.4) by parts yields _ (3.5) for any h Cof^M/M). Let s(x) = ^ (-l)i^d^^(x). Then for any N<n C, c < My, Set c j a«w,x) 4-6)s(x)dx < r < I f(x)s(x)dx. (3.6) i/[-7r,7r]* 7[-7r,7r]^ Then (3.6) implies (3.7) For any multi-integer j Z*, j ^ 0, let w^ = j -. Hence w, < 1. We obtain = I s(x) I e'u'*<r J[ TT.TT]^ J -re
8 342 Boundary Element Technology From (3.7), we have According to the definition of M/, we have / /(xkx)dx < = f, which contradicts (3.5). The proof of the lemma is finished. Proof of Proposition 4 First, by Lemmas 1 and 5, we conclude that there is a network m N(x) = CQ(/) + Y^Cfc<7((Wfc,x) 4-Ok), W& < 1, #& < 7T fc=l such that Observe that (w&,x) -f Ok\ < (s 4- I)TT for x [ 7r,7r]*, and for t G [ (5 + l)?r, (s4-l)7r], a(t) is a sum of at most 5+2 functions </>( 4-2n7r), 2n < s-\-l. Therefore, the conclusion of Propostion 4 follows. Acknowledgement: The author thanks referees' helpful comments. References [1] A. Barron, Universal approximation bounds for superpositions of a sigmoidal function, IEEE Trans. Inf. Theory, 39 (1993), [2] G. Davis, S. Mallat, and M. Avellaneda, Adaptive greedy approximations, Constr. Approx., 13 (1997), [3] M.J. Donahue, L. Gurvits, C. Darken, and E. Sontag, Rates of convex approximation in non-hilbert spaces, Constr. Approx., 13 (1997), [4] M. Daehlen, T. Lyche, and L.L. Schumaker, Mathematical Methods for Curves and Surfaces, Nashville & London, Vanderbilt University Press, 1995.
9 Boundary Element Technology 343 [5] R.A. DeVore and V.N. Temlyakov, Some remarks on greedy algorithms, Advances in Computational Mathematics, 5 (1996), [6] R. Franke, Scattered data interpolation: test of some methods, Mathematics of Computation, 38 (1982), [7] M.A. Golberg and C.S. Chen, Discrete Projection Methods in Integral Equations, Computational Mechanics Publications, Southampton, Boston, [8] L.K. Jones, A simple lemma on greedy approximation in Hilbert space and convergence rates for projection pursuit regression and neural network training, Ann. Statist., 20 (1992), [9] X. Li, Simultaneous approximations of multivariate functions and their derivatives by neural networks with one hidden layer, Neurocomputing, 12 (1996), [10] X. Li, On simultaneous approximations by radial basis function neural networks, Applied Mathematics and Computation, 95 (1998), [11] C.A. Micchelli, Interpolation of scattered data: distance matrices and conditionally positive definite functions, Constr. Approx., 1 (1986), [12] G. Pisier, Remarques sur un resultat non public de B. Maurey, Proc. Seminaire d' analyse fonctionelle, vol. 1-12, Ecole Polytechnique, Palaiseau, [13] W. Rudin, Functional Analysis, McGraw-Hill, New York, [14] R. Schaback, Improved error bounds for scattered data interpolation by radial basis function, Math. Comp., to appear. [15] J. Stewart, Positive definite functions and generalizations, an historical survey, Rocky Mountain J. Math., 6 (1976), [16] H. Wendland, Piecewise polynomial, positive definite and compactly supported radial basis functions of minimal degree, Advances in Computational Mathematics, 4 (1995), [17] H. Wendland, Error estimates for interpolation by compactly supported radial basis functions of minimal degree, J. Approx. Theory, 93 (1998), [18] Z. Wu, Compactly supported positive definite radial functions, Advances in Computational Mathematics, 4 (1995) [19] Z. Wu and R. Schaback, Local error estimates for radial basis function interpolation of scattered data, IMA Journal of Numerical Analysis, 13 (1993)
Approximation by Conditionally Positive Definite Functions with Finitely Many Centers
Approximation by Conditionally Positive Definite Functions with Finitely Many Centers Jungho Yoon Abstract. The theory of interpolation by using conditionally positive definite function provides optimal
