On the mean value of certain functions connected with the convergence of orthogonal series


 Charlene McBride
 2 years ago
 Views:
Transcription
1 Analysis Mathematica, 4 (1978), On the mean value of certain functions connected with the convergence of orthogonal series B. S. KASIN Dedicated to Professor P. L. UVjanov on his 50th birthday In the study of the almost everywhere (a.e.) convergence of orthogonal series often arises the following question. Let Ф = Ф(и) = {<^(х)}" =1 be an orthonormal system of functions defined on the segment [0, 1]. Let us define the operator 1 S%: lg>~l 2 (Q, 1) in the following way. If у = {у^=1 б 1\, then (1) St(y) =f(x) = sup Let s(<p) denote the norm of the operator S%, i.e., (2) 5(Ф)= sup \\Si(y)h4o.i) llyllm^l We would like to estimate the number,у(ф). The classical result of D.E. Mensov and H. Rademacher (cf. [3, p. 188]) says that for every Ф = Ф(п) 2 we have (3) s(^)^clnw. For every n^l Mensov (cf. [3, p. 192]) exhibited an example of a system Ф 0 =Ф 0 (п) such that (4) 5(Ф 0 )^с1пл. In the present paper we shall study the "mean values" of the norms,у(ф). To illuminate the precise meaning of the term "mean value" we consider the following set of orthonormal systems 2" = {Ф}. Received March 1, By 11 we denote the space R n with Euclidean norm. 2 In the sequel С, с, and В denote positive absolute constants.
2 28 В. S. Kasin If Ф б", then Ф = {(р 1 (х)}" ==1 and each function <р ( (х) is piecewise constant on the n subintervals of [0, 1] of equal length. There is a onetoone correspondance between the systems <P Q n and the elements of the group O n of orthogonal matrices of order n. Namely, to the system Ф = {Ф^(Х)} corresponds the matrix А = {а^}^о п of the following form: (5),,.,==L^i^/i) il*l,j*n). With the aid of this correspondance the Haar measure fx n defined on O n (cf. [6, p. 43]) may be carried over onto the set Q n. In the sequel the measure of a closed set Ее Q n shall be denoted also by ц п (Е). We have the following Theorem. There exist absolute constants С and y>0 such that for every пш\ and t^o we have (6) M* fi": *(*) s* t} ^ (Се* 2 )". The statement of this theorem is included in [4] without proof. From this theorem we immediately obtain the corollaries below. Corollary 1. There exists an absolute constant В such that (б 7 )!л п {Феа п :8(Ф)^В}^е п. Corollary 2. Let S(n) denote the set of all permutations of the numbers 1,2,...,TI. For G S(n). let Ф^ denote the rearrangement of the terms of Ф ( 6") according to a. Then [хлф^о 1 : max $(Ф а ) шс Yhin} < n" n. From the estimation (6') it follows that there are "very few" systems having the property discovered by Mensov (we emphasize that in (4) the system Ф 0 (п) may be chosen from Q n ). For the proof of Theorem 1 we need some lemmas. First we shall introduce some notations. The number of elements of any finite set E shall be denoted by \E\. Furthermore, if x R m, then (x)j (l^j^m) denotes the jth coordinate of the vector x, and, as usually, R\ ={x = {(х)$ =1 : (x)j ^ 0, 1 ^ j ^ n}; S» = {x R n : \\х\\ п = 1}. If L is a subspace in R n, dim L = q 9 then m L denotes the normed Lebesgue measure on the gdimensional sphere S n DL (m L (S n f]l) = l). The measure m Rn (E) shall be denoted simply by m(e).
