# In memory of Lars Hörmander

Size: px
Start display at page:

## Transcription

1 ON HÖRMANDER S SOLUTION OF THE -EQUATION. I HAAKAN HEDENMALM ABSTRAT. We explain how Hörmander s classical solution of the -equation in the plane with a weight which permits growth near infinity carries over to the rather opposite situation when we ask for decay near infinity. Here, however, a natural condition on the datum needs to be imposed. The condition is not only natural but also necessary to have the result at least in the Fock weight case. The norm identity which leads to the estimate is related to general area-type results in the theory of conformal mappings. In memory of Lars Hörmander 1.1. Basic notation. Let 1. INTRODUTION := 1 ( 2 ) 4 x + 2, da(z) := dxdy, 2 y 2 denote the normalized Laplacian and the area element, respectively. Here, z = x + iy is the standard decomposition into real and imaginary parts. We let denote the complex plane. We also need the standard complex differential operators z := 1 2( x + i ), z := 1 ( y 2 x i ), y so that factors as = z z. We sometimes drop indication of the differentiation variable z Hörmander s solution of the -problem. We present Hörmander s theorem [12], [13] in the simplest possible case, when the domain is the entire complex plane and the weight φ : R is 2 -smooth with φ > 0 everywhere. Theorem 1.1. (Hörmander) If the complex-valued function f is locally area L 2 -integrable in the plane, then there exists a solution to the -equation u = f with u 2 e 2φ da 1 f 2 e 2φ 2 φ da. Here, we remark that the assertion of the theorem is void unless the integral on the right hand side is finite Mathematics Subject lassification. Primary. Key words and phrases. -equation, weighted estimates. The research of the author was supported by the Göran Gustafsson Foundation (KVA) and by Vetenskapsrådet (VR). 1

2 2 HAAKAN HEDENMALM 1.3. The -equation with growing weights. While Theorem 1.1 essentially deals with decaying weights e 2φ, it is natural to ask what happens if we were to consider the growing weights e 2φ in place of e 2φ. So when could we say that there exists a solution to u = f with (1.1) u 2 e 2φ da f 2 e 2φ da, where the symbol is understood liberally? Here, we should have the (Fock weight) example φ(z) = 1 2 z 2 in mind. It is rather clear that we cannot hope to have an estimate of the type (1.1) without an additional condition on on the datum f. For instance, if φ(z) = 1 2 z 2 and f (z) = e z 2, it is not possible to find such a fast-decaying function u with u = f (cf. Section 3 below). It is natural to look for a class of data f that would come from functions u with compact support. Let c () denote the standard space of infinitely differentiable compactly supported test functions. The calculation f gda = g uda = u gda, u c (), shows that the datum f = u with u c () must satisfy, for entire functions g, (1.2) f gda = 0. We remark here that the calculation (1.2) is the basis for what is known as Havin s lemma [7] (see also, e.g., [8]). We let L 2 (, e 2φ ) and L 2 (, e 2φ ) be the weighted area L 2 -spaces with the indicated weights. The corresponding norms are g 2 = g 2 e 2φ da, g 2 = g 2 e 2φ da. L 2 (,e 2φ ) L 2 (,e 2φ ) The space of entire functions in L 2 (, e 2φ ) is denoted by A 2 (, e 2φ ). We recall the standing assumption that φ be 2 -smooth with φ > 0 everywhere. Theorem 1.2. Suppose f L 2 (, e 2φ ) meets the condition (1.2) for all g A 2 (, e 2φ ). Then there exists a solution to the -equation u = f with u 2 e 2φ φda 1 2 f 2 e 2φ da. Superficially, this theorem looks quite different than Hörmander s Theorem 1.1. However, it is in a sense which can be made precise dual to Theorem 1.1. In our presentation, we will derive it from the same rather elementary calculation which Hörmander uses in e.g. [13], p. 250; cf. also Section 2 of [5] on arleman inequalities, where essentially the same calculation is made. We might add that if the polynomials are dense in the space A 2 (, e 2φ ) of entire functions, as they are, e.g., for radial φ, then the condition (1.2) needs to be verified only for monomials g(z) = z j, j = 0, 1, 2,.... Remark 1.3. Naturally, Theorem 1.2 should generalize to the setting of several complex variables. Under a simple condition on the weight, the solution supplied by Theorem 1.2 is unique.