More information2014:05 Incremental Greedy Algorithm and its Applications in Numerical Integration. V. Temlyakov
INTERDISCIPLINARY MATHEMATICS INSTITUTE 2014:05 Incremental Greedy Algorithm and its Applications in Numerical Integration V. Temlyakov IMI PREPRINT SERIES COLLEGE OF ARTS AND SCIENCES UNIVERSITY OF SOUTH
More informationSolving the 3D Laplace Equation by Meshless Collocation via Harmonic Kernels
Solving the 3D Laplace Equation by Meshless Collocation via Harmonic Kernels Y.C. Hon and R. Schaback April 9, Abstract This paper solves the Laplace equation u = on domains Ω R 3 by meshless collocation
More informationKernel B Splines and Interpolation
Kernel B Splines and Interpolation M. Bozzini, L. Lenarduzzi and R. Schaback February 6, 5 Abstract This paper applies divided differences to conditionally positive definite kernels in order to generate
More informationMATH 590: Meshfree Methods
MATH 590: Meshfree Methods Chapter 9: Conditionally Positive Definite Radial Functions Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2010 fasshauer@iit.edu MATH
More informationMATH 590: Meshfree Methods
MATH 590: Meshfree Methods Chapter 5: Completely Monotone and Multiply Monotone Functions Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2010 fasshauer@iit.edu MATH
More informationINVERSE AND SATURATION THEOREMS FOR RADIAL BASIS FUNCTION INTERPOLATION
MATHEMATICS OF COMPUTATION Volume 71, Number 238, Pages 669 681 S 0025-5718(01)01383-7 Article electronically published on November 28, 2001 INVERSE AND SATURATION THEOREMS FOR RADIAL BASIS FUNCTION INTERPOLATION
More information290 J.M. Carnicer, J.M. Pe~na basis (u 1 ; : : : ; u n ) consisting of minimally supported elements, yet also has a basis (v 1 ; : : : ; v n ) which f
Numer. Math. 67: 289{301 (1994) Numerische Mathematik c Springer-Verlag 1994 Electronic Edition Least supported bases and local linear independence J.M. Carnicer, J.M. Pe~na? Departamento de Matematica
More informationMultivariate Interpolation with Increasingly Flat Radial Basis Functions of Finite Smoothness
Multivariate Interpolation with Increasingly Flat Radial Basis Functions of Finite Smoothness Guohui Song John Riddle Gregory E. Fasshauer Fred J. Hickernell Abstract In this paper, we consider multivariate
More informationStability of Kernel Based Interpolation
Stability of Kernel Based Interpolation Stefano De Marchi Department of Computer Science, University of Verona (Italy) Robert Schaback Institut für Numerische und Angewandte Mathematik, University of Göttingen
More informationPositive Definite Functions on Hilbert Space
Positive Definite Functions on Hilbert Space B. J. C. Baxter Department of Mathematics and Statistics, Birkbeck College, University of London, Malet Street, London WC1E 7HX, England b.baxter@bbk.ac.uk
More informationON VECTOR-VALUED INEQUALITIES FOR SIDON SETS AND SETS OF INTERPOLATION
C O L L O Q U I U M M A T H E M A T I C U M VOL. LXIV 1993 FASC. 2 ON VECTOR-VALUED INEQUALITIES FOR SIDON SETS AND SETS OF INTERPOLATION BY N. J. K A L T O N (COLUMBIA, MISSOURI) Let E be a Sidon subset
More informationData fitting by vector (V,f)-reproducing kernels
Data fitting by vector (V,f-reproducing kernels M-N. Benbourhim to appear in ESAIM.Proc 2007 Abstract In this paper we propose a constructive method to build vector reproducing kernels. We define the notion
More informationL p Approximation of Sigma Pi Neural Networks
IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 11, NO. 6, NOVEMBER 2000 1485 L p Approximation of Sigma Pi Neural Networks Yue-hu Luo and Shi-yi Shen Abstract A feedforward Sigma Pi neural networks with a
More informationDIRECT ERROR BOUNDS FOR SYMMETRIC RBF COLLOCATION
Meshless Methods in Science and Engineering - An International Conference Porto, 22 DIRECT ERROR BOUNDS FOR SYMMETRIC RBF COLLOCATION Robert Schaback Institut für Numerische und Angewandte Mathematik (NAM)
