On the law of large numbers for free identically distributed random variables 1

Size: px
Start display at page:

Download "On the law of large numbers for free identically distributed random variables 1"

Transcription

1 General Mathematics Vol. 5, No. 4 (2007), On the law of large numbers for free identically distributed random variables Bogdan Gheorghe Munteanu Abstract A version of law of large numbers for free identically distributed random variables is considering at this work. It shown that lim t µ(x : x > t) = 0 t is a sufficient and necessary condition for the weak law of large numbers for the sequence X, X 2,..., free random variables Mathematical Subject Classification: 45L54, 60F05, 47C5 Key words: Law of large numbers, free random variables, free convolution Introduction Analytic theory of free additive convolution is useful in frame of this article. The calculation of free additive convolution is based on an analoque of the Fourier transform first introduced by Voiculescu [6]. We need the version of Received 4 July, 2007 Accepted for publication (in revised form) 4 December,

2 56 Bogdan Gheorghe Munteanu this apparatus which is suitable for the convolution of arbitrary probability measures []. First, some notation. Let C denote the complex field, C + and C the upper and lower half plane. We consider Γ α = {z = x + iy : y > 0 and x < αy}, Γ α,β = {z Γ α : y > β}, α,β > 0. where α and β are positive numbers. Given a probability measure µ on R, its Cauchy transform G µ : C + C is defined as G µ (z) := µ dx z x = E( (z X) ) The reciprocal Cauchy transform is defined by F µ : C + C +, F µ (z) = /G µ (z). 2 Proof of the main result First, we would like to formulate the theorem in terms of free convolutions rather than random variables. To do this, observe that given a self-adjoint random variable X affiliated with some algebras A and a scalar λ > 0, we have µ λ X = D λ µ X where D λ is the dilation of a measure µ defined by D λ µ(a) = µ(λ A), (A R measurable). Theorem 2.. Let µ be a probability measure R. The following conditions are equivalent:

3 On the law of large numbers (i) There exist real constants M,M 2,... R such that lim ν n = δ 0, n where ν n = D /n µ... D /n µ δ Mn and δ 0 Dirac distribution; }{{} n ori (ii) The measure µ satisfies lim t µ (x : x > t) = 0. Moreover, if (ii) is t +n satisfied, the constants M n in (i) can be chosen to be M n = tdµ(t). For the proof of this theorem, we establish some preparatory lemmas. The foolowing result is related by Bercovici and Voiculescu (993) in Proposition 4.5. to (993,[7]). Lemma 2.. ([7]) Let µ be a probability measure on R. Given a truncated cone Γ α,β. Then exists a truncated cone Γ α,β n such that F µ(γ α,β ) Γ α,β. Proof. Fix a number α (0, ) such that γ = tan α, and choose β > 0 so large that (2.) F µ (u) u sin γ u, for all Iu > β and β > β. The relation (2.) is possible beacause α ug µ (u) = z u dµ(t), thus Fµ(u) u, forall u Γ z t u α. R We note the disk D u = {w w u < sin γ u }. We observe that F µ (u) D u if u Γ α,β Γ α, while the implication D u Γ α,β if u Γ α,β is justify in figure, where u Γ α,β, while Iu > Iu In the sequel we will use the following notation. If y 0, we denote = [ y,y] and = (, y) (y, + ). Proposition 2.. ([8]) Let {µ n } n= be a sequence of probability measures on R. The following assertions are equivalent (a) The sequence {µ n } n= converges weakly to a probability measure µ;

4 58 Bogdan Gheorghe Munteanu Figure : Graphic justfy of implication D u Γ α,β, u Γ α (b) There exist α,β > 0 such that the sequence (ψ νn ) n=, ψ, ψ Γ α,β and ψ νn (u) = 0( u ) if u, u Γ α,β ; (c) There exist α,β > 0 such that the functions ψ µn are defined on Γ α,β for every n, lim ψ µn (iy) exists for every y > β n uniformly in n as y. and ψ µn (iy) = o(y) Lemma 2.2. Let µ be a probability measure on R satisfying condition (i) of Theorem 2.. Then lim (IF µ(iy) y) = 0. Proof. By Proposition 2.,(ii) weak convergence of ν 0 to δ 0 implies that there exist α 0, β 0 > 0 such that ψ νn (u) 0 for u Γ α0,β 0. However, ψ νn (u) = nψ D/n µ(u) M n = n n ψ µ(nu) M n = ψ µ (nu) M n, which implies (2.2) lim n Iψ νn (u) = lim n Iψ µ (nu) = 0,

5 On the law of large numbers for all u Γ α0,β 0. By Lemma 2., exists α 2,α 3,β 2,β 3 > 0 such that α 0 > α 2 and β 0 < β 2 such that F µ has an inverse on Γ α2,β 2 and Γ α0,β 0 Γ α2,β 2 F µ (Γ α3,β 3 ),(α 2 > α 3 and β 3 > β 2 ). Therefore for any for all u Γ α3,β 3 it follows that (2.3) F µ (u) = u ψ µ (F µ (u)). Indeed, in condition ψ µ (u) = F (u) u replace u F µ (u) and we obtain eager relation. In particular, defining α,β such that α 0 > α > α 2 and β 0 < β < β 2 then the relation (2.2) holds u Γ α,β ( because Γ α0,β 0 Γ α,β ). Since nγ α,β = Γ α0,β 0, it is immediate now that n= (2.4) lim u ; u Γ α,β Iψ µ (u) = 0. Since Fµ(u) u u, u Γ α, we conclude that F µ (iy) iy, and hence F µ (iy). Then lim Iψ µ (F µ (iy)) = 0 (in relation (2.4) replace u by F µ (iy)). If in (2.3) we replace u by iy, we obtain IF µ (iy) y = Iψ µ (F µ (iy)), which concludes the proof. Lemma 2.3. Let σ be a finite positive measure on R such that lim yσ( ) = 0. Then the following hold (i) lim y y (ii) lim + +y y (iii) lim y k +y y y 2 +t 2 dσ(t) = 0; dσ(t) = 0; k+ dσ(t) = 0,k > 0. Proof. Consider the function y y, t [ y, y] y f y (t) = 2 +t 2 0, otherwise.