3 Convergence of orthogonal series 29 Lemma 1. For each rn^l one may choose a set of vectors Q m ={e} such that (i) GJ<C M ; (ii) if e Q m, then e R m + and e,sl/3; (Hi) for any vector y S m C)R"l there is a vector e=e(y) Q m such that (e(y))js(y)j (ls/sm). Proof. Suppose that y S m f}r1. Let us set for 5=0,1,2,... * E s (y) = {j : 2 1 < уj ё 2, 1 s ; = m}. It is easy to see that (7) a) b) ( i: г \E,(y)\ < 2* +», 2 /ST^j) 1/2 = ]^1 2 " For each j.s m n.r+ let us define the vector e(y) R m in the following way: (2* 1 if jсад, OSss [1 log 2 m] +1, (8) (e(y))j=\ (0 if ttl>e,(y), 0^ss[log 2 mj. From the construction of e{y) it is obvious that (9) 0^(e(y))j^(y)j. Besides, in virtue of the definition of E s (y) and part b) of relation (7) we have l f l J ' Let ««= U e(y). y S m nr On account of (9) and (9') to conclude the proof of Lemma 1 it is enough to show that O w ^C m. It is obvious that for any j;<es m ni^ and s^o we have \E s (y)\^m. Consequently, the number of the different systems Я (у) of the form 1 AGO = { 3,(y),..., GOD does not exceed (wf l) 2+(log2W)/2. From part a) of relation (7) it follows easily that the number of vectors e(y) of the system Q m with the same system k{y) does not exceed (O m П ст. l^$^(log 2 m)/2
4 30 В. S. Kasin Finally, for : the number \Q m \ we obtain the estimate l^s^(log 2w)/2 By using the elementary estimate C^^C q (m/q) q we obtain that a [(log 2 m)/2] ( m \ 22s [log 8 m] ( m \ ' sc " Ж Ш sc \g Ш 2q  This proves Lemma 1. { [log 2 m] / m \ 29 1 f[log 2 m] m 1 In Д (f j }= C m exp{ Д 2ЧпЩ g m\ 2 In [^C ffl exp{mc'}sc m. We shall use the following wellknown fact (cf, for example, [5, p. 335]). On the sphere S n there exists a set = {<} such that \Q' n \^C n and for any y S n there exists a vector e / = e / (y) Q / n for which (10) Ik'OOjlhsey. The estimate below is also wellknown. Lemma 2. For t^0 9 l^n<, and x 0 S n we have the inequality f(t) = m{x^s n : \(x,x 0 )\ <* t}^ e~^\ For t>l the statement of Lemma 2 is obvious, as in this case f(t)=0. For Q^t^l the set {x S n :(x, x 0 )^t} is contained in the hemisphere of radius (1 1 2 ) 1/2 and consequently, ДО (i./2)(ni)/2 = [(2 / 2 ) 1 /' 2 ]* аи (,,  1 >/ 2л ^ e~ f2n/4 (in obtaining the last inequality we used the estimation (1 xf^^e" 1, Lemma 2 has been proved. Lemma 2 immediately implies  0<x<:l). Consequence. Let X be a subspace of R\ d\mx=m, (Mx 0 i? w. Then for t^q we have (11) m x {* w fu: (x,x 0 )l ^/} ^ exp{ mr 2 W ^2}. Lemma 3. ITiere are absolute constants С and a>0 л/сй that for every subspace XaR n 0>3, diml^ 1+«/2), vector a={a f } *S r ", and number t^o we have the inequality (12) m L fj? 5"nL: max l^e, J ^ Л ^ Cfe" 1 ".
5 Convergence of orthogonal series 31 Proof. It is enough to prove Lemma 3 for t^20jn w, since for t^20n~ 1/2 the statement of our lemma is obtained because of the choice of the constant С in (12). Without loss of generality we may suppose that a^o (1^Шп). n For $=0,1,2,...' let us split the sum 2ад i nt0 the "pieces" {P*}^1 in the following way: n Hi 1 n Р =2 1Уи Pl= 2 ЪУг + Яп^т* Р1 = (а П1 ЯпОУт+ 2 а гугi=l i=l i=n 1 +l Analogously, the "piece" P* 1 is represented in the form ps 1 ps _j_ ps r v  r.2v't"' r 2v + l' A similar division is used in the proof of Erdos' theorem on the a.e. convergence of lacunary trigonometric series (cf. [1, p ]). It is not hard to see (cf. [1, p. 705, Lemma 2]) that the splitting {PUfJi 1, $=0, 1, 2,..., of the above form may be chosen in such a way that for each "piece" n P s v = 2 a \ v,s) yt (0^v<2") we have the estimation i=l (13) J[a/ V ' s) ] 2^2" s. 8 = 1. (13) implies that we can find an integer s 0 such that all "pieces" P* (0^v^2* 1) contain at most two terms. Then we have the inequality max 2^Уг so ^ 2 max \P S V\ + max а^,. 0^v<2 s=0 = v ' ля s 1^Шп Consequently, if for fixed vectors {a} and {y} and for s=0,l,... 9 s 0 inequalities (14) then Hence we have max \P S V 0^V<2 5 max VII r 2 i = l and max \a t y t \ ^, 5(s+l) 2 iiiiv ' "'  3 we have the (15) m L \ytst\l: max 24j#' = 2 2 may^s'^l.^^jt ^]^ + т ь {^ 5"П :шах в у, *4}.