More informationNotes for Functional Analysis
Notes for Functional Analysis Wang Zuoqin (typed by Xiyu Zhai) November 6, 2015 1 Lecture 18 1.1 The convex hull Let X be any vector space, and E X a subset. Definition 1.1. The convex hull of E is the
More informationAN ELEMENTARY PROOF OF THE OPTIMAL RECOVERY OF THE THIN PLATE SPLINE RADIAL BASIS FUNCTION
J. KSIAM Vol.19, No.4, 409 416, 2015 http://dx.doi.org/10.12941/jksiam.2015.19.409 AN ELEMENTARY PROOF OF THE OPTIMAL RECOVERY OF THE THIN PLATE SPLINE RADIAL BASIS FUNCTION MORAN KIM 1 AND CHOHONG MIN
More informationMATH 590: Meshfree Methods
MATH 590: Meshfree Methods Chapter 33: Adaptive Iteration Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2010 fasshauer@iit.edu MATH 590 Chapter 33 1 Outline 1 A
More informationMATH 590: Meshfree Methods
MATH 590: Meshfree Methods Chapter 33: Adaptive Iteration Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2010 fasshauer@iit.edu MATH 590 Chapter 33 1 Outline 1 A
More informationApproximation of Multivariate Functions
Approximation of Multivariate Functions Vladimir Ya. Lin and Allan Pinkus Abstract. We discuss one approach to the problem of approximating functions of many variables which is truly multivariate in character.
More informationDecomposition of Riesz frames and wavelets into a finite union of linearly independent sets
Decomposition of Riesz frames and wavelets into a finite union of linearly independent sets Ole Christensen, Alexander M. Lindner Abstract We characterize Riesz frames and prove that every Riesz frame
More informationInterpolation by Basis Functions of Different Scales and Shapes
Interpolation by Basis Functions of Different Scales and Shapes M. Bozzini, L. Lenarduzzi, M. Rossini and R. Schaback Abstract Under very mild additional assumptions, translates of conditionally positive
More informationOn the optimality of incremental neural network algorithms
On the optimality of incremental neural network algorithms Ron Meir* Department of Electrical Engineering Technion, Haifa 32000, Israel rmeir@dumbo.technion.ac.il Vitaly Maiorovt Department of Mathematics
More informationApproximating By Ridge Functions. Allan Pinkus. We hope it will also encourage some readers to consider researching
Approximating By Ridge Functions Allan Pinus Abstract. This paper surveys certain aspects of the study of ridge functions. We hope it will also encourage some readers to consider researching problems in
More informationD. Shepard, Shepard functions, late 1960s (application, surface modelling)
Chapter 1 Introduction 1.1 History and Outline Originally, the motivation for the basic meshfree approximation methods (radial basis functions and moving least squares methods) came from applications in
More informationof Orthogonal Matching Pursuit
A Sharp Restricted Isometry Constant Bound of Orthogonal Matching Pursuit Qun Mo arxiv:50.0708v [cs.it] 8 Jan 205 Abstract We shall show that if the restricted isometry constant (RIC) δ s+ (A) of the measurement
More informationOn Ridge Functions. Allan Pinkus. September 23, Technion. Allan Pinkus (Technion) Ridge Function September 23, / 27
On Ridge Functions Allan Pinkus Technion September 23, 2013 Allan Pinkus (Technion) Ridge Function September 23, 2013 1 / 27 Foreword In this lecture we will survey a few problems and properties associated
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 informationStability constants for kernel-based interpolation processes
Dipartimento di Informatica Università degli Studi di Verona Rapporto di ricerca Research report 59 Stability constants for kernel-based interpolation processes Stefano De Marchi Robert Schaback Dipartimento
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 informationOn interpolation by radial polynomials C. de Boor Happy 60th and beyond, Charlie!