6 60 Bogdan Gheorghe Munteanu Then lim f y (t) = 0, for all t R and for all y > 0 we have f y (t) = y y y 2 + t 2 y2 y 2 + y L (σ). This condition is the condition of the Lebesque convergence theorem (2000,[4], + Theoreme 3.8., page 76), which implies that lim f y (t) dσ(t) = 0. The function g(t) = The result now follows because (2.5) y y But y y y y 2 +t 2 is increasing for t [0, y] and y >. y 2 + t 2 dσ(t) = y y f y (t) dσ(t) + y y y y 2 + t 2 dσ(t). dσ(t) y 2 +t 2 2 yσ( y ) if to count on inequality y. y+ 2 In condition (2.5) through on limit when y, lim y y y 2 + t 2 dσ(t) = lim yσ( y ) = 0. To prove (ii) we integrate by parts and we count on σ ( ) = σ (R) σ( ) = σ ([ y,y]) = 2y, we then have σ( t ) = 2 σ(t). }{{} t We have y y dσ(t) = 2 dσ(t) = 2 tσ(t) y 0 σ(t) dt 0 0 = 2yσ(y) 2 y 0 σ(t) dt = [ σ( )]σ(y) + = yσ( ) + y 0 y 0 σ( t ) dt. σ( t ) dt y

7 On the law of large numbers... 6 Then dσ(t) = y σ() + y σ( t ) dt. 0 It is clear that y σ() = o() if y. Select ǫ > 0 and choose N > 0 large enough such that tσ( t ) < ǫ, t N. Then, as y, y σ( t ) dt = 0 N 0 Nσ(R) ǫ + o(). σ( t ) dt + y N + y N ǫ t dt We will use the notation M y = t dµ(t) if y Z. tσ( t ) t Lemma 2.4. Let µ be a probability measure on R such that lim yµ( ) = 0. Then F µ (iy) = iy M y + o() as y. Proof. We will prove the estimate G µ (iy) = iy My y 2 + y 2 o() as y. We can estimate separately the real and the imaginary parts of G µ (iy). For the real part we have thus G µ (z) = RG µ (iy) = = = M y y 2 + dµ(t) z t = G µ(iy) = t y 2 + t 2 dµ(t) = ty 2 t 3 y 2 (y 2 + t 2 ) dµ(t) + dµ(t) iy t = t y 2 + t dµ(t) + 2 t 3 y 2 (y 2 + t 2 ) dµ(t) + t 3 y 2 (y 2 + t 2 ) dµ(t) + dt iy + t y 2 t 2 dµ(t), t y 2 + t 2 dµ(t) t y 2 + t 2 dµ(t). t y 2 + t 2 dµ(t)

8 62 Bogdan Gheorghe Munteanu On the other side, t 3 y 2 (y 2 + t 2 ) dµ(t) + y 2 + t dµ(t) 2 y 4 t y 2 + t dµ(t) 2 3 dµ(t) + 2y }{{} y 2 y 2 2+ dµ(t) Iy 3 y 2 (y 2 + t 2 ) dµ(t)+ dµ(t) }{{} µ() Lemma 2.3 (iii) o() + yµ( y 2 2y 2 y ) = o(). y 2 Thus, RG µ (iy) My + o(). The imaginary part y 2 y 2 IG µ (iy) = y y 2 + t dµ(t) = 2 y = dµ(t) + y y }{{} µ(r)= t 2 y 2 ± t 2 y 2 + t 2 dµ(t) t 2 y 2 + t 2 dµ(t) = y + y y 2 + t dµ(t) + 2 y y + t 2 dµ(t) + y 3 Lemma 2.3 (iii) y + y 2 o(). 2y µ() t 2 y 2 + t 2 dµ(t) ). ( But = y 2 = iy iy = iy G µ(iy) iy+m y o() iy+m y o() + My o() iy We say the development in Mac-Laurent series of function x +x x +... Then the reciprocal Cauchy transform is F µ (iy) = ( iy My o() iy ) = iy M y + o(). G µ(iy) =

9 On the law of large numbers Lemma 2.5. Let µ be a probability measure on R such that lim yµ( ) = 0 and z Γ /4. Then as z in Γ /4. d dz F µ(z) = + z o(), Proof. By the Nevanlinna representation according to (963,[2]) of F µ (z), where η(y) = F(z) = a + z tz t z dσ(t), z C+ wich implies IF µ (iy) = y + η(y), y(+t 2 ) y 2 +t 2 dσ(t). By Lemma 2.4, F µ (iy) = iy M y +o() wich implies IF µ (iy) = y. Then, through identicaly results that η(y) = o() as y. Observe that, for y > 0, yt 2 + y, then, η(y) y 2 +t 2 2 yt 2 y 2 +t 2 dσ(t) 2 yσ(). Therefore yσ( ) = o() as y. Again by the Nevanlinna representation we get d dz F µ(z) = + For z = x + iy Γ /4 wich implies x < y 4 and + + t 2 (z t) dσ(t) t 2 z t dσ(t) + t 2 t 2 + y 2 y 2 dσ(t) 2 yt 2 y 2 +t 2 dσ(t) + +t 2 + t 2 (x t) 2 + y 2 dσ(t) since here use the inequality t 2 + y 2 y 2 t2 +y 2 2 for all t. (z t) 2 dσ(t). + t 2 t 2 + y 2 dσ(t),

10 64 Bogdan Gheorghe Munteanu Furthermore, notice that I y +t 2 dσ(t) +y σ( t 2 +y 2 y+y y) = 2 y σ( y) = y y σ( y) = yσ( y y ) = y y o(). Here we used the fact that the function t +t2 is increasing for t > 0 and y >. t 2 +y 2 + t 2 On the other side, dσ(t) σ( t 2 + y 2 y) = y o(). Since y y }{{} z < 7, for z Γ 4 /4, we get the desired estimate. Lemma 2.6. Let µ be a probability measure on R such that lim yµ( ) = 0. Then ψ µ (iy) = M y + o() as y. Proof. By Lemma 2.4, F µ (iy) = iy M y + h(y) with lim h(y) = 0, while by Lemma 2.3(ii), M y = o(lny) if y, follows that F µ (iy) Γ /4 for y large enough. Thus for y large enough, iy = F µ (F µ (iy)) = F µ (iy M y +h(y)). However, let z = iy M y +h(y), then F µ (z) = z+m y h(y) = z+o( z ) as z and z Γ /4. By Lemma 2.5 and Fµ (z) = z + o( z ), we have that d F dz µ (z) = + k(z) with k(z) = o() as z and z Γ /4. Therefore z Fµ (iy M y + h(y)) Fµ (iy) = (iy M y + h(y) iy) d dz F We get that ψ µ (iy) = F µ (iy) iy = [ M y + h(y)] ( ) + k(γ) γ µ (z) z=γ = ( M y + h(y))( + o()) as y. = Fµ (iy M y + h(y)) +M y h(y) + M y o() h(y) o() iy }{{} iy = M y + o() as y.

11 On the law of large numbers Proof of the Theorem 2. Proof. (i) (ii): assume that µ satisfies condition (i) of the theorem. The Nevanlinna representation of F µ (z) implies for y > 0 that, wich is equivalent with F µ (iy) = a + iy + F µ (iy) = a + iy + + iyz t iy dσ(t) t y 2 t + i(yt 2 + y) t 2 + y 2 dσ(t), which means IF µ (iy) = y + η(y), with η(y) = + y(+t 2 ) RF µ (iy) = a + ξ(y), t 2 +y 2 dσ(t) and ξ(y) = + t( y 2 ) t 2 +y 2 dσ(t). By Lemmas 2.2 and 2.4, F µ (iy) = iy M y + o(), IF µ (iy) = y, we have that η(y) = o() if y, and the same argument used in Lemma 2.5, yσ( ) = o() as y. This estimate along with Lemma 2.3(i) allows us to conclude that ξ(y) = o( y) as y. Indeed ξ(y) y y y y y y y y t( y 2 ) t 2 + y 2 dσ(t) = o(), t 2 + y2 dσ(t) = y y y 2 y 2 (t 2 + y 2 ) dσ(t) Here we used the inequality y2 y 2 < for y >. We can now estimate the imaginary part of G µ (iy). Is know the fact that I z = Iz z 2. Then

