Problem Set 4. Generalized Functions
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The Hilbert Transform
(February 14, 2017) The Hilbert transform Paul Garrett [email protected] http:=/www.math.umn.edu/egarrett/ [This document is http:=/www.math.umn.edu/egarrett/m/fun/notes 2016-17/hilbert transform.pdf] 1. The principal-value functional 2. Other descriptions of the principal-value functional 3. Uniqueness of odd/even homogeneous distributions 4. Hilbert transforms 5. Examples 6. ... 7. Appendix: uniqueness of equivariant functionals 8. Appendix: distributions supported at f0g 9. Appendix: the snake lemma Formulaically, the Cauchy principal-value functional η is Z 1 f(x) Z f(x) ηf = principal-value functional of f = P:V: dx = lim dx −∞ x "!0+ jxj>" x This is a somewhat fragile presentation. In particular, the apparent integral is not a literal integral! The uniqueness proven below helps prove plausible properties like the Sokhotski-Plemelj theorem from [Sokhotski 1871], [Plemelji 1908], with a possibly unexpected leading term: Z 1 f(x) Z 1 f(x) lim dx = −iπ f(0) + P:V: dx "!0+ −∞ x + i" −∞ x See also [Gelfand-Silov 1964]. Other plausible identities are best certified using uniqueness, such as Z f(x) − f(−x) ηf = 1 dx (for f 2 C1( ) \ L1( )) 2 x R R R f(x)−f(−x) using the canonical continuous extension of x at 0. Also, 2 Z f(x) − f(0) · e−x ηf = dx (for f 2 Co( ) \ L1( )) x R R R The Hilbert transform of a function f on R is awkwardly described as a principal-value integral 1 Z 1 f(t) 1 Z f(t) (Hf)(x) = P:V: dt = lim dt π −∞ x − t π "!0+ jt−xj>" x − t with the leading constant 1/π understandable with sufficient hindsight: we will see that this adjustment makes H extend to a unitary operator on L2(R). -
Analytic Continuation of Massless Two-Loop Four-Point Functions
CERN-TH/2002-145 hep-ph/0207020 July 2002 Analytic Continuation of Massless Two-Loop Four-Point Functions T. Gehrmanna and E. Remiddib a Theory Division, CERN, CH-1211 Geneva 23, Switzerland b Dipartimento di Fisica, Universit`a di Bologna and INFN, Sezione di Bologna, I-40126 Bologna, Italy Abstract We describe the analytic continuation of two-loop four-point functions with one off-shell external leg and internal massless propagators from the Euclidean region of space-like 1 3decaytoMinkowskian regions relevant to all 1 3and2 2 reactions with one space-like or time-like! off-shell external leg. Our results can be used! to derive two-loop! master integrals and unrenormalized matrix elements for hadronic vector-boson-plus-jet production and deep inelastic two-plus-one-jet production, from results previously obtained for three-jet production in electron{positron annihilation. 1 Introduction In recent years, considerable progress has been made towards the extension of QCD calculations of jet observables towards the next-to-next-to-leading order (NNLO) in perturbation theory. One of the main ingredients in such calculations are the two-loop virtual corrections to the multi leg matrix elements relevant to jet physics, which describe either 1 3 decay or 2 2 scattering reactions: two-loop four-point functions with massless internal propagators and→ up to one off-shell→ external leg. Using dimensional regularization [1, 2] with d = 4 dimensions as regulator for ultraviolet and infrared divergences, the large number of different integrals6 appearing in the two-loop Feynman amplitudes for 2 2 scattering or 1 3 decay processes can be reduced to a small number of master integrals. -
GENERALIZED FUNCTIONS LECTURES Contents 1. the Space
GENERALIZED FUNCTIONS LECTURES Contents 1. The space of generalized functions on Rn 2 1.1. Motivation 2 1.2. Basic definitions 3 1.3. Derivatives of generalized functions 5 1.4. The support of generalized functions 6 1.5. Products and convolutions of generalized functions 7 1.6. Generalized functions on Rn 8 1.7. Generalized functions and differential operators 9 1.8. Regularization of generalized functions 9 n 2. Topological properties of Cc∞(R ) 11 2.1. Normed spaces 11 2.2. Topological vector spaces 12 2.3. Defining completeness 14 2.4. Fréchet spaces 15 2.5. Sequence spaces 17 2.6. Direct limits of Fréchet spaces 18 2.7. Topologies on the space of distributions 19 n 3. Geometric properties of C−∞(R ) 21 3.1. Sheaf of distributions 21 3.2. Filtration on spaces of distributions 23 3.3. Functions and distributions on a Cartesian product 24 4. p-adic numbers and `-spaces 26 4.1. Defining p-adic numbers 26 4.2. Misc. -not sure what to do with them (add to an appendix about p-adic numbers?) 