On interpolation by radial polynomials C. de Boor Happy 60th and beyond, Charlie! Abstract A lemma of Micchelli s, concerning radial polynomials and weighted sums of point evaluations, is shown to hold
More informationScattered Data Interpolation with Polynomial Precision and Conditionally Positive Definite Functions
Chapter 3 Scattered Data Interpolation with Polynomial Precision and Conditionally Positive Definite Functions 3.1 Scattered Data Interpolation with Polynomial Precision Sometimes the assumption on the
More informationA sharp upper bound on the approximation order of smooth bivariate pp functions C. de Boor and R.Q. Jia
A sharp upper bound on the approximation order of smooth bivariate pp functions C. de Boor and R.Q. Jia Introduction It is the purpose of this note to show that the approximation order from the space Π
More informationNon-radial solutions to a bi-harmonic equation with negative exponent
Non-radial solutions to a bi-harmonic equation with negative exponent Ali Hyder Department of Mathematics, University of British Columbia, Vancouver BC V6TZ2, Canada ali.hyder@math.ubc.ca Juncheng Wei
More informationError formulas for divided difference expansions and numerical differentiation
Error formulas for divided difference expansions and numerical differentiation Michael S. Floater Abstract: We derive an expression for the remainder in divided difference expansions and use it to give
More informationLocally convex spaces, the hyperplane separation theorem, and the Krein-Milman theorem
56 Chapter 7 Locally convex spaces, the hyperplane separation theorem, and the Krein-Milman theorem Recall that C(X) is not a normed linear space when X is not compact. On the other hand we could use semi
More informationEstimates of the Number of Hidden Units and Variation with Respect to Half-spaces
Neural Networks 10: 1061--1068, 1997 Estimates of the Number of Hidden Units and Variation with Respect to Half-spaces Věra Kůrková Institute of Computer Science, Czech Academy of Sciences, Prague, Czechia
More informationMATH 590: Meshfree Methods
MATH 590: Meshfree Methods Chapter 7: Conditionally Positive Definite Functions Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2010 fasshauer@iit.edu MATH 590 Chapter
More informationBKW-Operators on the Interval and the Sequence Spaces
journal of approximation theory 87, 159169 (1996) article no. 98 BKW-Operators on the Interval and the Sequence Spaces Keiji Izuchi Department of Mathematics, Faculty of Science, Niigata University, Niigata
More informationBiorthogonal Spline Type Wavelets
PERGAMON Computers and Mathematics with Applications 0 (00 1 0 www.elsevier.com/locate/camwa Biorthogonal Spline Type Wavelets Tian-Xiao He Department of Mathematics and Computer Science Illinois Wesleyan
More informationL. Levaggi A. Tabacco WAVELETS ON THE INTERVAL AND RELATED TOPICS
Rend. Sem. Mat. Univ. Pol. Torino Vol. 57, 1999) L. Levaggi A. Tabacco WAVELETS ON THE INTERVAL AND RELATED TOPICS Abstract. We use an abstract framework to obtain a multilevel decomposition of a variety
More informationA NOTE ON MATRIX REFINEMENT EQUATIONS. Abstract. Renement equations involving matrix masks are receiving a lot of attention these days.