12 66 Bogdan Gheorghe Munteanu (2.6) IG µ (iy) = I F µ (iy) = IF µ(iy) F µ (iy) 2 = y + o() R 2 F µ (iy) + I 2 F µ (iy) y + o() = (a + o( y + o() = y)) 2 + (y + o()) 2 a 2 + y 2 y + o() < = y 2 y + y o(). 2 By Lemma 2.4, IG µ (iy) = y + y + t 2 y 2 +t 2 dµ(t) wich implies yt 2 dµ(t) = o(). y 2 + t2 By proof of Lemma 2.5 results that yt 2 y 2 + t 2 dµ(t) yt 2 y 2 + t 2 dµ(t) 2 yµ(), which lead at yµ( ) = o() as y that is (ii). (ii) (i): suppose that µ satisfies condition (ii) of the theorem. We have ψ νn (z) = ψ µ (nz) M n where the ν n are defined as in condition (i) of the theorem. Notice that the functions ψ νn are defined on a certain truncated cone Γ α,β for every n. By Lemma 2.6 for every fixed y > β, ψ νn (iy) = ψ µ (iny) M n = M ny M n + o(), if n and y. Assuming without loss of generality that β >, the argument used in

13 On the law of large numbers Lemma 2.3, gives the following estimate M ny M n = t dµ(t) t dµ(t) I ny Therefore, lim ψ νn (iy) = 0. n Moreover I n I ny I n dµ(t) = nyµ( ny) nµ( n ) + nyµ( ny ) nµ( n ) + sup tµ( t ) t [n,ny] ny n ny n t dt µ( t ) dt nyµ( ny ) nµ( n ) + sup tµ( t ) = o(). t [n,ny] lim sup ψ νn (iy) y 2k + k lim sup y 2k lim sup + k lim sup y y = 0. Hence Proposition 2.(c), ν n converges weakly to a measure ν and ψ ν (iy) = 0 for every y > β. The identicaly theorem then implies that ψ ν (z) = 0, for every that z Γ α,β which implies that ν = δ 0. = References [] Bercovici, H. and Voiculescu, D., Free convolution of measures with unbounded support, Indiana U. Math. J. 42, [2] Akhiezer, N.I. and Glazman, I.M., Theory of linear operators in Hilbert space, Frederick Ungar, 963. [3] Maassen, H., Addition of freely independent random variables, J.Funct. Anal., 06(2) (992),

14 68 Bogdan Gheorghe Munteanu [4] Orman, V.G., Măsura si procese aleatoare, Editura Universităţii Transilvania, Braşov, Romania, [5] Voiculescu, D., Symmetrics of some reduced free product C -algebras, Lecture Notes in Mathematics 32, Springer, Buşteni, Romania, 983. [6] Voiculescu, D., Addition of certain non-commuting random variables, J.Funct. Anal., 66 (986), [7] Voiculescu, D.,Free noncommutative random variables, random matrices and the II factors of free groups, University of California, Berkeley, 990. [8] Wigner, E.P., Characteristic vectors of bordered matrices with infinite dimensions, Ann. of Math. 62 (955), [9] Wigner, E.P., On the distributions of the roots of certain symmetric matrices, Ann. of Math. 67 (958), Faculty of Mathematics and Computer Science Transilvania University Str.Iuliu Maniu 50 RO Braşov,România b.munteanu@unitbv.ro

FREE PROBABILITY AND RANDOM MATRICES

FREE PROBABILITY AND RANDOM MATRICES FEE POBABILITY AND ANDOM MATICES Lecture 4: Free Harmonic Analysis, October 4, 007 The Cauchy Transform. Let C + = {z C Im(z > 0} denote the complex upper half plane, and C = {z Im(z < 0} denote the lower

More information

Failure of the Raikov Theorem for Free Random Variables

Failure of the Raikov Theorem for Free Random Variables Failure of the Raikov Theorem for Free Random Variables Florent Benaych-Georges DMA, École Normale Supérieure, 45 rue d Ulm, 75230 Paris Cedex 05 e-mail: benaych@dma.ens.fr http://www.dma.ens.fr/ benaych

More information

The Central Limit Theorem for Free Additive Convolution

The Central Limit Theorem for Free Additive Convolution journal of functional analsis 140, 359380 (1996) article no. 0112 The Central Limit Theorem for Free Additive Convolution Vittorino Pata*, - Dipartimento di Elettronica per l'automazione, Universita di

More information

Spectral law of the sum of random matrices

Spectral law of the sum of random matrices Spectral law of the sum of random matrices Florent Benaych-Georges benaych@dma.ens.fr May 5, 2005 Abstract The spectral distribution of a matrix is the uniform distribution on its spectrum with multiplicity.

More information

Free Convolution with A Free Multiplicative Analogue of The Normal Distribution

Free Convolution with A Free Multiplicative Analogue of The Normal Distribution Free Convolution with A Free Multiplicative Analogue of The Normal Distribution Ping Zhong Indiana University Bloomington July 23, 2013 Fields Institute, Toronto Ping Zhong free multiplicative analogue

More information

COMPUTING MOMENTS OF FREE ADDITIVE CONVOLUTION OF MEASURES

COMPUTING MOMENTS OF FREE ADDITIVE CONVOLUTION OF MEASURES COMPUTING MOMENTS OF FREE ADDITIVE CONVOLUTION OF MEASURES W LODZIMIERZ BRYC Abstract. This short note explains how to use ready-to-use components of symbolic software to convert between the free cumulants

More information

MULTIPLICATIVE MONOTONIC CONVOLUTION

MULTIPLICATIVE MONOTONIC CONVOLUTION Illinois Journal of Mathematics Volume 49, Number 3, Fall 25, Pages 929 95 S 9-282 MULTIPLICATIVE MONOTONIC CONVOLUTION HARI BERCOVICI Abstract. We show that the monotonic independence introduced by Muraki

More information

Analysis Comprehensive Exam Questions Fall 2008

Analysis Comprehensive Exam Questions Fall 2008 Analysis Comprehensive xam Questions Fall 28. (a) Let R be measurable with finite Lebesgue measure. Suppose that {f n } n N is a bounded sequence in L 2 () and there exists a function f such that f n (x)

More information

Cumulants of a convolution and applications to monotone probability theory

Cumulants of a convolution and applications to monotone probability theory Cumulants of a convolution and applications to monotone probability theory arxiv:95.3446v2 [math.pr] 24 May 29 Takahiro Hasebe JSPS Research Fellow (DC), Graduate School of Science, Kyoto University, Kyoto

More information

Inequalities in Hilbert Spaces

Inequalities in Hilbert Spaces Inequalities in Hilbert Spaces Jan Wigestrand Master of Science in Mathematics Submission date: March 8 Supervisor: Eugenia Malinnikova, MATH Norwegian University of Science and Technology Department of

More information

1 Math 241A-B Homework Problem List for F2015 and W2016

1 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 information

Part IB. Further Analysis. Year

Part IB. Further Analysis. Year Year 2004 2003 2002 2001 10 2004 2/I/4E Let τ be the topology on N consisting of the empty set and all sets X N such that N \ X is finite. Let σ be the usual topology on R, and let ρ be the topology on

More information

Homework 27. Homework 28. Homework 29. Homework 30. Prof. Girardi, Math 703, Fall 2012 Homework: Define f : C C and u, v : R 2 R by

Homework 27. Homework 28. Homework 29. Homework 30. Prof. Girardi, Math 703, Fall 2012 Homework: Define f : C C and u, v : R 2 R by Homework 27 Define f : C C and u, v : R 2 R by f(z) := xy where x := Re z, y := Im z u(x, y) = Re f(x + iy) v(x, y) = Im f(x + iy). Show that 1. u and v satisfies the Cauchy Riemann equations at (x, y)

More information

Reducing subspaces. Rowan Killip 1 and Christian Remling 2 January 16, (to appear in J. Funct. Anal.)