3 ON HÖRMANDER S SOLUTION OF THE -EQUATION. I 3 Theorem 1.4. If, in addition, φ is 4 -smooth and meets the curvature-type condition then the solution u in Theorem 1.2 is unique. 1 log φ 2 on, φ 2. THE PROOF OF THEOREM A norm identity. We denote by L 2 the norm in the space L 2 (). For a function F, we let M F denote the operator of multiplication by F. The first step is the following norm identity for v c (): (2.1) v v φ 2 v + v φ 2 = 2 v 2 φda. L 2 L 2 To arrive at (2.1), we do as follows. As for the first term, if we let, L 2 denote the standard sesquilinear inner product of L 2 (), we see that v v φ 2 L 2 = v v φ, v v φ L 2 = ( M φ )v, ( M φ )v L 2 = ( M φ ) ( M φ )v, v L 2 and together with the corresponding calculation for the second term, we find that (2.1) expresses that (2.2) ( M φ ) ( M φ ) ( + M φ ) ( + M φ ) = 2M φ. Here, the adjoints involved are readily expressed: =, =, and M F = M F. The product rule says that M F = M F + M F and MF = M F + M F. By identifying adjoints and carrying out the necessary algebraic manipulations, (2.2) is immediate. We note in passing that the left-hand side of (2.2) expresses a self-commutator Reduction to a norm inequality. While the norm identity (2.1) is interesting by itself, we observe here that it has the consequence that (2.3) 2 v 2 φda v v φ 2, v L 2 c (). We write T := M φ, and express (2.3) again: (2.4) 2 v 2 φda Tv 2, L 2 v c (). If v j is a sequence of functions such that Tv j converges in L 2 (), then by (2.4) the functions v j φ converge in L 2 () as well, and in particular, v j converges locally as an L 2 -function. It follows that if h L 2 () is in to the L 2 ()-closure of T c (), then there exists a function v L 2 (, φ) such that Tv = h in the sense of distribution theory, with (2.5) 2 v 2 φda h 2. L 2 It remains to identify the L 2 ()-closure of T c (). To this end, we identify the orthogonal complement of T c (). So, let k L 2 () be such that (2.6) k, Tv L 2 = 0, v c (). If we write T := M φ, distribution theory gives that (2.6) is the same as T k, v L 2 = 0, v c (),

4 4 HAAKAN HEDENMALM which in its turn expresses that T k = 0 holds in the sense of distributions. Let us write ker T for the space of all k L 2 () with T k = 0. We have arrived at the following result. Theorem 2.1. Suppose h L 2 () ker T. Then there exists a function v L 2 (, φ) such that Tv = h with 2 v 2 φda h 2. L 2 Since T = M e φ M e φ, k ker T holds if and only if e φ k A 2 (, e 2φ ). Proof of Theorem 1.2. We put h := e φ f L 2 (). By condition (1.2) for all g A 2 (, e 2φ ), we have if we write k = e φ ḡ that 0 = f gda = h kda. As the function g runs over A 2 (, e 2φ ), k runs over all elements of ker T. We conclude that h L 2 () ker T, so that Theorem 2.1 applies, and gives a v L 2 (, φ) with Tv = h with the norm control (2.7) 2 v 2 φda h 2. L 2 We put u := e φ v. The operator T factors T = M M e φ e φ, which means that the equation Tv = h is equivalent to u = f. Finally, the estimate (2.7) is equivalent to the estimate 2 u 2 e 2φ φda f 2 e 2φ da, which concludes the proof of the theorem. Proof of Theorem 1.4. We first observe that any two solutions of the -equation differ by an entire function. Moreover, under the given curvature-type condition, an entire function F L 2 (, e 2φ φ) necessarily must vanish everywhere, by the following argument. The function F 2 e 2φ φ is clearly nonnegative and it is also subharmonic in. Indeed, if F is nontrivial, we have that [in the sense of distribution theory] log F 0, and consequently log [ F 2 e 2φ φ ] = 2 log F + 2 φ + log φ 0, which gives that the exponentiated function F 2 e 2φ φ is subharmonic, as claimed. In the remaining case when F vanishes identically the claim is trivial. Next, by the estimate of each solution of the -problem supplied by Theorem 1.2, it is given that the function F 2 e 2φ φ is in L 2 (). If D(z 0, r) denotes the open disk of radius r about z 0, the sub-mean vakue property of subharmonic functions gives that F(z 0 ) 2 e 2φ(z0) φ(z 0 ) 1 F 2 e 2φ φda 1 F 2 e 2φ φda. πr 2 D(z 0,r) πr 2 Letting r +, we see that the left hand side vanishes. As z 0 is arbitrary, it follows that F(z) 0. This completes the proof. Remark 2.2. In Lemma 3.1 of [2] appears a condition which is analogous to (1.2) in the context of distributions with compact support. Also, in the related paper [15], the Fock weight case φ(z) = 1 2 z 2 is considered in the soft-topology setting of solutions which are distributions.