6 .32 В, S. Kasin Due to (13), the consequence of Lemma 2 (cf. (11)), and the inequality diml^ ^ l+n/2 we obtain that <16> У^{у ЬП8':\Р'\^1^}^^{ 1^^У Besides, by using (11) we get that {16') m L \yzst\l: max \a,y t \ ё 4"} = Si Wt {,^nx: W>^} S iexp{0}. Comparing inequalities (15), (16), and (16') we arrive at <17) mjye^nl: max 24yJ = Л = io Г 2 s / 2 ra 1 " ( t 2 n] If г 2 и>200, then it is easy to show that s o Г 2 s / 2 n 1 and, moreover, ^ех Н~^Л~^< ^ ехр ^4 j?2«exp{~2^с'еу* п (у > 0). By putting the last inequalities into (17) we obtain the estimation (12). Lemma 3 is proved. We are going to use the following property of the measure /л п on the group O n (cf. [6, p. 55]). Let usfixan arbitrary vector e 0 S n and denote by О й ~ 1 = {Л) the subgroup of the group O n whose elements are those matrices A for which Ae 0 =e 0. The symbol fi n^l9 as before, denotes Haar measure on О п ~ г. For e S" let us choose an arbitrary matrix A e from O n with A e e 0 = e. Then we have {18) ff(a) dfi n = J dm f f(a e A) dii n. x $n 0nl for any measurable bounded function /04), A O n.
7 Convergence of orthogonal series 33 Proof of Theorem. We may suppose that л^10, as for ж 10 the statement of the theorem shall be satisfied if we choose the constant С in (6) large enough. Let a system Ф () п be given. Suppose that the matrix А(Ф)=А = {а (] } О п is defined by the equality (5). Then it is easy to see that 2 \2у* а Л\ = sup 2 2 y + 2 \2УЫ\ # y={ yi }<=S*,{Nj}4 = l \i = l t i=[!t/2] + l Vi = l /J Consequently, it is enough to prove that for m=[n/2] and m=n [n/2] and for all t^o we have 2л 1/2 (20) M(t) = ti{aco»: '^ sup [J (i y ; a y j 2 ] 1/2 is L} ^ (Cei**)" Let the sequence {Д/}" =1 of numbers be fixed. By using the properties of the system of vectors Q' n (cf. (10)) we obtain that [ m / AT, \2i"ii/a Г m f N J /2 2 2w s sup 2\2(уе'(у)Ш,^_ j=l \i = l /.J ye^lj^l 4=1 ^2, 1/2 + supfi{f(e'0')) J a y } 8 ] 3,^S n Lj=i lj=i J J f и ч1/2 Taking into account that Z(^~^(7)) 2 =1/2 we arrive at 1 г ш г NJ i 2 ii/ 2 J J As ^ ^С И (cf. (10)), from the last inequality and from the definition of the function M{t) (cf. (20)) we obtain that (21) M(t) * {СУ sup JASO*: sup f 2 [2 *,<J1 =" 4=}. By virtue of Lemma 1 from inequality (21) it follows that (22) M(t) ^C n sup /*jv< ц \ A O n : sup zes n,e Q m I {#,} J? 2ъ<*! j i=l j=(e)j,l*j*m Let the vectors z^s", e Q m, and the number t^0 befixed.then (23) /iu 0":sup 1 3 Analysis Mathematica t 1 =(e) y, lijsm = / {ty} V2 * nn o n J /=i = m } nx )dn,
8 34 В. S. Kasin where A = {a ij }eo n, l^j^rn, and I 1 if sup {0 otherwise. N j 2 z^ij i=l Jf^> As \\e\\i^l/3 9 е О т (cf. Lemma 1), a multiple application of Lemma 3 and equality (18) show that f л m m ( а л (24) f IIxM)dfi n = ncap\ T t*n(ejn* on j=i j=i { z J ^ C m Qxpljt 2 n 2(e)j\ ^ C n exp(^2) 5 where y>0 is an absolute constant. From (22) (24) we obtain that M(t)^ Cexp{ynt 2 }. Thus inequality (20) is proved. As it was noted earlier, the statement of our theorem follows from (20). Remark. We estimated the mean value of the function я(ф) on the set Q n consisting of orthonormal systems. The estimation of the mean value of the function,у(ф) on the set of all (not necessarily orthogonal) systems such that Ф {(Pi(x)}" =lr> \<p.(x)\ = l 9 the function (р г (х) being constant on the intervals ((j l)/n, j/n), t^i, j=n, x [0, 1], may be carried out following the method of [2]. References [1] H. К. Бари, Тригонометрические ряды, Физматгиз (Москва, 1961). [2] G. BENNETT, V. GOODMAN, С. М. NEWMAN, Norms of random matrices, Pacific J. Math., 59 (1975), [3] S. KACZMARZ und H. STEINHAUS, Theorie der Orthogonalreihen (Warszawa Lwow, 1935) С. Качмаж и Г. Штейнгауз, Теория ортогональных рядов, Физматгиз (Москва, 1958). [4] Б. С. Кашин, О некоторых свойствах ортогональных систем сходимости, Труды матем. инта им. В. А. Стеклова АН СССР, 143 (1977), [5] L. FEJES TOTH, Lagerungen in der Ebene, aufder Kugel und im Raum, Springer (Berlin Gottingen Heidelberg, 1953) Л. Фейеш Т от, Расположения на плоскости, на сфере и в пространстве, Физматгиз (Москва, 1958). [6] A. WEIL, LHntegration dans les groupes topologiques et ses applications (Paris, 1940) А. Вей ль, Интегрирование в топологических группах и его применения, Иностранная литература (Москва, 1950).
