Sequences and Series of Functions
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Exercise Set 13 Top04-E013 ; November 26, 2004 ; 9:45 A.M
Prof. D. P.Patil, Department of Mathematics, Indian Institute of Science, Bangalore August-December 2004 MA-231 Topology 13. Function spaces1) —————————— — Uniform Convergence, Stone-Weierstrass theorem, Arzela-Ascoli theorem ———————————————————————————————– November 22, 2004 Karl Theodor Wilhelm Weierstrass† Marshall Harvey Stone†† (1815-1897) (1903-1989) First recall the following definitions and results : Our overall aim in this section is the study of the compactness and completeness properties of a subset F of the set Y X of all maps from a space X into a space Y . To do this a usable topology must be introduced on F (presumably related to the structures of X and Y ) and when this is has been done F is called a f unction space. N13.1 . (The Topology of Pointwise Convergence 2) Let X be any set, Y be any topological space and let fn : X → Y , n ∈ N, be a sequence of maps. We say that the sequence (fn)n∈N is pointwise convergent,ifforeverypoint x ∈X, the sequence fn(x) , n∈N,inY is convergent. a). The (Tychonoff) product topology on Y X is determined solely by the topology of Y (even if X is a topological X X space) the structure on X plays no part. A sequence fn : X →Y , n∈N,inY converges to a function f in Y if and only if for every point x ∈X, the sequence fn(x) , n∈N,inY is convergent. This provides the reason for the name the topology of pointwise convergence;this topology is also simply called the pointwise topology andisdenoted by Tptc . -
Ch. 15 Power Series, Taylor Series
Ch. 15 Power Series, Taylor Series 서울대학교 조선해양공학과 서유택 2017.12 ※ 본 강의 자료는 이규열, 장범선, 노명일 교수님께서 만드신 자료를 바탕으로 일부 편집한 것입니다. Seoul National 1 Univ. 15.1 Sequences (수열), Series (급수), Convergence Tests (수렴판정) Sequences: Obtained by assigning to each positive integer n a number zn z . Term: zn z1, z 2, or z 1, z 2 , or briefly zn N . Real sequence (실수열): Sequence whose terms are real Convergence . Convergent sequence (수렴수열): Sequence that has a limit c limznn c or simply z c n . For every ε > 0, we can find N such that Convergent complex sequence |zn c | for all n N → all terms zn with n > N lie in the open disk of radius ε and center c. Divergent sequence (발산수열): Sequence that does not converge. Seoul National 2 Univ. 15.1 Sequences, Series, Convergence Tests Convergence . Convergent sequence: Sequence that has a limit c Ex. 1 Convergent and Divergent Sequences iin 11 Sequence i , , , , is convergent with limit 0. n 2 3 4 limznn c or simply z c n Sequence i n i , 1, i, 1, is divergent. n Sequence {zn} with zn = (1 + i ) is divergent. Seoul National 3 Univ. 15.1 Sequences, Series, Convergence Tests Theorem 1 Sequences of the Real and the Imaginary Parts . A sequence z1, z2, z3, … of complex numbers zn = xn + iyn converges to c = a + ib . if and only if the sequence of the real parts x1, x2, … converges to a . and the sequence of the imaginary parts y1, y2, … converges to b. Ex. -
Topic 7 Notes 7 Taylor and Laurent Series
Topic 7 Notes Jeremy Orloff 7 Taylor and Laurent series 7.1 Introduction We originally defined an analytic function as one where the derivative, defined as a limit of ratios, existed. We went on to prove Cauchy's theorem and Cauchy's integral formula. These revealed some deep properties of analytic functions, e.g. the existence of derivatives of all orders. Our goal in this topic is to express analytic functions as infinite power series. This will lead us to Taylor series. When a complex function has an isolated singularity at a point we will replace Taylor series by Laurent series. Not surprisingly we will derive these series from Cauchy's integral formula. Although we come to power series representations after exploring other properties of analytic functions, they will be one of our main tools in understanding and computing with analytic functions. 7.2 Geometric series Having a detailed understanding of geometric series will enable us to use Cauchy's integral formula to understand power series representations of analytic functions. We start with the definition: Definition. A finite geometric series has one of the following (all equivalent) forms. 2 3 n Sn = a(1 + r + r + r + ::: + r ) = a + ar + ar2 + ar3 + ::: + arn n X = arj j=0 n X = a rj j=0 The number r is called the ratio of the geometric series because it is the ratio of consecutive terms of the series. Theorem. The sum of a finite geometric series is given by a(1 − rn+1) S = a(1 + r + r2 + r3 + ::: + rn) = : (1) n 1 − r Proof. -