5 ON HÖRMANDER S SOLUTION OF THE -EQUATION. I 5 Remark 2.3. With φ = 0, the norm identity (2.1) expresses that the Beurling operator is an isometry on L 2 (), which implies the Grunsky inequalities in the theory of conformal mapping (see, e.g., [3]; cf. [4]). In the (somewhat singular) case when φ(z) = θ log z, (2.1) provides the main norm identity of [9], which leads to a Prawitz-Grunsky type inequality for conformal maps. We remark here that the Prawitz inequality was the key input estimate in the analysis of the universal integral means spectrum in [10], [11]. The most general Prawitz-Grunsky type inequality (with multiple branch points ) for conformal maps was obtained by Shimorin [16] (see also [1]). It would be of interest to see if the results of Shimorin may be obtained from the general norm identity (2.1). 3. DISUSSION OF THE NEESSITY OF THE ORTHOGONALITY ONDITION ON THE DATUM 3.1. The Fock weight case. We now narrow down the discussion to the Fock weight φ(z) = 1 2 z 2. Since the weight is radial, the polynomials are dense in A 2 (, e z 2 ). Also, the curvaturetype condition of Theorem 1.4 is readily checked. It follows that Theorems 1.2 and 1.4 combine to give the following result. Theorem 3.1. Suppose f L 2 (, e z 2 ). If the datum f satisfies the moment condition z j f (z)da(z) = 0, j = 0, 1, 2,..., then there exists a unique solution u to the equation u = f with u(z) 2 e z 2 da(z) f (z) 2 e z 2 da(z). We may ask what would happen if the orthogonality condition is not satisfied. Maybe there still exists some rapidly decaying solution u anyway? The answer is definitely no. Theorem 3.2. Suppose f L 2 (, e z 2 ), and that u solves the equation u = f while u L 2 (, e ɛ z 2 ) for some positive real ɛ. Then the datum f has z j f (z)da(z) = 0, j = 0, 1, 2,.... Before we turn to the proof, we observe that if f L 2 (R) with f (x) e x2 /β dx < + R for some real β > 0, then its Fourier transform ˆ f (ξ) := e iξx f (x)dx R extends to an entire function with f ˆ (ξ) 2 e β Im ξ 2 da(ξ) = 2π3/2 f (x) e x2 /β dx < +. β R In fact, the standard Bargmann transform theory asserts that this integrability condition characterizes the Fourier image of this weighted L 2 space on R (cf. [6]). The two-variable extension of the above-mentioned result maintains that if f L 2 (, e z 2 /β ), then its Fourier transform ˆ f (ξ, η) := e i(ξx+ηy) f (x + iy)dxdy

6 6 HAAKAN HEDENMALM is an entire function of two variables, with (3.1) f ˆ (ξ, η) 2 e β( Im ξ 2 + Im η 2) da(ξ)da(η) = 4π3 f (z) e z 2 /β da(z) < +. β Proof of Theorem 3.2. Since f L 2 (, e z 2 ), the function ˆ f is an entire function of two variables with the norm bound (3.1) with β = 1. Likewise, as u L 2 (, e ɛ z 2 ) for some positive ɛ, û is an entire function of two variables. After Fourier transformation, the relation u = f reads (iξ η)û(ξ, η) = 2 ˆ f (ξ, η). This is only possible if f ˆ (ξ, η) vanishes when iξ η = 0, i.e., f ˆ (ξ, iξ) 0. By the definition of the Fourier transform, this means that + 0 = f ˆ ( iξ) j (ξ, iξ) = e iξ(x+iy) f (x + iy)da(x + iy) = e iξz f (z)da(z) = By Taylor s formula, then, this implies that z j f (z)da(z) = 0, j = 0, 1, 2,..., as needed. j=0 j! z j f (z)da(z). Remark 3.3. The constant in Theorem 3.1 is sharp. Indeed, we may consider the datum f (z) = ze z 2 for which the solution is u(z) = e z 2. We calculate that u(z) 2 e z 2 da(z) = while f (z) 2 e z 2 da(z) = which gives the desired sharpness. e z 2 da(z) = π z 2 e z 2 da(z) = π, 4. AKNOWLEDGEMENTS The author wishes to thank Ioannis Parissis and Serguei Shimorin [14] for stimulating discussions related to the norm identity (2.1). The author also wishes to thank Grigori Rozenblum for an enlightening conversion on the topic of this paper at the Euler Institute in St-Petersburg in July, REFERENES [1] Abuzyarova, N., Hedenmalm, H., Branch point area methods in conformal mapping. J. Anal. Math. 99 (2006), [2] Alexandrov, A., Rozenblum, G., Finite rank Toeplitz operators: some extensions of D. Luecking s theorem. J. Funct. Anal. 256 (2009), no. 7, [3] Baranov, A., Hedenmalm, H., Boundary properties of Green functions in the plane. Duke Math. J. 145 (2008), [4] Bergman, S., Schiffer, M., Kernel functions and conformal mapping. ompositio Math. 8 (1951), [5] Donnelly, H., Fefferman,., Nodal sets for eigenfunctions of the Laplacian on surfaces. J. Amer. Math. Soc. 3 (1990), [6] Gröchenig, K. Foundations of time-frequency analysis. Applied and Numerical Harmonic Analysis. Birkhäuser Boston, Inc., Boston, MA, [7] Havin, V. P., Approximation in the mean by analytic functions. Dokl. Akad. Nauk SSSR 178 (1968), [8] Hedenmalm, H., The dual of a Bergman space on simply connected domains. J. Anal. Math. 88 (2002), [9] Hedenmalm, H., Planar Beurling transform and Grunsky inequalities. Ann. Acad. Sci. Fenn. Math. 33 (2008), [10] Hedenmalm, H., Shimorin, S., Weighted Bergman spaces and the integral means spectrum of conformal mappings. Duke Math. J. 127 (2005), no. 2,