Reducing subspaces. Rowan Killip 1 and Christian Remling 2 January 16, (to appear in J. Funct. Anal.) Reducing subspaces Rowan Killip 1 and Christian Remling 2 January 16, 2001 (to appear in J. Funct. Anal.) 1. University of Pennsylvania, 209 South 33rd Street, Philadelphia PA 19104-6395, USA. On leave

More information

Complex Analysis. Travis Dirle. December 4, 2016

Complex Analysis. Travis Dirle. December 4, 2016 Complex Analysis 2 Complex Analysis Travis Dirle December 4, 2016 2 Contents 1 Complex Numbers and Functions 1 2 Power Series 3 3 Analytic Functions 7 4 Logarithms and Branches 13 5 Complex Integration

More information

13 Maximum Modulus Principle

13 Maximum Modulus Principle 3 Maximum Modulus Principle Theorem 3. (maximum modulus principle). If f is non-constant and analytic on an open connected set Ω, then there is no point z 0 Ω such that f(z) f(z 0 ) for all z Ω. Remark

More information

Considering our result for the sum and product of analytic functions, this means that for (a 0, a 1,..., a N ) C N+1, the polynomial.

Considering our result for the sum and product of analytic functions, this means that for (a 0, a 1,..., a N ) C N+1, the polynomial. Lecture 3 Usual complex functions MATH-GA 245.00 Complex Variables Polynomials. Construction f : z z is analytic on all of C since its real and imaginary parts satisfy the Cauchy-Riemann relations and

More information

Department of Mathematics, University of California, Berkeley. GRADUATE PRELIMINARY EXAMINATION, Part A Fall Semester 2014

Department of Mathematics, University of California, Berkeley. GRADUATE PRELIMINARY EXAMINATION, Part A Fall Semester 2014 Department of Mathematics, University of California, Berkeley YOUR 1 OR 2 DIGIT EXAM NUMBER GRADUATE PRELIMINARY EXAMINATION, Part A Fall Semester 2014 1. Please write your 1- or 2-digit exam number on

More information

Some Inequalities for Commutators of Bounded Linear Operators in Hilbert Spaces

Some Inequalities for Commutators of Bounded Linear Operators in Hilbert Spaces Some Inequalities for Commutators of Bounded Linear Operators in Hilbert Spaces S.S. Dragomir Abstract. Some new inequalities for commutators that complement and in some instances improve recent results

More information

Qualifying Exam Complex Analysis (Math 530) January 2019

Qualifying Exam Complex Analysis (Math 530) January 2019 Qualifying Exam Complex Analysis (Math 53) January 219 1. Let D be a domain. A function f : D C is antiholomorphic if for every z D the limit f(z + h) f(z) lim h h exists. Write f(z) = f(x + iy) = u(x,

More information

Solutions to Complex Analysis Prelims Ben Strasser

Solutions to Complex Analysis Prelims Ben Strasser Solutions to Complex Analysis Prelims Ben Strasser In preparation for the complex analysis prelim, I typed up solutions to some old exams. This document includes complete solutions to both exams in 23,

More information

MORE NOTES FOR MATH 823, FALL 2007

MORE NOTES FOR MATH 823, FALL 2007 MORE NOTES FOR MATH 83, FALL 007 Prop 1.1 Prop 1. Lemma 1.3 1. The Siegel upper half space 1.1. The Siegel upper half space and its Bergman kernel. The Siegel upper half space is the domain { U n+1 z C

More information

MATH 423 Linear Algebra II Lecture 3: Subspaces of vector spaces. Review of complex numbers. Vector space over a field.

MATH 423 Linear Algebra II Lecture 3: Subspaces of vector spaces. Review of complex numbers. Vector space over a field. MATH 423 Linear Algebra II Lecture 3: Subspaces of vector spaces. Review of complex numbers. Vector space over a field. Vector space A vector space is a set V equipped with two operations, addition V V

More information

. Then g is holomorphic and bounded in U. So z 0 is a removable singularity of g. Since f(z) = w 0 + 1

. Then g is holomorphic and bounded in U. So z 0 is a removable singularity of g. Since f(z) = w 0 + 1 Now we describe the behavior of f near an isolated singularity of each kind. We will always assume that z 0 is a singularity of f, and f is holomorphic on D(z 0, r) \ {z 0 }. Theorem 4.2.. z 0 is a removable

More information

Chapter 4: Open mapping theorem, removable singularities

Chapter 4: Open mapping theorem, removable singularities Chapter 4: Open mapping theorem, removable singularities Course 44, 2003 04 February 9, 2004 Theorem 4. (Laurent expansion) Let f : G C be analytic on an open G C be open that contains a nonempty annulus

More information

Solutions to Tutorial 11 (Week 12)

Solutions to Tutorial 11 (Week 12) THE UIVERSITY OF SYDEY SCHOOL OF MATHEMATICS AD STATISTICS Solutions to Tutorial 11 (Week 12) MATH3969: Measure Theory and Fourier Analysis (Advanced) Semester 2, 2017 Web Page: http://sydney.edu.au/science/maths/u/ug/sm/math3969/

More information

MATH MEASURE THEORY AND FOURIER ANALYSIS. Contents

MATH MEASURE THEORY AND FOURIER ANALYSIS. Contents MATH 3969 - MEASURE THEORY AND FOURIER ANALYSIS ANDREW TULLOCH Contents 1. Measure Theory 2 1.1. Properties of Measures 3 1.2. Constructing σ-algebras and measures 3 1.3. Properties of the Lebesgue measure

More information

Physics 307. Mathematical Physics. Luis Anchordoqui. Wednesday, August 31, 16

Physics 307. Mathematical Physics. Luis Anchordoqui. Wednesday, August 31, 16 Physics 307 Mathematical Physics Luis Anchordoqui 1 Bibliography L. A. Anchordoqui and T. C. Paul, ``Mathematical Models of Physics Problems (Nova Publishers, 2013) G. F. D. Duff and D. Naylor, ``Differential

More information

Complex Analysis Qualifying Exam Solutions

Complex Analysis Qualifying Exam Solutions Complex Analysis Qualifying Exam Solutions May, 04 Part.. Let log z be the principal branch of the logarithm defined on G = {z C z (, 0]}. Show that if t > 0, then the equation log z = t has exactly one