9 Convergence of orthogonal series 35 О среднем значении некоторых функций, связанных со сходимостью ортогональных рядов Б. С. КАШИН В статье рассматриваются множества Q n, 1 ^ п < «*>, ортонормйрованных систем Ф = = i<pi ( *)}"= 1» состоящих из фушспжйэ постоянных на интервалах I» I» 1 ^j^n. На 6 й естественно переносится с группы ортогональных матриц порядка п мера Хаара. Изучается поведение # на Q n функции 8(Ф)= sup (/ sup (iyiviidfdxf*. Доказывается, что при / > 0 и п = 1, 2,... /*{Ф Й П : л(ф) ^ /} ^ (Се у **) я. Б. С. КАШИН СССР, МОСКВА УЛ. ВАВИЛОВА 42 МАТЕМАТИЧЕСКИЙ ИНСТИТУТ ИМ. В. А. СТЕКЛОВА АН СССР 3*
OPTIMAL SELECTION BASED ON RELATIVE RANK* (the "Secretary Problem")
OPTIMAL SELECTION BASED ON RELATIVE RANK* (the "Secretary Problem") BY Y. S. CHOW, S. MORIGUTI, H. ROBBINS AND S. M. SAMUELS ABSTRACT n rankable persons appear sequentially in random order. At the ith
More informationInner Product Spaces and Orthogonality
Inner Product Spaces and Orthogonality week 34 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,
More informationMATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets.
MATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets. Norm The notion of norm generalizes the notion of length of a vector in R n. Definition. Let V be a vector space. A function α
More informationT ( a i x i ) = a i T (x i ).
Chapter 2 Defn 1. (p. 65) Let V and W be vector spaces (over F ). We call a function T : V W a linear transformation form V to W if, for all x, y V and c F, we have (a) T (x + y) = T (x) + T (y) and (b)
More informationAn example of a computable
An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the concept
More informationTwo classes of ternary codes and their weight distributions
Two classes of ternary codes and their weight distributions Cunsheng Ding, Torleiv Kløve, and Francesco Sica Abstract In this paper we describe two classes of ternary codes, determine their minimum weight
More informationTail inequalities for order statistics of logconcave vectors and applications
Tail inequalities for order statistics of logconcave vectors and applications Rafał Latała Based in part on a joint work with R.Adamczak, A.E.Litvak, A.Pajor and N.TomczakJaegermann Banff, May 2011 Basic
More informationNOTES ON LINEAR TRANSFORMATIONS
NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all
More informationBANACH AND HILBERT SPACE REVIEW
BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but
More informationα = u v. In other words, Orthogonal Projection
Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v
More informationERDOS PROBLEMS ON IRREGULARITIES OF LINE SIZES AND POINT DEGREES A. GYARFAS*
BOLYAI SOCIETY MATHEMATICAL STUDIES, 11 Paul Erdos and his Mathematics. II, Budapest, 2002, pp. 367373. ERDOS PROBLEMS ON IRREGULARITIES OF LINE SIZES AND POINT DEGREES A. GYARFAS* Problems and results
More informationInner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
More information1 Norms and Vector Spaces
008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)
More informationLet H and J be as in the above lemma. The result of the lemma shows that the integral
Let and be as in the above lemma. The result of the lemma shows that the integral ( f(x, y)dy) dx is well defined; we denote it by f(x, y)dydx. By symmetry, also the integral ( f(x, y)dx) dy is well defined;
More informationIRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction
IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible
More informationINDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS
INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem
More informationThe Ideal Class Group
Chapter 5 The Ideal Class Group We will use Minkowski theory, which belongs to the general area of geometry of numbers, to gain insight into the ideal class group of a number field. We have already mentioned
More information1 Sets and Set Notation.
LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most
More informationMath 312 Homework 1 Solutions
Math 31 Homework 1 Solutions Last modified: July 15, 01 This homework is due on Thursday, July 1th, 01 at 1:10pm Please turn it in during class, or in my mailbox in the main math office (next to 4W1) Please
More information1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationNotes on Symmetric Matrices
CPSC 536N: Randomized Algorithms 201112 Term 2 Notes on Symmetric Matrices Prof. Nick Harvey University of British Columbia 1 Symmetric Matrices We review some basic results concerning symmetric matrices.
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationTAKEAWAY GAMES. ALLEN J. SCHWENK California Institute of Technology, Pasadena, California INTRODUCTION
TAKEAWAY GAMES ALLEN J. SCHWENK California Institute of Technology, Pasadena, California L INTRODUCTION Several games of Tf takeaway?f have become popular. The purpose of this paper is to determine the
More informationA new continuous dependence result for impulsive retarded functional differential equations
CADERNOS DE MATEMÁTICA 11, 37 47 May (2010) ARTIGO NÚMERO SMA#324 A new continuous dependence result for impulsive retarded functional differential equations M. Federson * Instituto de Ciências Matemáticas
More informationModern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh
Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh Peter Richtárik Week 3 Randomized Coordinate Descent With Arbitrary Sampling January 27, 2016 1 / 30 The Problem
More informationOrder statistics and concentration of l r norms for logconcave vectors
Order statistics and concentration of l r norms for logconcave vectors Rafa l Lata la Abstract We establish upper bounds for tails of order statistics of isotropic logconcave vectors and apply them to
More informationSimilarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
More informationPROPERTIES OF SOME NEW SEMINORMED SEQUENCE SPACES DEFINED BY A MODULUS FUNCTION
STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume L, Number 3, September 2005 PROPERTIES OF SOME NEW SEMINORMED SEQUENCE SPACES DEFINED BY A MODULUS FUNCTION YAVUZ ALTIN AYŞEGÜL GÖKHAN HIFSI ALTINOK Abstract.
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number
More informationCHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.
CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,
More information3. INNER PRODUCT SPACES
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.
More informationLINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,
More information2.3 Convex Constrained Optimization Problems
42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions
More informationGENERATING SETS KEITH CONRAD
GENERATING SETS KEITH CONRAD 1 Introduction In R n, every vector can be written as a unique linear combination of the standard basis e 1,, e n A notion weaker than a basis is a spanning set: a set of vectors
More informationINTEGRAL OPERATORS ON THE PRODUCT OF C(K) SPACES
INTEGRAL OPERATORS ON THE PRODUCT OF C(K) SPACES FERNANDO BOMBAL AND IGNACIO VILLANUEVA Abstract. We study and characterize the integral multilinear operators on a product of C(K) spaces in terms of the
More informationMetric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some
More informationI. GROUPS: BASIC DEFINITIONS AND EXAMPLES
I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called
More informationMathematical Methods of Engineering Analysis
Mathematical Methods of Engineering Analysis Erhan Çinlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets................................... 1 Subsets.............................
More informationLecture 11. Shuanglin Shao. October 2nd and 7th, 2013
Lecture 11 Shuanglin Shao October 2nd and 7th, 2013 Matrix determinants: addition. Determinants: multiplication. Adjoint of a matrix. Cramer s rule to solve a linear system. Recall that from the previous
More informationAbout the inverse football pool problem for 9 games 1
Seventh International Workshop on Optimal Codes and Related Topics September 61, 013, Albena, Bulgaria pp. 15133 About the inverse football pool problem for 9 games 1 Emil Kolev Tsonka Baicheva Institute
More information1. Let X and Y be normed spaces and let T B(X, Y ).
Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: NVP, Frist. 20050314 Skrivtid: 9 11.30 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok
More informationMATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix.
MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted
More informationON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE. 1. Introduction
ON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE J.M. CALABUIG, J. RODRÍGUEZ, AND E.A. SÁNCHEZPÉREZ Abstract. Let m be a vector measure taking values in a Banach space X. We prove that
More informationMath 315: Linear Algebra Solutions to Midterm Exam I
Math 35: Linear Algebra s to Midterm Exam I # Consider the following two systems of linear equations (I) ax + by = k cx + dy = l (II) ax + by = 0 cx + dy = 0 (a) Prove: If x = x, y = y and x = x 2, y =
More informationVector Spaces II: Finite Dimensional Linear Algebra 1
John Nachbar September 2, 2014 Vector Spaces II: Finite Dimensional Linear Algebra 1 1 Definitions and Basic Theorems. For basic properties and notation for R N, see the notes Vector Spaces I. Definition
More informationLinear Algebra Notes for Marsden and Tromba Vector Calculus
Linear Algebra Notes for Marsden and Tromba Vector Calculus ndimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of
More informationAN ALGORITHM FOR DETERMINING WHETHER A GIVEN BINARY MATROID IS GRAPHIC
AN ALGORITHM FOR DETERMINING WHETHER A GIVEN BINARY MATROID IS GRAPHIC W. T. TUTTE. Introduction. In a recent series of papers [l4] on graphs and matroids I used definitions equivalent to the following.
More informationMA651 Topology. Lecture 6. Separation Axioms.
MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples
More informationNo: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics
No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results
More information{f 1 (U), U F} is an open cover of A. Since A is compact there is a finite subcover of A, {f 1 (U 1 ),...,f 1 (U n )}, {U 1,...
44 CHAPTER 4. CONTINUOUS FUNCTIONS In Calculus we often use arithmetic operations to generate new continuous functions from old ones. In a general metric space we don t have arithmetic, but much of it
More informationFactorization Theorems
Chapter 7 Factorization Theorems This chapter highlights a few of the many factorization theorems for matrices While some factorization results are relatively direct, others are iterative While some factorization
More informationWe shall turn our attention to solving linear systems of equations. Ax = b
59 Linear Algebra We shall turn our attention to solving linear systems of equations Ax = b where A R m n, x R n, and b R m. We already saw examples of methods that required the solution of a linear system
More informationCONTRIBUTIONS TO ZERO SUM PROBLEMS
CONTRIBUTIONS TO ZERO SUM PROBLEMS S. D. ADHIKARI, Y. G. CHEN, J. B. FRIEDLANDER, S. V. KONYAGIN AND F. PAPPALARDI Abstract. A prototype of zero sum theorems, the well known theorem of Erdős, Ginzburg
More informationDiagonal, Symmetric and Triangular Matrices
Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by
More informationMatrix Norms. Tom Lyche. September 28, Centre of Mathematics for Applications, Department of Informatics, University of Oslo
Matrix Norms Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo September 28, 2009 Matrix Norms We consider matrix norms on (C m,n, C). All results holds for
More informationThe Dirichlet Unit Theorem
Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if
More informationISOMETRIES OF R n KEITH CONRAD
ISOMETRIES OF R n KEITH CONRAD 1. Introduction An isometry of R n is a function h: R n R n that preserves the distance between vectors: h(v) h(w) = v w for all v and w in R n, where (x 1,..., x n ) = x
More informationChapter 7. Continuity
Chapter 7 Continuity There are many processes and eects that depends on certain set of variables in such a way that a small change in these variables acts as small change in the process. Changes of this
More informationChapter 7. BANDIT PROBLEMS.
Chapter 7. BANDIT PROBLEMS. Bandit problems are problems in the area of sequential selection of experiments, and they are related to stopping rule problems through the theorem of Gittins and Jones (974).