7 ON HÖRMANDER S SOLUTION OF THE -EQUATION. I 7 [11] Hedenmalm, H., Shimorin, S., On the universal integral means spectrum of conformal mappings near the origin. Proc. Amer. Math. Soc. 135 (2007), no. 7, [12] Hörmander, L., An introduction to complex analysis in several variables. D. Van Nostrand o., Inc., Princeton, N.J.- Toronto, Ont.-London [13] Hörmander, L., Notions of convexity. Progress in Mathematics, 127. Birkhäuser Boston, Inc., Boston, MA, [14] Parissis, I., Shimorin, S., Private communication. [15] Rozenblum, G., Shirokov, N., Some weighted estimates for the -equation and a finite rank theorem for Toeplitz operators in the Fock space. Proc. Lond. Math. Soc. (3) 109 (2014), no. 5, [16] Shimorin, S., Branching points area theorems for univalent functions. St. Petersburg Math. J. 18 (2007), HEDENMALM: DEPARTMENT OF MATHEMATIS, KTH ROYAL INSTITUTE OF TEHNOLOGY, S STOKHOLM, SWEDEN address:

### MICROLOCAL ANALYSIS OF THE BOCHNER-MARTINELLI INTEGRAL

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX)0000-0 MICROLOCAL ANALYSIS OF THE BOCHNER-MARTINELLI INTEGRAL NIKOLAI TARKHANOV AND NIKOLAI VASILEVSKI

### CHAPTER 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.,

### IRREDUCIBLE 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

### 1. Introduction ON D -EXTENSION PROPERTY OF THE HARTOGS DOMAINS

Publ. Mat. 45 (200), 42 429 ON D -EXTENSION PROPERTY OF THE HARTOGS DOMAINS Do Duc Thai and Pascal J. Thomas Abstract A complex analytic space is said to have the D -extension property if and only if any

### F. 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

### Numerical 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

### A PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS. In memory of Rou-Huai Wang

A PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS XAVIER CABRÉ, MANEL SANCHÓN, AND JOEL SPRUCK In memory of Rou-Huai Wang 1. Introduction In this note we consider semistable

### Reference: Introduction to Partial Differential Equations by G. Folland, 1995, Chap. 3.

5 Potential Theory Reference: Introduction to Partial Differential Equations by G. Folland, 995, Chap. 3. 5. Problems of Interest. In what follows, we consider Ω an open, bounded subset of R n with C 2

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### The 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

### 1. Introduction. PROPER HOLOMORPHIC MAPPINGS BETWEEN RIGID POLYNOMIAL DOMAINS IN C n+1

Publ. Mat. 45 (2001), 69 77 PROPER HOLOMORPHIC MAPPINGS BETWEEN RIGID POLYNOMIAL DOMAINS IN C n+1 Bernard Coupet and Nabil Ourimi Abstract We describe the branch locus of proper holomorphic mappings between