A NOTE ON MATRI REFINEMENT EQUATIONS THOMAS A. HOGAN y Abstract. Renement equations involving matrix masks are receiving a lot of attention these days. They can play a central role in the study of renable
More informationNONTRIVIAL SOLUTIONS FOR SUPERQUADRATIC NONAUTONOMOUS PERIODIC SYSTEMS. Shouchuan Hu Nikolas S. Papageorgiou. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 34, 29, 327 338 NONTRIVIAL SOLUTIONS FOR SUPERQUADRATIC NONAUTONOMOUS PERIODIC SYSTEMS Shouchuan Hu Nikolas S. Papageorgiou
More informationBernstein-Szegö Inequalities in Reproducing Kernel Hilbert Spaces ABSTRACT 1. INTRODUCTION
Malaysian Journal of Mathematical Sciences 6(2): 25-36 (202) Bernstein-Szegö Inequalities in Reproducing Kernel Hilbert Spaces Noli N. Reyes and Rosalio G. Artes Institute of Mathematics, University of
More informationReal Analysis, 2nd Edition, G.B.Folland Elements of Functional Analysis
Real Analysis, 2nd Edition, G.B.Folland Chapter 5 Elements of Functional Analysis Yung-Hsiang Huang 5.1 Normed Vector Spaces 1. Note for any x, y X and a, b K, x+y x + y and by ax b y x + b a x. 2. It
More informationk-dimensional INTERSECTIONS OF CONVEX SETS AND CONVEX KERNELS
Discrete Mathematics 36 (1981) 233-237 North-Holland Publishing Company k-dimensional INTERSECTIONS OF CONVEX SETS AND CONVEX KERNELS Marilyn BREEN Department of Mahematics, Chiversify of Oklahoma, Norman,
More informationThe Dirichlet s P rinciple. In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation:
Oct. 1 The Dirichlet s P rinciple In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation: 1. Dirichlet s Principle. u = in, u = g on. ( 1 ) If we multiply
More informationMeshfree Approximation Methods with MATLAB
Interdisciplinary Mathematical Sc Meshfree Approximation Methods with MATLAB Gregory E. Fasshauer Illinois Institute of Technology, USA Y f? World Scientific NEW JERSEY LONDON SINGAPORE BEIJING SHANGHAI
More informationExistence of Solutions for a Class of p(x)-biharmonic Problems without (A-R) Type Conditions
International Journal of Mathematical Analysis Vol. 2, 208, no., 505-55 HIKARI Ltd, www.m-hikari.com https://doi.org/0.2988/ijma.208.886 Existence of Solutions for a Class of p(x)-biharmonic Problems without
More informationMultiplication Operators with Closed Range in Operator Algebras
J. Ana. Num. Theor. 1, No. 1, 1-5 (2013) 1 Journal of Analysis & Number Theory An International Journal Multiplication Operators with Closed Range in Operator Algebras P. Sam Johnson Department of Mathematical
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 informationON A CERTAIN GENERALIZATION OF THE KRASNOSEL SKII THEOREM
Journal of Applied Analysis Vol. 9, No. 1 23, pp. 139 147 ON A CERTAIN GENERALIZATION OF THE KRASNOSEL SKII THEOREM M. GALEWSKI Received July 3, 21 and, in revised form, March 26, 22 Abstract. We provide
More informationDiscrete Projection Methods for Integral Equations
SUB Gttttingen 7 208 427 244 98 A 5141 Discrete Projection Methods for Integral Equations M.A. Golberg & C.S. Chen TM Computational Mechanics Publications Southampton UK and Boston USA Contents Sources
More informationA Note on Nonconvex Minimax Theorem with Separable Homogeneous Polynomials
A Note on Nonconvex Minimax Theorem with Separable Homogeneous Polynomials G. Y. Li Communicated by Harold P. Benson Abstract The minimax theorem for a convex-concave bifunction is a fundamental theorem
More informationOrthogonal Matching Pursuit for Sparse Signal Recovery With Noise
Orthogonal Matching Pursuit for Sparse Signal Recovery With Noise The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation As Published
More informationEXISTENCE OF SET-INTERPOLATING AND ENERGY- MINIMIZING CURVES
EXISTENCE OF SET-INTERPOLATING AND ENERGY- MINIMIZING CURVES JOHANNES WALLNER Abstract. We consider existence of curves c : [0, 1] R n which minimize an energy of the form c (k) p (k = 1, 2,..., 1 < p
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 informationAdaptive Control with Multiresolution Bases