More informationInner product. Definition of inner product
Math 20F Linear Algebra Lecture 25 1 Inner product Review: Definition of inner product. Slide 1 Norm and distance. Orthogonal vectors. Orthogonal complement. Orthogonal basis. Definition of inner product
More informationCARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE
CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE With the notion of bijection at hand, it is easy to formalize the idea that two finite sets have the same number of elements: we just need to verify
More informationMathematics for Econometrics, Fourth Edition
Mathematics for Econometrics, Fourth Edition Phoebus J. Dhrymes 1 July 2012 1 c Phoebus J. Dhrymes, 2012. Preliminary material; not to be cited or disseminated without the author s permission. 2 Contents
More informationLinear Maps. Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007)
MAT067 University of California, Davis Winter 2007 Linear Maps Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007) As we have discussed in the lecture on What is Linear Algebra? one of
More informationOn the greatest and least prime factors of n! + 1, II
Publ. Math. Debrecen Manuscript May 7, 004 On the greatest and least prime factors of n! + 1, II By C.L. Stewart In memory of Béla Brindza Abstract. Let ε be a positive real number. We prove that 145 1
More informationSection 1.4. Lines, Planes, and Hyperplanes. The Calculus of Functions of Several Variables
The Calculus of Functions of Several Variables Section 1.4 Lines, Planes, Hyperplanes In this section we will add to our basic geometric understing of R n by studying lines planes. If we do this carefully,
More informationAdaptive Online Gradient Descent
Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650
More informationSection 5.3. Section 5.3. u m ] l jj. = l jj u j + + l mj u m. v j = [ u 1 u j. l mj
Section 5. l j v j = [ u u j u m ] l jj = l jj u j + + l mj u m. l mj Section 5. 5.. Not orthogonal, the column vectors fail to be perpendicular to each other. 5..2 his matrix is orthogonal. Check that
More informationPermanents, Order Statistics, Outliers, and Robustness
Permanents, Order Statistics, Outliers, and Robustness N. BALAKRISHNAN Department of Mathematics and Statistics McMaster University Hamilton, Ontario, Canada L8S 4K bala@mcmaster.ca Received: November
More informationOn the generation of elliptic curves with 16 rational torsion points by Pythagorean triples
On the generation of elliptic curves with 16 rational torsion points by Pythagorean triples Brian Hilley Boston College MT695 Honors Seminar March 3, 2006 1 Introduction 1.1 Mazur s Theorem Let C be a
More informationMethods for Finding Bases
Methods for Finding Bases Bases for the subspaces of a matrix Rowreduction methods can be used to find bases. Let us now look at an example illustrating how to obtain bases for the row space, null space,
More informationTensor product of vector spaces
Tensor product of vector spaces Construction Let V,W be vector spaces over K = R or C. Let F denote the vector space freely generated by the set V W and let N F denote the subspace spanned by the elements
More informationSection 4.4 Inner Product Spaces
Section 4.4 Inner Product Spaces In our discussion of vector spaces the specific nature of F as a field, other than the fact that it is a field, has played virtually no role. In this section we no longer
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationRecall that two vectors in are perpendicular or orthogonal provided that their dot
Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal
More informationLINEAR DIFFERENTIAL EQUATIONS IN DISTRIBUTIONS
LINEAR DIFFERENTIAL EQUATIONS IN DISTRIBUTIONS LESLIE D. GATES, JR.1 1. Introduction. The main result of this paper is an extension of the analysis of Laurent Schwartz [l ] concerning the primitive of
More informationx1 x 2 x 3 y 1 y 2 y 3 x 1 y 2 x 2 y 1 0.
Cross product 1 Chapter 7 Cross product We are getting ready to study integration in several variables. Until now we have been doing only differential calculus. One outcome of this study will be our ability
More informationFacts About Eigenvalues
Facts About Eigenvalues By Dr David Butler Definitions Suppose A is an n n matrix An eigenvalue of A is a number λ such that Av = λv for some nonzero vector v An eigenvector of A is a nonzero vector v
More informationRi and. i=1. S i N. and. R R i
The subset R of R n is a closed rectangle if there are n nonempty closed intervals {[a 1, b 1 ], [a 2, b 2 ],..., [a n, b n ]} so that R = [a 1, b 1 ] [a 2, b 2 ] [a n, b n ]. The subset R of R n is an
More informationMATH 4330/5330, Fourier Analysis Section 11, The Discrete Fourier Transform
MATH 433/533, Fourier Analysis Section 11, The Discrete Fourier Transform Now, instead of considering functions defined on a continuous domain, like the interval [, 1) or the whole real line R, we wish
More informationFinite dimensional C algebras
Finite dimensional C algebras S. Sundar September 14, 2012 Throughout H, K stand for finite dimensional Hilbert spaces. 1 Spectral theorem for selfadjoint opertors Let A B(H) and let {ξ 1, ξ 2,, ξ n
More informationBasic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011
Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely
More informationF. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein)
Journal of Algerian Mathematical Society Vol. 1, pp. 1 6 1 CONCERNING THE l p CONJECTURE FOR DISCRETE SEMIGROUPS F. ABTAHI and M. ZARRIN (Communicated by J. Goldstein) Abstract. For 2 < p
More informationn k=1 k=0 1/k! = e. Example 6.4. The series 1/k 2 converges in R. Indeed, if s n = n then k=1 1/k, then s 2n s n = 1 n + 1 +...