More information

CESARO OPERATORS ON THE HARDY SPACES OF THE HALF-PLANE

CESARO OPERATORS ON THE HARDY SPACES OF THE HALF-PLANE CESARO OPERATORS ON THE HARDY SPACES OF THE HALF-PLANE ATHANASIOS G. ARVANITIDIS AND ARISTOMENIS G. SISKAKIS Abstract. In this article we study the Cesàro operator C(f)() = d, and its companion operator

More information

ABSOLUTELY CONTINUOUS SPECTRUM OF A TYPICAL SCHRÖDINGER OPERATOR WITH A SLOWLY DECAYING POTENTIAL

ABSOLUTELY CONTINUOUS SPECTRUM OF A TYPICAL SCHRÖDINGER OPERATOR WITH A SLOWLY DECAYING POTENTIAL ABSOLUTELY CONTINUOUS SPECTRUM OF A TYPICAL SCHRÖDINGER OPERATOR WITH A SLOWLY DECAYING POTENTIAL OLEG SAFRONOV 1. Main results We study the absolutely continuous spectrum of a Schrödinger operator 1 H

More information

November 27 Lecturer Dmitri Zaitsev Michaelmas Term Course S h e e t 2. Due: after the lecture. + O(4)) (z 3! + O(5)) = c z3 + O(4),

November 27 Lecturer Dmitri Zaitsev Michaelmas Term Course S h e e t 2. Due: after the lecture. + O(4)) (z 3! + O(5)) = c z3 + O(4), November 7 Lecturer Dmitri Zaitsev Michaelmas Term 017 Course 343 017 S h e e t Due: after the lecture Exercise 1 Determine the zero order of f at 0: (i) f(z) = z cosz sinz; Solution Use power series expansion

More information

Part IB Complex Analysis

Part IB Complex Analysis Part IB Complex Analysis Theorems Based on lectures by I. Smith Notes taken by Dexter Chua Lent 2016 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after

More information

Linear Algebra and Dirac Notation, Pt. 1

Linear Algebra and Dirac Notation, Pt. 1 Linear Algebra and Dirac Notation, Pt. 1 PHYS 500 - Southern Illinois University February 1, 2017 PHYS 500 - Southern Illinois University Linear Algebra and Dirac Notation, Pt. 1 February 1, 2017 1 / 13

More information

APPROXIMATION OF MOORE-PENROSE INVERSE OF A CLOSED OPERATOR BY A SEQUENCE OF FINITE RANK OUTER INVERSES

APPROXIMATION OF MOORE-PENROSE INVERSE OF A CLOSED OPERATOR BY A SEQUENCE OF FINITE RANK OUTER INVERSES APPROXIMATION OF MOORE-PENROSE INVERSE OF A CLOSED OPERATOR BY A SEQUENCE OF FINITE RANK OUTER INVERSES S. H. KULKARNI AND G. RAMESH Abstract. Let T be a densely defined closed linear operator between

More information

Lecture I: Asymptotics for large GUE random matrices

Lecture I: Asymptotics for large GUE random matrices Lecture I: Asymptotics for large GUE random matrices Steen Thorbjørnsen, University of Aarhus andom Matrices Definition. Let (Ω, F, P) be a probability space and let n be a positive integer. Then a random

More information

CRITERIA FOR IRREDUCIBILITY OF REGULAR REPRESENTATIONS CORRESPONDING TO PRODUCT MEASURES OF GROUP OF FINITE UPPER-TRIANGULAR MATRICES. A. V.

CRITERIA FOR IRREDUCIBILITY OF REGULAR REPRESENTATIONS CORRESPONDING TO PRODUCT MEASURES OF GROUP OF FINITE UPPER-TRIANGULAR MATRICES. A. V. CRITERIA FOR IRREDUCIBILITY OF REGULAR REPRESENTATIONS CORRESPONDING TO PRODUCT MEASURES OF GROUP OF FINITE UPPER-TRIANGULAR MATRICES A. V. Kosyak Received: 2..99 Abstract. We define the analog of the

More information

CHAPTER VIII HILBERT SPACES

CHAPTER VIII HILBERT SPACES CHAPTER VIII HILBERT SPACES DEFINITION Let X and Y be two complex vector spaces. A map T : X Y is called a conjugate-linear transformation if it is a reallinear transformation from X into Y, and if T (λx)

More information

Course 214 Basic Properties of Holomorphic Functions Second Semester 2008

Course 214 Basic Properties of Holomorphic Functions Second Semester 2008 Course 214 Basic Properties of Holomorphic Functions Second Semester 2008 David R. Wilkins Copyright c David R. Wilkins 1989 2008 Contents 7 Basic Properties of Holomorphic Functions 72 7.1 Taylor s Theorem

More information

Probability and Measure

Probability and Measure Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 84 Paper 4, Section II 26J Let (X, A) be a measurable space. Let T : X X be a measurable map, and µ a probability

More information

Hilbert Spaces. Hilbert space is a vector space with some extra structure. We start with formal (axiomatic) definition of a vector space.

Hilbert Spaces. Hilbert space is a vector space with some extra structure. We start with formal (axiomatic) definition of a vector space. Hilbert Spaces Hilbert space is a vector space with some extra structure. We start with formal (axiomatic) definition of a vector space. Vector Space. Vector space, ν, over the field of complex numbers,

More information

Harnack s theorem for harmonic compact operator-valued functions

Harnack s theorem for harmonic compact operator-valued functions Linear Algebra and its Applications 336 (2001) 21 27 www.elsevier.com/locate/laa Harnack s theorem for harmonic compact operator-valued functions Per Enflo, Laura Smithies Department of Mathematical Sciences,

More information

CONTINUITY OF CP-SEMIGROUPS IN THE POINT-STRONG OPERATOR TOPOLOGY

CONTINUITY OF CP-SEMIGROUPS IN THE POINT-STRONG OPERATOR TOPOLOGY J. OPERATOR THEORY 64:1(21), 149 154 Copyright by THETA, 21 CONTINUITY OF CP-SEMIGROUPS IN THE POINT-STRONG OPERATOR TOPOLOGY DANIEL MARKIEWICZ and ORR MOSHE SHALIT Communicated by William Arveson ABSTRACT.

More information

NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA4247 Complex Analysis II Lecture Notes Part II

NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA4247 Complex Analysis II Lecture Notes Part II NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA4247 Complex Analysis II Lecture Notes Part II Chapter 2 Further properties of analytic functions 21 Local/Global behavior of analytic functions;

More information

the convolution of f and g) given by

the convolution of f and g) given by 09:53 /5/2000 TOPIC Characteristic functions, cont d This lecture develops an inversion formula for recovering the density of a smooth random variable X from its characteristic function, and uses that

More information

ON HARMONIC FUNCTIONS ON SURFACES WITH POSITIVE GAUSS CURVATURE AND THE SCHWARZ LEMMA

ON HARMONIC FUNCTIONS ON SURFACES WITH POSITIVE GAUSS CURVATURE AND THE SCHWARZ LEMMA ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 44, Number 5, 24 ON HARMONIC FUNCTIONS ON SURFACES WITH POSITIVE GAUSS CURVATURE AND THE SCHWARZ LEMMA DAVID KALAJ ABSTRACT. We prove some versions of the Schwarz