Adaptive Control with Multiresolution Bases Christophe P. Bernard Centre de Mathématiques Appliquées Ecole Polytechnique 91128 Palaiseau cedex, FRANCE bernard@cmapx.polytechnique.fr Jean-Jacques E. Slotine
More informationViscosity Iterative Approximating the Common Fixed Points of Non-expansive Semigroups in Banach Spaces
Viscosity Iterative Approximating the Common Fixed Points of Non-expansive Semigroups in Banach Spaces YUAN-HENG WANG Zhejiang Normal University Department of Mathematics Yingbing Road 688, 321004 Jinhua
More informationFundamentality of Ridge Functions
Fundamentality of Ridge Functions Vladimir Ya. Lin and Allan Pinkus Abstract. For a given integer d, 1 d n 1, let Ω be a subset of the set of all d n real matrices. Define the subspace M(Ω) = span{g(ax)
More informationApproximation theory in neural networks
Approximation theory in neural networks Yanhui Su yanhui su@brown.edu March 30, 2018 Outline 1 Approximation of functions by a sigmoidal function 2 Approximations of continuous functionals by a sigmoidal
More informationOptimal data-independent point locations for RBF interpolation
Optimal dataindependent point locations for RF interpolation S De Marchi, R Schaback and H Wendland Università di Verona (Italy), Universität Göttingen (Germany) Metodi di Approssimazione: lezione dell
More informationViewed From Cubic Splines. further clarication and improvement. This can be described by applying the
Radial Basis Functions Viewed From Cubic Splines Robert Schaback Abstract. In the context of radial basis function interpolation, the construction of native spaces and the techniques for proving error
More informationON AN INEQUALITY OF KOLMOGOROV AND STEIN
BULL. AUSTRAL. MATH. SOC. VOL. 61 (2000) [153-159] 26B35, 26D10 ON AN INEQUALITY OF KOLMOGOROV AND STEIN HA HUY BANG AND HOANG MAI LE A.N. Kolmogorov showed that, if /,/',..., /'"' are bounded continuous
More informationFunctional Analysis, Stein-Shakarchi Chapter 1
Functional Analysis, Stein-Shakarchi Chapter 1 L p spaces and Banach Spaces Yung-Hsiang Huang 018.05.1 Abstract Many problems are cited to my solution files for Folland [4] and Rudin [6] post here. 1 Exercises
More information1 Continuity Classes C m (Ω)
0.1 Norms 0.1 Norms A norm on a linear space X is a function : X R with the properties: Positive Definite: x 0 x X (nonnegative) x = 0 x = 0 (strictly positive) λx = λ x x X, λ C(homogeneous) x + y x +
More informationOn the computation of Hermite-Humbert constants for real quadratic number fields
Journal de Théorie des Nombres de Bordeaux 00 XXXX 000 000 On the computation of Hermite-Humbert constants for real quadratic number fields par Marcus WAGNER et Michael E POHST Abstract We present algorithms
More informationMultiresolution analysis by infinitely differentiable compactly supported functions. N. Dyn A. Ron. September 1992 ABSTRACT
Multiresolution analysis by infinitely differentiable compactly supported functions N. Dyn A. Ron School of of Mathematical Sciences Tel-Aviv University Tel-Aviv, Israel Computer Sciences Department University
More informationON THE SUPPORT OF CERTAIN SYMMETRIC STABLE PROBABILITY MEASURES ON TVS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 63, Number 2, April 1977 ON THE SUPPORT OF CERTAIN SYMMETRIC STABLE PROBABILITY MEASURES ON TVS BALRAM S. RAJPUT Abstract. Let be a LCTVS, and let
More informationMetric Spaces and Topology
Chapter 2 Metric Spaces and Topology From an engineering perspective, the most important way to construct a topology on a set is to define the topology in terms of a metric on the set. This approach underlies
More informationOverview of normed linear spaces
20 Chapter 2 Overview of normed linear spaces Starting from this chapter, we begin examining linear spaces with at least one extra structure (topology or geometry). We assume linearity; this is a natural
More informationThe Infinity Norm of a Certain Type of Symmetric Circulant Matrix