6 Series We call a normed space (X, ) a Banach space provided that every Cauchy sequence (x n ) in X converges. For example, R with the norm = is an example of Banach space. Now let (x n ) be a sequence
More information4.6 Null Space, Column Space, Row Space
NULL SPACE, COLUMN SPACE, ROW SPACE Null Space, Column Space, Row Space In applications of linear algebra, subspaces of R n typically arise in one of two situations: ) as the set of solutions of a linear
More informationMath 4310 Handout  Quotient Vector Spaces
Math 4310 Handout  Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable
More informationCOMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS
COMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS IGOR V. EROVENKO AND B. SURY ABSTRACT. We compute commutativity degrees of wreath products A B of finite abelian groups A and B. When B
More informationStudia Scientiarum Mathematicarum Hungarica 41 (2), 243 266 (2004)
Studia Scientiarum Mathematicarum Hungarica 4 (), 43 66 (004) PLANAR POINT SETS WITH A SMALL NUMBER OF EMPTY CONVEX POLYGONS I. BÁRÁNY and P. VALTR Communicated by G. Fejes Tóth Abstract A subset A of
More informationLectures 56: Taylor Series
Math 1d Instructor: Padraic Bartlett Lectures 5: Taylor Series Weeks 5 Caltech 213 1 Taylor Polynomials and Series As we saw in week 4, power series are remarkably nice objects to work with. In particular,
More informationSeries Convergence Tests Math 122 Calculus III D Joyce, Fall 2012
Some series converge, some diverge. Series Convergence Tests Math 22 Calculus III D Joyce, Fall 202 Geometric series. We ve already looked at these. We know when a geometric series converges and what it
More informationChapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.
Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize
More informationand s n (x) f(x) for all x and s.t. s n is measurable if f is. REAL ANALYSIS Measures. A (positive) measure on a measurable space
RAL ANALYSIS A survey of MA 641643, UAB 19992000 M. Griesemer Throughout these notes m denotes Lebesgue measure. 1. Abstract Integration σalgebras. A σalgebra in X is a nonempty collection of subsets
More informationCONTROLLABILITY. Chapter 2. 2.1 Reachable Set and Controllability. Suppose we have a linear system described by the state equation
Chapter 2 CONTROLLABILITY 2 Reachable Set and Controllability Suppose we have a linear system described by the state equation ẋ Ax + Bu (2) x() x Consider the following problem For a given vector x in
More information18.06 Problem Set 4 Solution Due Wednesday, 11 March 2009 at 4 pm in 2106. Total: 175 points.
806 Problem Set 4 Solution Due Wednesday, March 2009 at 4 pm in 206 Total: 75 points Problem : A is an m n matrix of rank r Suppose there are righthandsides b for which A x = b has no solution (a) What
More informationIdeal Class Group and Units
Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals
More informationLinear Codes. In the V[n,q] setting, the terms word and vector are interchangeable.
Linear Codes Linear Codes In the V[n,q] setting, an important class of codes are the linear codes, these codes are the ones whose code words form a subvector space of V[n,q]. If the subspace of V[n,q]
More informationPolynomial Degree and Lower Bounds in Quantum Complexity: Collision and Element Distinctness with Small Range
THEORY OF COMPUTING, Volume 1 (2005), pp. 37 46 http://theoryofcomputing.org Polynomial Degree and Lower Bounds in Quantum Complexity: Collision and Element Distinctness with Small Range Andris Ambainis
More informationDuality of linear conic problems
Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least
More informationMATH 240 Fall, Chapter 1: Linear Equations and Matrices
MATH 240 Fall, 2007 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 9th Ed. written by Prof. J. Beachy Sections 1.1 1.5, 2.1 2.3, 4.2 4.9, 3.1 3.5, 5.3 5.5, 6.1 6.3, 6.5, 7.1 7.3 DEFINITIONS
More informationHOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!
Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following
More information