### 6. Define log(z) so that π < I log(z) π. Discuss the identities e log(z) = z and log(e w ) = w.

hapter omplex integration. omplex number quiz. Simplify 3+4i. 2. Simplify 3+4i. 3. Find the cube roots of. 4. Here are some identities for complex conjugate. Which ones need correction? z + w = z + w,

### Recall 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

### Duality 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

### SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH

31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,

### BANACH 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

### A STABILITY RESULT ON MUCKENHOUPT S WEIGHTS

Publicacions Matemàtiques, Vol 42 (998), 53 63. A STABILITY RESULT ON MUCKENHOUPT S WEIGHTS Juha Kinnunen Abstract We prove that Muckenhoupt s A -weights satisfy a reverse Hölder inequality with an explicit

### Basic 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

### EXISTENCE AND NON-EXISTENCE RESULTS FOR A NONLINEAR HEAT EQUATION

Sixth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 5 (7), pp. 5 65. ISSN: 7-669. UL: http://ejde.math.txstate.edu

### ON 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. SÁNCHEZ-PÉREZ Abstract. Let m be a vector measure taking values in a Banach space X. We prove that

### 4. Complex integration: Cauchy integral theorem and Cauchy integral formulas. Definite integral of a complex-valued function of a real variable

4. Complex integration: Cauchy integral theorem and Cauchy integral formulas Definite integral of a complex-valued function of a real variable Consider a complex valued function f(t) of a real variable

### arxiv:1409.7492v1 [math.cv] 26 Sep 2014

NECESSARY CONDITION FOR COMPACTNESS OF A DIFFERENCE OF COMPOSITION OPERATORS ON THE DIRICHLET SPACE MA LGORZATA MICHALSKA, ANDRZEJ M. MICHALSKI arxiv:409.749v [math.cv] 6 Sep 04 Abstract. Let ϕ be a self-map

### {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

### Properties of BMO functions whose reciprocals are also BMO

Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a non-negative BMO-function w, whose reciprocal is also in BMO, belongs to p> A p,and

### Analytic cohomology groups in top degrees of Zariski open sets in P n

Analytic cohomology groups in top degrees of Zariski open sets in P n Gabriel Chiriacescu, Mihnea Colţoiu, Cezar Joiţa Dedicated to Professor Cabiria Andreian Cazacu on her 80 th birthday 1 Introduction

### DIRICHLET S PROBLEM WITH ENTIRE DATA POSED ON AN ELLIPSOIDAL CYLINDER. 1. Introduction

DIRICHLET S PROBLEM WITH ENTIRE DATA POSED ON AN ELLIPSOIDAL CYLINDER DMITRY KHAVINSON, ERIK LUNDBERG, HERMANN RENDER. Introduction A function u is said to be harmonic if u := n j= 2 u = 0. Given a domain

### Mean Value Coordinates

Mean Value Coordinates Michael S. Floater Abstract: We derive a generalization of barycentric coordinates which allows a vertex in a planar triangulation to be expressed as a convex combination of its

### Geometrical Characterization of RN-operators between Locally Convex Vector Spaces

Geometrical Characterization of RN-operators between Locally Convex Vector Spaces OLEG REINOV St. Petersburg State University Dept. of Mathematics and Mechanics Universitetskii pr. 28, 198504 St, Petersburg

### Some Notes on Taylor Polynomials and Taylor Series

Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited

### Boundary value problems in complex analysis I

Boletín de la Asociación Matemática Venezolana, Vol. XII, No. (2005) 65 Boundary value problems in complex analysis I Heinrich Begehr Abstract A systematic investigation of basic boundary value problems

### Prime Numbers and Irreducible Polynomials

Prime Numbers and Irreducible Polynomials M. Ram Murty The similarity between prime numbers and irreducible polynomials has been a dominant theme in the development of number theory and algebraic geometry.

### Complex Function Theory. Second Edition. Donald Sarason >AMS AMERICAN MATHEMATICAL SOCIETY

Complex Function Theory Second Edition Donald Sarason >AMS AMERICAN MATHEMATICAL SOCIETY Contents Preface to the Second Edition Preface to the First Edition ix xi Chapter I. Complex Numbers 1 1.1. Definition

### MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

### MATHEMATICAL METHODS OF STATISTICS

MATHEMATICAL METHODS OF STATISTICS By HARALD CRAMER TROFESSOK IN THE UNIVERSITY OF STOCKHOLM Princeton PRINCETON UNIVERSITY PRESS 1946 TABLE OF CONTENTS. First Part. MATHEMATICAL INTRODUCTION. CHAPTERS

### FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

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

### Some remarks on Phragmén-Lindelöf theorems for weak solutions of the stationary Schrödinger operator

Wan Boundary Value Problems (2015) 2015:239 DOI 10.1186/s13661-015-0508-0 R E S E A R C H Open Access Some remarks on Phragmén-Lindelöf theorems for weak solutions of the stationary Schrödinger operator

### 16.3 Fredholm Operators

Lectures 16 and 17 16.3 Fredholm Operators A nice way to think about compact operators is to show that set of compact operators is the closure of the set of finite rank operator in operator norm. In this

### RAJALAKSHMI ENGINEERING COLLEGE MA 2161 UNIT I - ORDINARY DIFFERENTIAL EQUATIONS PART A

RAJALAKSHMI ENGINEERING COLLEGE MA 26 UNIT I - ORDINARY DIFFERENTIAL EQUATIONS. Solve (D 2 + D 2)y = 0. 2. Solve (D 2 + 6D + 9)y = 0. PART A 3. Solve (D 4 + 4)x = 0 where D = d dt 4. Find Particular Integral:

### INVARIANT SUBSPACES ON MULTIPLY CONNECTED DOMAINS

Publicacions Matemàtiques, Vol 42 (1998), 521 557. INVARIANT SUBSPACES ON MULTIPLY CONNECTED DOMAINS Ali Abkar and Håkan Hedenmalm Abstract The lattice of invariant subspaces of several Banach spaces of

### U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory

### 8.1 Examples, definitions, and basic properties

8 De Rham cohomology Last updated: May 21, 211. 8.1 Examples, definitions, and basic properties A k-form ω Ω k (M) is closed if dω =. It is exact if there is a (k 1)-form σ Ω k 1 (M) such that dσ = ω.

### Ideal 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

### 1 Inner Products and Norms on Real Vector Spaces

Math 373: Principles Techniques of Applied Mathematics Spring 29 The 2 Inner Product 1 Inner Products Norms on Real Vector Spaces Recall that an inner product on a real vector space V is a function from

### Metric Spaces. Chapter 1

Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence

### The analysis of Toeplitz operators, commutative Toeplitz algebras and applications to heat kernel constructions

The analysis of Toeplitz operators, commutative Toeplitz algebras and applications to heat kernel constructions Dissertation zur Erlangung des mathematisch-naturwissenschaftlichen Doktorgrades Doctor rerum

### A simple criterion on degree sequences of graphs

Discrete Applied Mathematics 156 (2008) 3513 3517 Contents lists available at ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam Note A simple criterion on degree

### The 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

### About the inverse football pool problem for 9 games 1

Seventh International Workshop on Optimal Codes and Related Topics September 6-1, 013, Albena, Bulgaria pp. 15-133 About the inverse football pool problem for 9 games 1 Emil Kolev Tsonka Baicheva Institute

### MATH 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

### Easy reading on topology of real plane algebraic curves

Easy reading on topology of real plane algebraic curves Viatcheslav Kharlamov and Oleg Viro This is a shortened version of introduction to book Topological Properties of Real Plane Algebraic Curves by

### No: 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

### Seminario Dottorato 2009/10

Università di Padova Dipartimento di Matematica Pura ed Applicata Scuole di Dottorato in Matematica Pura e Matematica Computazionale Seminario Dottorato 2009/10 Preface 2 Abstracts (from Seminario Dottorato

### On duality of modular G-Riesz bases and G-Riesz bases in Hilbert C*-modules

Journal of Linear and Topological Algebra Vol. 04, No. 02, 2015, 53-63 On duality of modular G-Riesz bases and G-Riesz bases in Hilbert C*-modules M. Rashidi-Kouchi Young Researchers and Elite Club Kahnooj

### 3 Contour integrals and Cauchy s Theorem

3 ontour integrals and auchy s Theorem 3. Line integrals of complex functions Our goal here will be to discuss integration of complex functions = u + iv, with particular regard to analytic functions. Of

### RESONANCES AND BALLS IN OBSTACLE SCATTERING WITH NEUMANN BOUNDARY CONDITIONS

RESONANCES AND BALLS IN OBSTACLE SCATTERING WITH NEUMANN BOUNDARY CONDITIONS T. J. CHRISTIANSEN Abstract. We consider scattering by an obstacle in R d, d 3 odd. We show that for the Neumann Laplacian if

### Method of Green s Functions

Method of Green s Functions 8.303 Linear Partial ifferential Equations Matthew J. Hancock Fall 006 We introduce another powerful method of solving PEs. First, we need to consider some preliminary definitions