More information

First Order Differential Equations Lecture 3

First Order Differential Equations Lecture 3 First Order Differential Equations Lecture 3 Dibyajyoti Deb 3.1. Outline of Lecture Differences Between Linear and Nonlinear Equations Exact Equations and Integrating Factors 3.. Differences between Linear

More information

MA3111S COMPLEX ANALYSIS I

MA3111S COMPLEX ANALYSIS I MA3111S COMPLEX ANALYSIS I 1. The Algebra of Complex Numbers A complex number is an expression of the form a + ib, where a and b are real numbers. a is called the real part of a + ib and b the imaginary

More information

MA30056: Complex Analysis. Revision: Checklist & Previous Exam Questions I

MA30056: Complex Analysis. Revision: Checklist & Previous Exam Questions I MA30056: Complex Analysis Revision: Checklist & Previous Exam Questions I Given z C and r > 0, define B r (z) and B r (z). Define what it means for a subset A C to be open/closed. If M A C, when is M said

More information

Theorem [Mean Value Theorem for Harmonic Functions] Let u be harmonic on D(z 0, R). Then for any r (0, R), u(z 0 ) = 1 z z 0 r

Theorem [Mean Value Theorem for Harmonic Functions] Let u be harmonic on D(z 0, R). Then for any r (0, R), u(z 0 ) = 1 z z 0 r 2. A harmonic conjugate always exists locally: if u is a harmonic function in an open set U, then for any disk D(z 0, r) U, there is f, which is analytic in D(z 0, r) and satisfies that Re f u. Since such

More information

Irina Pchelintseva A FIRST-ORDER SPECTRAL PHASE TRANSITION IN A CLASS OF PERIODICALLY MODULATED HERMITIAN JACOBI MATRICES

Irina Pchelintseva A FIRST-ORDER SPECTRAL PHASE TRANSITION IN A CLASS OF PERIODICALLY MODULATED HERMITIAN JACOBI MATRICES Opuscula Mathematica Vol. 8 No. 008 Irina Pchelintseva A FIRST-ORDER SPECTRAL PHASE TRANSITION IN A CLASS OF PERIODICALLY MODULATED HERMITIAN JACOBI MATRICES Abstract. We consider self-adjoint unbounded

More information

1. The COMPLEX PLANE AND ELEMENTARY FUNCTIONS: Complex numbers; stereographic projection; simple and multiple connectivity, elementary functions.

1. The COMPLEX PLANE AND ELEMENTARY FUNCTIONS: Complex numbers; stereographic projection; simple and multiple connectivity, elementary functions. Complex Analysis Qualifying Examination 1 The COMPLEX PLANE AND ELEMENTARY FUNCTIONS: Complex numbers; stereographic projection; simple and multiple connectivity, elementary functions 2 ANALYTIC FUNCTIONS:

More information

Positive definite solutions of some matrix equations

Positive definite solutions of some matrix equations Available online at www.sciencedirect.com Linear Algebra and its Applications 49 (008) 40 44 www.elsevier.com/locate/laa Positive definite solutions of some matrix equations Aleksandar S. Cvetković, Gradimir

More information

SOME INEQUALITIES FOR COMMUTATORS OF BOUNDED LINEAR OPERATORS IN HILBERT SPACES. S. S. Dragomir

SOME INEQUALITIES FOR COMMUTATORS OF BOUNDED LINEAR OPERATORS IN HILBERT SPACES. S. S. Dragomir Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Filomat 5: 011), 151 16 DOI: 10.98/FIL110151D SOME INEQUALITIES FOR COMMUTATORS OF BOUNDED LINEAR

More information

SPACES ENDOWED WITH A GRAPH AND APPLICATIONS. Mina Dinarvand. 1. Introduction

SPACES ENDOWED WITH A GRAPH AND APPLICATIONS. Mina Dinarvand. 1. Introduction MATEMATIČKI VESNIK MATEMATIQKI VESNIK 69, 1 (2017), 23 38 March 2017 research paper originalni nauqni rad FIXED POINT RESULTS FOR (ϕ, ψ)-contractions IN METRIC SPACES ENDOWED WITH A GRAPH AND APPLICATIONS

More information

Fall f(x)g(x) dx. The starting place for the theory of Fourier series is that the family of functions {e inx } n= is orthonormal, that is

Fall f(x)g(x) dx. The starting place for the theory of Fourier series is that the family of functions {e inx } n= is orthonormal, that is 18.103 Fall 2013 1. Fourier Series, Part 1. We will consider several function spaces during our study of Fourier series. When we talk about L p ((, π)), it will be convenient to include the factor 1/ in

More information

Solutions for Problem Set #5 due October 17, 2003 Dustin Cartwright and Dylan Thurston

Solutions for Problem Set #5 due October 17, 2003 Dustin Cartwright and Dylan Thurston Solutions for Problem Set #5 due October 17, 23 Dustin Cartwright and Dylan Thurston 1 (B&N 6.5) Suppose an analytic function f agrees with tan x, x 1. Show that f(z) = i has no solution. Could f be entire?

More information

Vectors in Function Spaces

Vectors in Function Spaces Jim Lambers MAT 66 Spring Semester 15-16 Lecture 18 Notes These notes correspond to Section 6.3 in the text. Vectors in Function Spaces We begin with some necessary terminology. A vector space V, also

More information

Zeros of lacunary random polynomials

Zeros of lacunary random polynomials Zeros of lacunary random polynomials Igor E. Pritsker Dedicated to Norm Levenberg on his 60th birthday Abstract We study the asymptotic distribution of zeros for the lacunary random polynomials. It is

More information

ON MATRIX VALUED SQUARE INTEGRABLE POSITIVE DEFINITE FUNCTIONS

ON MATRIX VALUED SQUARE INTEGRABLE POSITIVE DEFINITE FUNCTIONS 1 2 3 ON MATRIX VALUED SQUARE INTERABLE POSITIVE DEFINITE FUNCTIONS HONYU HE Abstract. In this paper, we study matrix valued positive definite functions on a unimodular group. We generalize two important

More information

A functional model for commuting pairs of contractions and the symmetrized bidisc

A functional model for commuting pairs of contractions and the symmetrized bidisc A functional model for commuting pairs of contractions and the symmetrized bidisc Nicholas Young Leeds and Newcastle Universities Lecture 3 A functional model for commuting pairs of contractions St Petersburg,

More information

General Mathematics Vol. 16, No. 1 (2008), A. P. Madrid, C. C. Peña

General Mathematics Vol. 16, No. 1 (2008), A. P. Madrid, C. C. Peña General Mathematics Vol. 16, No. 1 (2008), 41-50 On X - Hadamard and B- derivations 1 A. P. Madrid, C. C. Peña Abstract Let F be an infinite dimensional complex Banach space endowed with a bounded shrinking

More information

Random Process Lecture 1. Fundamentals of Probability

Random Process Lecture 1. Fundamentals of Probability Random Process Lecture 1. Fundamentals of Probability Husheng Li Min Kao Department of Electrical Engineering and Computer Science University of Tennessee, Knoxville Spring, 2016 1/43 Outline 2/43 1 Syllabus

More information

18.03 LECTURE NOTES, SPRING 2014

18.03 LECTURE NOTES, SPRING 2014 18.03 LECTURE NOTES, SPRING 2014 BJORN POONEN 7. Complex numbers Complex numbers are expressions of the form x + yi, where x and y are real numbers, and i is a new symbol. Multiplication of complex numbers

More information

Journal of Inequalities in Pure and Applied Mathematics

Journal of Inequalities in Pure and Applied Mathematics Journal of Inequalities in Pure and Applied Mathematics ON A HYBRID FAMILY OF SUMMATION INTEGRAL TYPE OPERATORS VIJAY GUPTA AND ESRA ERKUŞ School of Applied Sciences Netaji Subhas Institute of Technology

More information

Bernstein s analytic continuation of complex powers

Bernstein s analytic continuation of complex powers (April 3, 20) Bernstein s analytic continuation of complex powers Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/. Analytic continuation of distributions 2. Statement of the theorems

More information

Conformal maps. Lent 2019 COMPLEX METHODS G. Taylor. A star means optional and not necessarily harder.