11.3-11.4 Integral and Comparison Tests
11.3-11.4 Integral and Comparison Tests The Integral Test: Suppose a function f(x) is continuous, positive, and decreasing on [1; 1). Let an 1 P R 1 be defined by an = f(n). Then, the series an and the improper integral 1 f(x) dx either BOTH n=1 CONVERGE OR BOTH DIVERGE. Notes: • For the integral test, when we say that f must be decreasing, it is actually enough that f is EVENTUALLY ALWAYS DECREASING. In other words, as long as f is always decreasing after a certain point, the \decreasing" requirement is satisfied. • If the improper integral converges to a value A, this does NOT mean the sum of the series is A. Why? The integral of a function will give us all the area under a continuous curve, while the series is a sum of distinct, separate terms. • The index and interval do not always need to start with 1. Examples: Determine whether the following series converge or diverge. 1 n2 • X n2 + 9 n=1 1 2 • X n2 + 9 n=3 1 1 n • X n2 + 1 n=1 1 ln n • X n n=2 Z 1 1 p-series: We saw in Section 8.9 that the integral p dx converges if p > 1 and diverges if p ≤ 1. So, by 1 x 1 1 the Integral Test, the p-series X converges if p > 1 and diverges if p ≤ 1. np n=1 Notes: 1 1 • When p = 1, the series X is called the harmonic series. n n=1 • Any constant multiple of a convergent p-series is also convergent. -
Formal Power Series - Wikipedia, the Free Encyclopedia
Formal power series - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Formal_power_series Formal power series From Wikipedia, the free encyclopedia In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates. This perspective contrasts with that of power series, whose variables designate numerical values, and which series therefore only have a definite value if convergence can be established. Formal power series are often used merely to represent the whole collection of their coefficients. In combinatorics, they provide representations of numerical sequences and of multisets, and for instance allow giving concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved; this is known as the method of generating functions. Contents 1 Introduction 2 The ring of formal power series 2.1 Definition of the formal power series ring 2.1.1 Ring structure 2.1.2 Topological structure 2.1.3 Alternative topologies 2.2 Universal property 3 Operations on formal power series 3.1 Multiplying series 3.2 Power series raised to powers 3.3 Inverting series 3.4 Dividing series 3.5 Extracting coefficients 3.6 Composition of series 3.6.1 Example 3.7 Composition inverse 3.8 Formal differentiation of series 4 Properties 4.1 Algebraic properties of the formal power series ring 4.2 Topological properties of the formal power series -
7. Properties of Uniformly Convergent Sequences
46 1. THE THEORY OF CONVERGENCE 7. Properties of uniformly convergent sequences Here a relation between continuity, differentiability, and Riemann integrability of the sum of a functional series or the limit of a functional sequence and uniform convergence is studied. 7.1. Uniform convergence and continuity. Theorem 7.1. (Continuity of the sum of a series) N The sum of the series un(x) of terms continuous on D R is continuous if the series converges uniformly on D. ⊂ P Let Sn(x)= u1(x)+u2(x)+ +un(x) be a sequence of partial sum. It converges to some function S···(x) because every uniformly convergent series converges pointwise. Continuity of S at a point x means (by definition) lim S(y)= S(x) y→x Fix a number ε> 0. Then one can find a number δ such that S(x) S(y) <ε whenever 0 < x y <δ | − | | − | In other words, the values S(y) can get arbitrary close to S(x) and stay arbitrary close to it for all points y = x that are sufficiently close to x. Let us show that this condition follows6 from the hypotheses. Owing to the uniform convergence of the series, given ε > 0, one can find an integer m such that ε S(x) Sn(x) sup S Sn , x D, n m | − |≤ D | − |≤ 3 ∀ ∈ ∀ ≥ Note that m is independent of x. By continuity of Sn (as a finite sum of continuous functions), for n m, one can also find a number δ > 0 such that ≥ ε Sn(x) Sn(y) < whenever 0 < x y <δ | − | 3 | − | So, given ε> 0, the integer m is found. -