MATHEMATICS OF COMPUTATION, VOLUME 31, NUMBER 139 JULY 1977, PAGES 733-737 The Infinity Norm of a Certain Type of Symmetric Circulant Matrix By W. D. Hoskins and D. S. Meek Abstract. An attainable bound
More informationENGEL SERIES EXPANSIONS OF LAURENT SERIES AND HAUSDORFF DIMENSIONS
J. Aust. Math. Soc. 75 (2003), 1 7 ENGEL SERIES EXPANSIONS OF LAURENT SERIES AND HAUSDORFF DIMENSIONS JUN WU (Received 11 September 2001; revised 22 April 2002) Communicated by W. W. L. Chen Abstract For
More informationIntroduction to Bases in Banach Spaces
Introduction to Bases in Banach Spaces Matt Daws June 5, 2005 Abstract We introduce the notion of Schauder bases in Banach spaces, aiming to be able to give a statement of, and make sense of, the Gowers
More informationNonlinear tensor product approximation
ICERM; October 3, 2014 1 Introduction 2 3 4 Best multilinear approximation We are interested in approximation of a multivariate function f (x 1,..., x d ) by linear combinations of products u 1 (x 1 )
More informationGinés López 1, Miguel Martín 1 2, and Javier Merí 1
NUMERICAL INDEX OF BANACH SPACES OF WEAKLY OR WEAKLY-STAR CONTINUOUS FUNCTIONS Ginés López 1, Miguel Martín 1 2, and Javier Merí 1 Departamento de Análisis Matemático Facultad de Ciencias Universidad de
More informationi=1 β i,i.e. = β 1 x β x β 1 1 xβ d
66 2. Every family of seminorms on a vector space containing a norm induces ahausdorff locally convex topology. 3. Given an open subset Ω of R d with the euclidean topology, the space C(Ω) of real valued
More information3. Some tools for the analysis of sequential strategies based on a Gaussian process prior
3. Some tools for the analysis of sequential strategies based on a Gaussian process prior E. Vazquez Computer experiments June 21-22, 2010, Paris 21 / 34 Function approximation with a Gaussian prior Aim:
More informationKernels for Multi task Learning
Kernels for Multi task Learning Charles A Micchelli Department of Mathematics and Statistics State University of New York, The University at Albany 1400 Washington Avenue, Albany, NY, 12222, USA Massimiliano
More informationA Proof of Markov s Theorem for Polynomials on Banach spaces
A Proof of Markov s Theorem for Polynomials on Banach spaces Lawrence A. Harris Department of Mathematics, University of Kentucky Lexington, Kentucky 40506-007 larry@ms.uky.edu Dedicated to my teachers
More informationToward Approximate Moving Least Squares Approximation with Irregularly Spaced Centers
Toward Approximate Moving Least Squares Approximation with Irregularly Spaced Centers Gregory E. Fasshauer Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 6066, U.S.A.
More informationContinuous functions that are nowhere differentiable
Continuous functions that are nowhere differentiable S. Kesavan The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai - 600113. e-mail: kesh @imsc.res.in Abstract It is shown that the existence
More informationINFINITUDE OF MINIMALLY SUPPORTED TOTALLY INTERPOLATING BIORTHOGONAL MULTIWAVELET SYSTEMS WITH LOW APPROXIMATION ORDERS. Youngwoo Choi and Jaewon Jung
Korean J. Math. (0) No. pp. 7 6 http://dx.doi.org/0.68/kjm.0...7 INFINITUDE OF MINIMALLY SUPPORTED TOTALLY INTERPOLATING BIORTHOGONAL MULTIWAVELET SYSTEMS WITH LOW APPROXIMATION ORDERS Youngwoo Choi and
More informationLower Bounds for Approximation by MLP Neural Networks
Lower Bounds for Approximation by MLP Neural Networks Vitaly Maiorov and Allan Pinkus Abstract. The degree of approximation by a single hidden layer MLP model with n units in the hidden layer is bounded
More informationExercise Solutions to Functional Analysis
Exercise Solutions to Functional Analysis Note: References refer to M. Schechter, Principles of Functional Analysis Exersize that. Let φ,..., φ n be an orthonormal set in a Hilbert space H. Show n f n
More informationarxiv: v2 [cs.ne] 20 May 2016
A SINGLE HIDDEN LAYER FEEDFORWARD NETWORK WITH ONLY ONE NEURON IN THE HIDDEN LAYER CAN APPROXIMATE ANY UNIVARIATE FUNCTION arxiv:1601.00013v [cs.ne] 0 May 016 NAMIG J. GULIYEV AND VUGAR E. ISMAILOV Abstract.