Conformal maps. Lent 2019 COMPLEX METHODS G. Taylor. A star means optional and not necessarily harder. Lent 29 COMPLEX METHODS G. Taylor A star means optional and not necessarily harder. Conformal maps. (i) Let f(z) = az + b, with ad bc. Where in C is f conformal? cz + d (ii) Let f(z) = z +. What are the

More information

1 Orthonormal sets in Hilbert space

1 Orthonormal sets in Hilbert space Math 857 Fall 15 1 Orthonormal sets in Hilbert space Let S H. We denote by [S] the span of S, i.e., the set of all linear combinations of elements from S. A set {u α : α A} is called orthonormal, if u

More information

On an iteration leading to a q-analogue of the Digamma function

On an iteration leading to a q-analogue of the Digamma function On an iteration leading to a q-analogue of the Digamma function Christian Berg and Helle Bjerg Petersen May, 22 Abstract We show that the q-digamma function ψ q for < q < appears in an iteration studied

More information

Functional Analysis. Martin Brokate. 1 Normed Spaces 2. 2 Hilbert Spaces The Principle of Uniform Boundedness 32

Functional Analysis. Martin Brokate. 1 Normed Spaces 2. 2 Hilbert Spaces The Principle of Uniform Boundedness 32 Functional Analysis Martin Brokate Contents 1 Normed Spaces 2 2 Hilbert Spaces 2 3 The Principle of Uniform Boundedness 32 4 Extension, Reflexivity, Separation 37 5 Compact subsets of C and L p 46 6 Weak

More information

On non-expansivity of topical functions by a new pseudo-metric

On non-expansivity of topical functions by a new pseudo-metric https://doi.org/10.1007/s40065-018-0222-8 Arabian Journal of Mathematics H. Barsam H. Mohebi On non-expansivity of topical functions by a new pseudo-metric Received: 9 March 2018 / Accepted: 10 September

More information

An introduction to some aspects of functional analysis

An introduction to some aspects of functional analysis An introduction to some aspects of functional analysis Stephen Semmes Rice University Abstract These informal notes deal with some very basic objects in functional analysis, including norms and seminorms

More information

III. Consequences of Cauchy s Theorem

III. Consequences of Cauchy s Theorem MTH6 Complex Analysis 2009-0 Lecture Notes c Shaun Bullett 2009 III. Consequences of Cauchy s Theorem. Cauchy s formulae. Cauchy s Integral Formula Let f be holomorphic on and everywhere inside a simple

More information

Weak convergence and averaging for ODE

Weak convergence and averaging for ODE Weak convergence and averaging for ODE Lawrence C. Evans and Te Zhang Department of Mathematics University of California, Berkeley Abstract This mostly expository paper shows how weak convergence methods

More information

BOUNDARY CORRESPONDENCE UNDER QUASICONFORMAL HARMONIC DIFFEO- MORPHISMS OF A HALF-PLANE

BOUNDARY CORRESPONDENCE UNDER QUASICONFORMAL HARMONIC DIFFEO- MORPHISMS OF A HALF-PLANE Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 3, 25, 159 165 BOUNDARY CORRESPONDENCE UNDER QUASICONFORMAL HARMONIC DIFFEO- MORPHISMS OF A HALF-PLANE David Kalaj and Miroslav Pavlović Prirodno-matematički

More information

Deficiency And Relative Deficiency Of E-Valued Meromorphic Functions

Deficiency And Relative Deficiency Of E-Valued Meromorphic Functions Applied Mathematics E-Notes, 323, -8 c ISSN 67-25 Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/ Deficiency And Relative Deficiency Of E-Valued Meromorphic Functions Zhaojun Wu, Zuxing

More information

Finite-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

Finite-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 information

Exercises for Part 1

Exercises for Part 1 MATH200 Complex Analysis. Exercises for Part Exercises for Part The following exercises are provided for you to revise complex numbers. Exercise. Write the following expressions in the form x + iy, x,y

More information

CONVERGENCE OF THE FRACTIONAL PARTS OF THE RANDOM VARIABLES TO THE TRUNCATED EXPONENTIAL DISTRIBUTION

CONVERGENCE OF THE FRACTIONAL PARTS OF THE RANDOM VARIABLES TO THE TRUNCATED EXPONENTIAL DISTRIBUTION Ann. Acad. Rom. Sci. Ser. Math. Appl. ISSN 2066-6594 Vol. 4, No. 2 / 202 CONVERGENCE OF THE FRACTIONAL PARTS OF THE RANDOM VARIABLES TO THE TRUNCATED EXPONENTIAL DISTRIBUTION Bogdan Gheorghe Munteanu Abstract

More information

THEOREMS, ETC., FOR MATH 515

THEOREMS, ETC., FOR MATH 515 THEOREMS, ETC., FOR MATH 515 Proposition 1 (=comment on page 17). If A is an algebra, then any finite union or finite intersection of sets in A is also in A. Proposition 2 (=Proposition 1.1). For every

More information

On Power Series Analytic in the Open Unit Disk with Finite Doble-Logarithmic Order

On Power Series Analytic in the Open Unit Disk with Finite Doble-Logarithmic Order Applied Mathematical Sciences, Vol 2, 2008, no 32, 549-569 On Power Series Analytic in the Open Unit Disk with Finite Doble-Logarithmic Order Mehmet Açıkgöz University of Gaziantep, Faculty of Science

More information

A WEDGE-OF-THE-EDGE THEOREM: ANALYTIC CONTINUATION OF MULTIVARIABLE PICK FUNCTIONS IN AND AROUND THE BOUNDARY

A WEDGE-OF-THE-EDGE THEOREM: ANALYTIC CONTINUATION OF MULTIVARIABLE PICK FUNCTIONS IN AND AROUND THE BOUNDARY A WEDGE-OF-THE-EDGE THEOREM: ANALYTIC CONTINUATION OF MULTIVARIABLE PICK FUNCTIONS IN AND AROUND THE BOUNDARY J E PASCOE Abstract In 1956, quantum physicist N Bogoliubov discovered the edge-ofthe-wedge

More information

Math 185 Fall 2015, Sample Final Exam Solutions

Math 185 Fall 2015, Sample Final Exam Solutions Math 185 Fall 2015, Sample Final Exam Solutions Nikhil Srivastava December 12, 2015 1. True or false: (a) If f is analytic in the annulus A = {z : 1 < z < 2} then there exist functions g and h such that

More information

Applied Analysis (APPM 5440): Final exam 1:30pm 4:00pm, Dec. 14, Closed books.