1.1 Constructing the Real Numbers
18.095 Lecture Series in Mathematics IAP 2015 Lecture #1 01/05/2015 What is number theory? The study of numbers of course! But what is a number? • N = f0; 1; 2; 3;:::g can be defined in several ways: { Finite ordinals (0 := fg, n + 1 := n [ fng). { Finite cardinals (isomorphism classes of finite sets). { Strings over a unary alphabet (\", \1", \11", \111", . ). N is a commutative semiring: addition and multiplication satisfy the usual commu- tative/associative/distributive properties with identities 0 and 1 (and 0 annihilates). Totally ordered (as ordinals/cardinals), making it a (positive) ordered semiring. • Z = {±n : n 2 Ng.A commutative ring (commutative semiring, additive inverses). Contains Z>0 = N − f0g closed under +; × with Z = −Z>0 t f0g t Z>0. This makes Z an ordered ring (in fact, an ordered domain). • Q = fa=b : a; 2 Z; b 6= 0g= ∼, where a=b ∼ c=d if ad = bc. A field (commutative ring, multiplicative inverses, 0 6= 1) containing Z = fn=1g. Contains Q>0 = fa=b : a; b 2 Z>0g closed under +; × with Q = −Q>0 t f0g t Q>0. This makes Q an ordered field. • R is the completion of Q, making it a complete ordered field. Each of the algebraic structures N; Z; Q; R is canonical in the following sense: every non-trivial algebraic structure of the same type (ordered semiring, ordered ring, ordered field, complete ordered field) contains a copy of N; Z; Q; R inside it. 1.1 Constructing the real numbers What do we mean by the completion of Q? There are two possibilities: 1. -
Uniform Convergence and Differentiation Theorem 6.3.1
Math 341 Lecture #29 x6.3: Uniform Convergence and Differentiation We have seen that a pointwise converging sequence of continuous functions need not have a continuous limit function; we needed uniform convergence to get continuity of the limit function. What can we say about the differentiability of the limit function of a pointwise converging sequence of differentiable functions? 1+1=(2n−1) The sequence of differentiable hn(x) = x , x 2 [−1; 1], converges pointwise to the nondifferentiable h(x) = x; we will need to assume more about the pointwise converging sequence of differentiable functions to ensure that the limit function is differentiable. Theorem 6.3.1 (Differentiable Limit Theorem). Let fn ! f pointwise on the 0 closed interval [a; b], and assume that each fn is differentiable. If (fn) converges uniformly on [a; b] to a function g, then f is differentiable and f 0 = g. Proof. Let > 0 and fix c 2 [a; b]. Our goal is to show that f 0(c) exists and equals g(c). To this end, we will show the existence of δ > 0 such that for all 0 < jx − cj < δ, with x 2 [a; b], we have f(x) − f(c) − g(c) < x − c which implies that f(x) − f(c) f 0(c) = lim x!c x − c exists and is equal to g(c). The way forward is to replace (f(x) − f(c))=(x − c) − g(c) with expressions we can hopefully control: f(x) − f(c) f(x) − f(c) fn(x) − fn(c) fn(x) − fn(c) − g(c) = − + x − c x − c x − c x − c 0 0 − f (c) + f (c) − g(c) n n f(x) − f(c) fn(x) − fn(c) ≤ − x − c x − c fn(x) − fn(c) 0 0 + − f (c) + jf (c) − g(c)j: x − c n n The second and third expressions we can control respectively by the differentiability of 0 fn and the uniformly convergence of fn to g. -
Be a Metric Space
2 The University of Sydney show that Z is closed in R. The complement of Z in R is the union of all the Pure Mathematics 3901 open intervals (n, n + 1), where n runs through all of Z, and this is open since every union of open sets is open. So Z is closed. Metric Spaces 2000 Alternatively, let (an) be a Cauchy sequence in Z. Choose an integer N such that d(xn, xm) < 1 for all n ≥ N. Put x = xN . Then for all n ≥ N we have Tutorial 5 |xn − x| = d(xn, xN ) < 1. But xn, x ∈ Z, and since two distinct integers always differ by at least 1 it follows that xn = x. This holds for all n > N. 1. Let X = (X, d) be a metric space. Let (xn) and (yn) be two sequences in X So xn → x as n → ∞ (since for all ε > 0 we have 0 = d(xn, x) < ε for all such that (yn) is a Cauchy sequence and d(xn, yn) → 0 as n → ∞. Prove that n > N). (i)(xn) is a Cauchy sequence in X, and 4. (i) Show that if D is a metric on the set X and f: Y → X is an injective (ii)(xn) converges to a limit x if and only if (yn) also converges to x. function then the formula d(a, b) = D(f(a), f(b)) defines a metric d on Y , and use this to show that d(m, n) = |m−1 − n−1| defines a metric Solution. -