More informationSome Properties of the Augmented Lagrangian in Cone Constrained Optimization
MATHEMATICS OF OPERATIONS RESEARCH Vol. 29, No. 3, August 2004, pp. 479 491 issn 0364-765X eissn 1526-5471 04 2903 0479 informs doi 10.1287/moor.1040.0103 2004 INFORMS Some Properties of the Augmented
More informationConvergence of greedy approximation I. General systems
STUDIA MATHEMATICA 159 (1) (2003) Convergence of greedy approximation I. General systems by S. V. Konyagin (Moscow) and V. N. Temlyakov (Columbia, SC) Abstract. We consider convergence of thresholding
More information1.4 The Jacobian of a map
1.4 The Jacobian of a map Derivative of a differentiable map Let F : M n N m be a differentiable map between two C 1 manifolds. Given a point p M we define the derivative of F at p by df p df (p) : T p
More informationON A CLASS OF NONSMOOTH COMPOSITE FUNCTIONS
MATHEMATICS OF OPERATIONS RESEARCH Vol. 28, No. 4, November 2003, pp. 677 692 Printed in U.S.A. ON A CLASS OF NONSMOOTH COMPOSITE FUNCTIONS ALEXANDER SHAPIRO We discuss in this paper a class of nonsmooth
More informationLinear Independence of Finite Gabor Systems
Linear Independence of Finite Gabor Systems 1 Linear Independence of Finite Gabor Systems School of Mathematics Korea Institute for Advanced Study Linear Independence of Finite Gabor Systems 2 Short trip
More informationOn John type ellipsoids
On John type ellipsoids B. Klartag Tel Aviv University Abstract Given an arbitrary convex symmetric body K R n, we construct a natural and non-trivial continuous map u K which associates ellipsoids to
More informationA Perron-type theorem on the principal eigenvalue of nonsymmetric elliptic operators
A Perron-type theorem on the principal eigenvalue of nonsymmetric elliptic operators Lei Ni And I cherish more than anything else the Analogies, my most trustworthy masters. They know all the secrets of
More informationGeometry and topology of continuous best and near best approximations
Journal of Approximation Theory 105: 252 262, Geometry and topology of continuous best and near best approximations Paul C. Kainen Dept. of Mathematics Georgetown University Washington, D.C. 20057 Věra
More informationThe Rademacher Cotype of Operators from l N
The Rademacher Cotype of Operators from l N SJ Montgomery-Smith Department of Mathematics, University of Missouri, Columbia, MO 65 M Talagrand Department of Mathematics, The Ohio State University, 3 W
More informationWeak-Star Convergence of Convex Sets
Journal of Convex Analysis Volume 13 (2006), No. 3+4, 711 719 Weak-Star Convergence of Convex Sets S. Fitzpatrick A. S. Lewis ORIE, Cornell University, Ithaca, NY 14853, USA aslewis@orie.cornell.edu www.orie.cornell.edu/
More informationOn the usage of lines in GC n sets
On the usage of lines in GC n sets Hakop Hakopian, Vahagn Vardanyan arxiv:1807.08182v3 [math.co] 16 Aug 2018 Abstract A planar node set X, with X = ( ) n+2 2 is called GCn set if each node possesses fundamental
More informationProblem Set 6: Solutions Math 201A: Fall a n x n,
Problem Set 6: Solutions Math 201A: Fall 2016 Problem 1. Is (x n ) n=0 a Schauder basis of C([0, 1])? No. If f(x) = a n x n, n=0 where the series converges uniformly on [0, 1], then f has a power series
More informationINDUSTRIAL MATHEMATICS INSTITUTE. B.S. Kashin and V.N. Temlyakov. IMI Preprint Series. Department of Mathematics University of South Carolina
INDUSTRIAL MATHEMATICS INSTITUTE 2007:08 A remark on compressed sensing B.S. Kashin and V.N. Temlyakov IMI Preprint Series Department of Mathematics University of South Carolina A remark on compressed
More informationPLEASE SCROLL DOWN FOR ARTICLE
This article was downloaded by: [Ingrid, Beltita] On: 2 March 29 Access details: Access Details: [subscription number 9976743] Publisher Taylor & Francis Informa Ltd Registered in England and Wales Registered
More information