29 4.3. p-adic expansions 31 4.4. Inverse limits 32 4.5. Haar measure and local fields 32 4.6. Some basic properties of `-spaces. 33 4.7. Distributions on `-spaces 35 4.8. Distributions supported on a subspace 35 5. Vector valued distributions 37 Date: October 7, 2018. 1 GENERALIZED FUNCTIONS LECTURES 2 5.1. Smooth measures 37 5.2. Generalized functions versus distributions 38 5.3. Some linear algebra 39 5.4. Generalized functions supported on a subspace 41 6. -
An Explicit Formula for Dirichlet's L-Function
University of Tennessee at Chattanooga UTC Scholar Student Research, Creative Works, and Honors Theses Publications 5-2018 An explicit formula for Dirichlet's L-Function Shannon Michele Hyder University of Tennessee at Chattanooga, [email protected] Follow this and additional works at: https://scholar.utc.edu/honors-theses Part of the Mathematics Commons Recommended Citation Hyder, Shannon Michele, "An explicit formula for Dirichlet's L-Function" (2018). Honors Theses. This Theses is brought to you for free and open access by the Student Research, Creative Works, and Publications at UTC Scholar. It has been accepted for inclusion in Honors Theses by an authorized administrator of UTC Scholar. For more information, please contact [email protected]. An Explicit Formula for Dirichlet's L-Function Shannon M. Hyder Departmental Honors Thesis The University of Tennessee at Chattanooga Department of Mathematics Thesis Director: Dr. Andrew Ledoan Examination Date: April 9, 2018 Members of Examination Committee Dr. Andrew Ledoan Dr. Cuilan Gao Dr. Roger Nichols c 2018 Shannon M. Hyder ALL RIGHTS RESERVED i Abstract An Explicit Formula for Dirichlet's L-Function by Shannon M. Hyder The Riemann zeta function has a deep connection to the distribution of primes. In 1911 Landau proved that, for every fixed x > 1, X T xρ = − Λ(x) + O(log T ) 2π 0<γ≤T as T ! 1. Here ρ = β + iγ denotes a complex zero of the zeta function and Λ(x) is an extension of the usual von Mangoldt function, so that Λ(x) = log p if x is a positive integral power of a prime p and Λ(x) = 0 for all other real values of x. -
Bernstein's Analytic Continuation of Complex Powers of Polynomials
Bernsteins analytic continuation of complex p owers c Paul Garrett garrettmathumnedu version January Analytic continuation of distributions Statement of the theorems on analytic continuation Bernsteins pro ofs Pro of of the Lemma the Bernstein p olynomial Pro of of the Prop osition estimates on zeros Garrett Bernsteins analytic continuation of complex p owers Let f b e a p olynomial in x x with real co ecients For complex s let n s f b e the function dened by s s f x f x if f x s f x if f x s Certainly for s the function f is lo cally integrable For s in this range s we can dened a distribution denoted by the same symb ol f by Z s s f x x dx f n R n R the space of compactlysupp orted smo oth realvalued where is in C c n functions on R s The ob ject is to analytically continue the distribution f as a meromorphic distributionvalued function of s This typ e of question was considered in several provo cative examples in IM Gelfand and GE Shilovs Generalized Functions volume I One should also ask ab out analytic continuation as a temp ered distribution In a lecture at the Amsterdam Congress IM Gelfand rened this question to require further that one show that the p oles lie in a nite numb er of arithmetic progressions Bernstein proved the result in under a certain regularity hyp othesis on the zeroset of the p olynomial f Published in Journal of Functional Analysis and Its Applications translated from Russian The present discussion includes some background material from complex function theory and -
[Math.AG] 2 Jul 2015 Ic Oetime: Some and Since Time, of People