Applied Analysis (APPM 5440): Final exam 1:30pm 4:00pm, Dec. 14, Closed books. Applied Analysis APPM 44: Final exam 1:3pm 4:pm, Dec. 14, 29. Closed books. Problem 1: 2p Set I = [, 1]. Prove that there is a continuous function u on I such that 1 ux 1 x sin ut 2 dt = cosx, x I. Define

More information

1/12/05: sec 3.1 and my article: How good is the Lebesgue measure?, Math. Intelligencer 11(2) (1989),

1/12/05: sec 3.1 and my article: How good is the Lebesgue measure?, Math. Intelligencer 11(2) (1989), Real Analysis 2, Math 651, Spring 2005 April 26, 2005 1 Real Analysis 2, Math 651, Spring 2005 Krzysztof Chris Ciesielski 1/12/05: sec 3.1 and my article: How good is the Lebesgue measure?, Math. Intelligencer

More information

Applicable Analysis and Discrete Mathematics available online at

Applicable Analysis and Discrete Mathematics available online at Applicable Analysis and Discrete Mathematics available online at http://pefmath.etf.bg.ac.yu Appl. Anal. Discrete Math. 3 (2009), 236 241. doi:10.2298/aadm0902236a BANACH CONTRACTION PRINCIPLE ON CONE

More information

The Hilbert Transform and Fine Continuity

The Hilbert Transform and Fine Continuity Irish Math. Soc. Bulletin 58 (2006), 8 9 8 The Hilbert Transform and Fine Continuity J. B. TWOMEY Abstract. It is shown that the Hilbert transform of a function having bounded variation in a finite interval

More information

Fréchet differentiability of the norm of L p -spaces associated with arbitrary von Neumann algebras

Fréchet differentiability of the norm of L p -spaces associated with arbitrary von Neumann algebras Fréchet differentiability of the norm of L p -spaces associated with arbitrary von Neumann algebras (Part I) Fedor Sukochev (joint work with D. Potapov, A. Tomskova and D. Zanin) University of NSW, AUSTRALIA

More information

TOEPLITZ OPERATORS. Toeplitz studied infinite matrices with NW-SE diagonals constant. f e C :

TOEPLITZ OPERATORS. Toeplitz studied infinite matrices with NW-SE diagonals constant. f e C : TOEPLITZ OPERATORS EFTON PARK 1. Introduction to Toeplitz Operators Otto Toeplitz lived from 1881-1940 in Goettingen, and it was pretty rough there, so he eventually went to Palestine and eventually contracted

More information

ζ(u) z du du Since γ does not pass through z, f is defined and continuous on [a, b]. Furthermore, for all t such that dζ

ζ(u) z du du Since γ does not pass through z, f is defined and continuous on [a, b]. Furthermore, for all t such that dζ Lecture 6 Consequences of Cauchy s Theorem MATH-GA 45.00 Complex Variables Cauchy s Integral Formula. Index of a point with respect to a closed curve Let z C, and a piecewise differentiable closed curve

More information

A trigonometric orthogonality with respect to a nonnegative Borel measure

A trigonometric orthogonality with respect to a nonnegative Borel measure Filomat 6:4 01), 689 696 DOI 10.98/FIL104689M Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat A trigonometric orthogonality with

More information

ON THE FEKETE-SZEGÖ INEQUALITY FOR A CLASS OF ANALYTIC FUNCTIONS DEFINED BY USING GENERALIZED DIFFERENTIAL OPERATOR

ON THE FEKETE-SZEGÖ INEQUALITY FOR A CLASS OF ANALYTIC FUNCTIONS DEFINED BY USING GENERALIZED DIFFERENTIAL OPERATOR Acta Universitatis Apulensis ISSN: 58-539 No. 6/0 pp. 67-78 ON THE FEKETE-SZEGÖ INEQUALITY FOR A CLASS OF ANALYTIC FUNCTIONS DEFINED BY USING GENERALIZED DIFFERENTIAL OPERATOR Salma Faraj Ramadan, Maslina

More information

A SYNOPSIS OF HILBERT SPACE THEORY

A SYNOPSIS OF HILBERT SPACE THEORY A SYNOPSIS OF HILBERT SPACE THEORY Below is a summary of Hilbert space theory that you find in more detail in any book on functional analysis, like the one Akhiezer and Glazman, the one by Kreiszig or

More information

Math 411, Complex Analysis Definitions, Formulas and Theorems Winter y = sinα

Math 411, Complex Analysis Definitions, Formulas and Theorems Winter y = sinα Math 411, Complex Analysis Definitions, Formulas and Theorems Winter 014 Trigonometric Functions of Special Angles α, degrees α, radians sin α cos α tan α 0 0 0 1 0 30 π 6 45 π 4 1 3 1 3 1 y = sinα π 90,

More information

2 Lebesgue integration

2 Lebesgue integration 2 Lebesgue integration 1. Let (, A, µ) be a measure space. We will always assume that µ is complete, otherwise we first take its completion. The example to have in mind is the Lebesgue measure on R n,

More information

CHAPTER 1. AFFINE ALGEBRAIC VARIETIES

CHAPTER 1. AFFINE ALGEBRAIC VARIETIES CHAPTER 1. AFFINE ALGEBRAIC VARIETIES During this first part of the course, we will establish a correspondence between various geometric notions and algebraic ones. Some references for this part of the

More information

DIMENSIONS OF JULIA SETS OF HYPERBOLIC ENTIRE FUNCTIONS

DIMENSIONS OF JULIA SETS OF HYPERBOLIC ENTIRE FUNCTIONS Bull. London Math. Soc. 36 2004 263 270 C 2004 London Mathematical Society DOI: 10.1112/S0024609303002698 DIMENSIONS OF JULIA SETS OF HYPERBOLIC ENTIRE FUNCTIONS GWYNETH M. STALLARD Abstract It is known

More information

Bessel Functions Michael Taylor. Lecture Notes for Math 524

Bessel Functions Michael Taylor. Lecture Notes for Math 524 Bessel Functions Michael Taylor Lecture Notes for Math 54 Contents 1. Introduction. Conversion to first order systems 3. The Bessel functions J ν 4. The Bessel functions Y ν 5. Relations between J ν and

More information

Complex Analysis MATH 6300 Fall 2013 Homework 4

Complex Analysis MATH 6300 Fall 2013 Homework 4 Complex Analysis MATH 6300 Fall 2013 Homework 4 Due Wednesday, December 11 at 5 PM Note that to get full credit on any problem in this class, you must solve the problems in an efficient and elegant manner,

More information

5 Measure theory II. (or. lim. Prove the proposition. 5. For fixed F A and φ M define the restriction of φ on F by writing.

5 Measure theory II. (or. lim. Prove the proposition. 5. For fixed F A and φ M define the restriction of φ on F by writing. 5 Measure theory II 1. Charges (signed measures). Let (Ω, A) be a σ -algebra. A map φ: A R is called a charge, (or signed measure or σ -additive set function) if φ = φ(a j ) (5.1) A j for any disjoint

More information