### CURVES WHOSE SECANT DEGREE IS ONE IN POSITIVE CHARACTERISTIC. 1. Introduction

Acta Math. Univ. Comenianae Vol. LXXXI, 1 (2012), pp. 71 77 71 CURVES WHOSE SECANT DEGREE IS ONE IN POSITIVE CHARACTERISTIC E. BALLICO Abstract. Here we study (in positive characteristic) integral curves

### THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS

THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear

### Taylor and Maclaurin Series

Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions

### LINEAR 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

### RINGS WITH A POLYNOMIAL IDENTITY

RINGS WITH A POLYNOMIAL IDENTITY IRVING KAPLANSKY 1. Introduction. In connection with his investigation of projective planes, M. Hall [2, Theorem 6.2]* proved the following theorem: a division ring D in

### Section 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

### Recall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula:

Chapter 7 Div, grad, and curl 7.1 The operator and the gradient: Recall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula: ( ϕ ϕ =, ϕ,..., ϕ. (7.1 x 1

### THE DIMENSION OF A CLOSED SUBSET OF R n AND RELATED FUNCTION SPACES

THE DIMENSION OF A CLOSED SUBSET OF R n AND RELATED FUNCTION SPACES HANS TRIEBEL and HEIKE WINKELVOSS (Jena) Dedicated to Professor Károly Tandori on his seventieth birthday 1 Introduction Let F be a closed

### 7. Cauchy s integral theorem and Cauchy s integral formula

7. Cauchy s integral theorem and Cauchy s integral formula 7.. Independence of the path of integration Theorem 6.3. can be rewritten in the following form: Theorem 7. : Let D be a domain in C and suppose

### A SURGERY RESULT FOR THE SPECTRUM OF THE DIRICHLET LAPLACIAN. Keywords: shape optimization, eigenvalues, Dirichlet Laplacian

A SURGERY RESULT FOR THE SPECTRUM OF THE DIRICHLET LAPLACIA DORI BUCUR AD DARIO MAZZOLEI Abstract. In this paper we give a method to geometrically modify an open set such that the first k eigenvalues of

### ON CERTAIN DOUBLY INFINITE SYSTEMS OF CURVES ON A SURFACE

i93 c J SYSTEMS OF CURVES 695 ON CERTAIN DOUBLY INFINITE SYSTEMS OF CURVES ON A SURFACE BY C H. ROWE. Introduction. A system of co 2 curves having been given on a surface, let us consider a variable curvilinear

### WHICH LINEAR-FRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE?

WHICH LINEAR-FRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE? JOEL H. SHAPIRO Abstract. These notes supplement the discussion of linear fractional mappings presented in a beginning graduate course

### There is no degree invariant half-jump

There is no degree invariant half-jump Rod Downey Mathematics Department Victoria University of Wellington P O Box 600 Wellington New Zealand Richard A. Shore Mathematics Department Cornell University

### DIFFERENTIABILITY OF COMPLEX FUNCTIONS. Contents

DIFFERENTIABILITY OF COMPLEX FUNCTIONS Contents 1. Limit definition of a derivative 1 2. Holomorphic functions, the Cauchy-Riemann equations 3 3. Differentiability of real functions 5 4. A sufficient condition

### MA651 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

### Classification of Cartan matrices

Chapter 7 Classification of Cartan matrices In this chapter we describe a classification of generalised Cartan matrices This classification can be compared as the rough classification of varieties in terms

### Every Positive Integer is the Sum of Four Squares! (and other exciting problems)

Every Positive Integer is the Sum of Four Squares! (and other exciting problems) Sophex University of Texas at Austin October 18th, 00 Matilde N. Lalín 1. Lagrange s Theorem Theorem 1 Every positive integer

### 11 Ideals. 11.1 Revisiting Z

11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(

### Notes on metric spaces

Notes on metric spaces 1 Introduction The purpose of these notes is to quickly review some of the basic concepts from Real Analysis, Metric Spaces and some related results that will be used in this course.