G: Uniform Convergence of Fourier Series
G: Uniform Convergence of Fourier Series From previous work on the prototypical problem (and other problems) 8 < ut = Duxx 0 < x < l ; t > 0 u(0; t) = 0 = u(l; t) t > 0 (1) : u(x; 0) = f(x) 0 < x < l we developed a (formal) series solution 1 1 X X 2 2 2 nπx u(x; t) = u (x; t) = b e−n π Dt=l sin( ) ; (2) n n l n=1 n=1 2 R l nπy with bn = l 0 f(y) sin( l )dy. These are the Fourier sine coefficients for the initial data function f(x) on [0; l]. We have no real way to check that the series representation (2) is a solution to (1) because we do not know we can interchange differentiation and infinite summation. We have only assumed that up to now. In actuality, (2) makes sense as a solution to (1) if the series is uniformly convergent on [0; l] (and its derivatives also converges uniformly1). So we first discuss conditions for an infinite series to be differentiated (and integrated) term-by-term. This can be done if the infinite series and its derivatives converge uniformly. We list some results here that will establish this, but you should consult Appendix B on calculus facts, and review definitions of convergence of a series of numbers, absolute convergence of such a series, and uniform convergence of sequences and series of functions. Proofs of the following results can be found in any reasonable real analysis or advanced calculus textbook. 0.1 Differentiation and integration of infinite series Let I = [a; b] be any real interval. -
Pointwise Convergence Ofofof Complex Fourier Series
Pointwise convergence ofofof complex Fourier series Let f(x) be a periodic function with period 2 l defined on the interval [- l,l]. The complex Fourier coefficients of f( x) are l -i n π s/l cn = 1 ∫ f(s) e d s 2 l -l This leads to a Fourier series representation for f(x) ∞ i n π x/l f(x) = ∑ cn e n = -∞ We have two important questions to pose here. 1. For a given x, does the infinite series converge? 2. If it converges, does it necessarily converge to f(x)? We can begin to address both of these issues by introducing the partial Fourier series N i n π x/l fN(x) = ∑ cn e n = -N In terms of this function, our two questions become 1. For a given x, does lim fN(x) exist? N→∞ 2. If it does, is lim fN(x) = f(x)? N→∞ The Dirichlet kernel To begin to address the questions we posed about fN(x) we will start by rewriting fN(x). Initially, fN(x) is defined by N i n π x/l fN(x) = ∑ cn e n = -N If we substitute the expression for the Fourier coefficients l -i n π s/l cn = 1 ∫ f(s) e d s 2 l -l into the expression for fN(x) we obtain N l 1 -i n π s/l i n π x/l fN(x) = ∑ ∫ f(s) e d s e n = -N(2 l -l ) l N = ∫ 1 ∑ (e-i n π s/l ei n π x/l) f(s) d s -l (2 l n = -N ) 1 l N = ∫ 1 ∑ ei n π (x-s)/l f(s) d s -l (2 l n = -N ) The expression in parentheses leads us to make the following definition. -
Uniform Convexity and Variational Convergence
Trudy Moskov. Matem. Obw. Trans. Moscow Math. Soc. Tom 75 (2014), vyp. 2 2014, Pages 205–231 S 0077-1554(2014)00232-6 Article electronically published on November 5, 2014 UNIFORM CONVEXITY AND VARIATIONAL CONVERGENCE V. V. ZHIKOV AND S. E. PASTUKHOVA Dedicated to the Centennial Anniversary of B. M. Levitan Abstract. Let Ω be a domain in Rd. We establish the uniform convexity of the Γ-limit of a sequence of Carath´eodory integrands f(x, ξ): Ω×Rd → R subjected to a two-sided power-law estimate of coercivity and growth with respect to ξ with expo- nents α and β,1<α≤ β<∞, and having a common modulus of convexity with respect to ξ. In particular, the Γ-limit of a sequence of power-law integrands of the form |ξ|p(x), where the variable exponent p:Ω→ [α, β] is a measurable function, is uniformly convex. We prove that one can assign a uniformly convex Orlicz space to the Γ-limit of a sequence of power-law integrands. A natural Γ-closed extension of the class of power-law integrands is found. Applications to the homogenization theory for functionals of the calculus of vari- ations and for monotone operators are given. 1. Introduction 1.1. The notion of uniformly convex Banach space was introduced by Clarkson in his famous paper [1]. Recall that a norm ·and the Banach space V equipped with this norm are said to be uniformly convex if for each ε ∈ (0, 2) there exists a δ = δ(ε)such that ξ + η ξ − η≤ε or ≤ 1 − δ 2 for all ξ,η ∈ V with ξ≤1andη≤1.