### Fiber sums of genus 2 Lefschetz fibrations

Proceedings of 9 th Gökova Geometry-Topology Conference, pp, 1 10 Fiber sums of genus 2 Lefschetz fibrations Denis Auroux Abstract. Using the recent results of Siebert and Tian about the holomorphicity

### An optimal transportation problem with import/export taxes on the boundary

An optimal transportation problem with import/export taxes on the boundary Julián Toledo Workshop International sur les Mathématiques et l Environnement Essaouira, November 2012..................... Joint

### Moving Least Squares Approximation

Chapter 7 Moving Least Squares Approimation An alternative to radial basis function interpolation and approimation is the so-called moving least squares method. As we will see below, in this method the

### 1 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

### Section 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

### A class of infinite dimensional stochastic processes

A class of infinite dimensional stochastic processes John Karlsson Linköping University CRM Barcelona, July 7, 214 Joint work with Jörg-Uwe Löbus John Karlsson (Linköping University) Infinite dimensional

### On the number-theoretic functions ν(n) and Ω(n)

ACTA ARITHMETICA LXXVIII.1 (1996) On the number-theoretic functions ν(n) and Ω(n) by Jiahai Kan (Nanjing) 1. Introduction. Let d(n) denote the divisor function, ν(n) the number of distinct prime factors,

### On Chebyshev interpolation of analytic functions

On Chebyshev interpolation of analytic functions Laurent Demanet Department of Mathematics Massachusetts Institute of Technology Lexing Ying Department of Mathematics University of Texas at Austin March

### CONSTANT-SIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE p-laplacian. Pasquale Candito and Giuseppina D Aguí

Opuscula Math. 34 no. 4 2014 683 690 http://dx.doi.org/10.7494/opmath.2014.34.4.683 Opuscula Mathematica CONSTANT-SIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE p-laplacian Pasquale

### Row Ideals and Fibers of Morphisms

Michigan Math. J. 57 (2008) Row Ideals and Fibers of Morphisms David Eisenbud & Bernd Ulrich Affectionately dedicated to Mel Hochster, who has been an inspiration to us for many years, on the occasion

### THE BANACH CONTRACTION PRINCIPLE. Contents

THE BANACH CONTRACTION PRINCIPLE ALEX PONIECKI Abstract. This paper will study contractions of metric spaces. To do this, we will mainly use tools from topology. We will give some examples of contractions,

### University of Ostrava. Fuzzy Transforms

University of Ostrava Institute for Research and Applications of Fuzzy Modeling Fuzzy Transforms Irina Perfilieva Research report No. 58 2004 Submitted/to appear: Fuzzy Sets and Systems Supported by: Grant

### Math212a1010 Lebesgue measure.

Math212a1010 Lebesgue measure. October 19, 2010 Today s lecture will be devoted to Lebesgue measure, a creation of Henri Lebesgue, in his thesis, one of the most famous theses in the history of mathematics.

### MATH 425, PRACTICE FINAL EXAM SOLUTIONS.

MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator

### FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES

FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied

### MATRIX 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 +

### 2 Fourier Analysis and Analytic Functions

2 Fourier Analysis and Analytic Functions 2.1 Trigonometric Series One of the most important tools for the investigation of linear systems is Fourier analysis. Let f L 1 be a complex-valued Lebesgue-integrable

### An Introduction to Partial Differential Equations

An Introduction to Partial Differential Equations Andrew J. Bernoff LECTURE 2 Cooling of a Hot Bar: The Diffusion Equation 2.1. Outline of Lecture An Introduction to Heat Flow Derivation of the Diffusion

### An Introduction to Partial Differential Equations in the Undergraduate Curriculum

An Introduction to Partial Differential Equations in the Undergraduate Curriculum J. Tolosa & M. Vajiac LECTURE 11 Laplace s Equation in a Disk 11.1. Outline of Lecture The Laplacian in Polar Coordinates

### On the Eigenvalues of Integral Operators

Çanaya Üniversitesi Fen-Edebiyat Faültesi, Journal of Arts and Sciences Say : 6 / Aral 006 On the Eigenvalues of Integral Operators Yüsel SOYKAN Abstract In this paper, we obtain asymptotic estimates of

### Subsets of Euclidean domains possessing a unique division algorithm

Subsets of Euclidean domains possessing a unique division algorithm Andrew D. Lewis 2009/03/16 Abstract Subsets of a Euclidean domain are characterised with the following objectives: (1) ensuring uniqueness

### ON FIBER DIAMETERS OF CONTINUOUS MAPS

ON FIBER DIAMETERS OF CONTINUOUS MAPS PETER S. LANDWEBER, EMANUEL A. LAZAR, AND NEEL PATEL Abstract. We present a surprisingly short proof that for any continuous map f : R n R m, if n > m, then there

### ISU Department of Mathematics. Graduate Examination Policies and Procedures

ISU Department of Mathematics Graduate Examination Policies and Procedures There are four primary criteria to be used in evaluating competence on written or oral exams. 1. Knowledge Has the student demonstrated