MONODROMY AND NORMAL FORMS FABRIZIO CATANESE Abstract. We discuss the history of the monodromy theorem, starting from Weierstraß, and the concept of monodromy group. From this viewpoint we compare then the Weierstraß, the Le- gendre and other normal forms for elliptic curves, explaining their geometric meaning and distinguishing them by their stabilizer in PSL(2, Z) and their monodromy. Then we focus on the birth of the concept of the Jacobian variety, and the geometrization of the theory of Abelian functions and integrals. We end illustrating the methods of complex analysis in the simplest issue, the difference equation f(z)= g(z + 1) g(z) on C. − Contents Introduction 1 1. The monodromy theorem 3 1.1. Riemann domain and sheaves 6 1.2. Monodromy or polydromy? 6 2. Normalformsandmonodromy 8 3. Periodic functions and Abelian varieties 16 3.1. Cohomology as difference equations 21 References 23 Introduction In Jules Verne’s novel of 1874, ‘Le Tour du monde en quatre-vingts jours’ , Phileas Fogg is led to his remarkable adventure by a bet made arXiv:1507.00711v1 [math.AG] 2 Jul 2015 in his Club: is it possible to make a tour of the world in 80 days? Idle questions and bets can be very stimulating, but very difficult to answer when they deal with the history of mathematics, and one asks how certain ideas, which have been a common knowledge for long time, did indeed evolve and mature through a long period of time, and through the contributions of many people. In short, there are three idle questions which occupy my attention since some time: Date: July 3, 2015. -
Chapter 4 the Riemann Zeta Function and L-Functions
Chapter 4 The Riemann zeta function and L-functions 4.1 Basic facts We prove some results that will be used in the proof of the Prime Number Theorem (for arithmetic progressions). The L-function of a Dirichlet character χ modulo q is defined by 1 X L(s; χ) = χ(n)n−s: n=1 P1 −s We view ζ(s) = n=1 n as the L-function of the principal character modulo 1, (1) (1) more precisely, ζ(s) = L(s; χ0 ), where χ0 (n) = 1 for all n 2 Z. We first prove that ζ(s) has an analytic continuation to fs 2 C : Re s > 0gnf1g. We use an important summation formula, due to Euler. Lemma 4.1 (Euler's summation formula). Let a; b be integers with a < b and f :[a; b] ! C a continuously differentiable function. Then b X Z b Z b f(n) = f(x)dx + f(a) + (x − [x])f 0(x)dx: n=a a a 105 Remark. This result often occurs in the more symmetric form b Z b Z b X 1 1 0 f(n) = f(x)dx + 2 (f(a) + f(b)) + (x − [x] − 2 )f (x)dx: n=a a a Proof. Let n 2 fa; a + 1; : : : ; b − 1g. Then Z n+1 Z n+1 x − [x]f 0(x)dx = (x − n)f 0(x)dx n n h in+1 Z n+1 Z n+1 = (x − n)f(x) − f(x)dx = f(n + 1) − f(x)dx: n n n By summing over n we get b Z b X Z b (x − [x])f 0(x)dx = f(n) − f(x)dx; a n=a+1 a which implies at once Lemma 4.1. -
Introduction to Analytic Number Theory More About the Gamma Function We Collect Some More Facts About Γ(S)
Math 259: Introduction to Analytic Number Theory More about the Gamma function We collect some more facts about Γ(s) as a function of a complex variable that will figure in our treatment of ζ(s) and L(s, χ). All of these, and most of the Exercises, are standard textbook fare; one basic reference is Ch. XII (pp. 235–264) of [WW 1940]. One reason for not just citing Whittaker & Watson is that some of the results concerning Euler’s integrals B and Γ have close analogues in the Gauss and Jacobi sums associated to Dirichlet characters, and we shall need these analogues before long. The product formula for Γ(s). Recall that Γ(s) has simple poles at s = 0, −1, −2,... and no zeros. We readily concoct a product that has the same behavior: let ∞ 1 Y . s g(s) := es/k 1 + , s k k=1 the product converging uniformly in compact subsets of C − {0, −1, −2,...} because ex/(1 + x) = 1 + O(x2) for small x. Then Γ/g is an entire function with neither poles nor zeros, so it can be written as exp α(s) for some entire function α. We show that α(s) = −γs, where γ = 0.57721566490 ... is Euler’s constant: N X 1 γ := lim − log N + . N→∞ k k=1 That is, we show: Lemma. The Gamma function has the product formulas ∞ N ! e−γs Y . s 1 Y k Γ(s) = e−γsg(s) = es/k 1 + = lim N s . (1) s k s N→∞ s + k k=1 k=1 Proof : For s 6= 0, −1, −2,..., the quotient g(s + 1)/g(s) is the limit as N→∞ of N N ! N s Y 1 + s s X 1 Y k + s e1/k k = exp s + 1 1 + s+1 s + 1 k k + s + 1 k=1 k k=1 k=1 N ! N X 1 = s · · exp − log N + . -
Appendix B the Fourier Transform of Causal Functions
Appendix A Distribution Theory We expect the reader to have some familiarity with the topics of distribu tion theory and Fourier transforms, as well as the theory of functions of a complex variable. However, to provide a bridge between texts that deal with these topics and this textbook, we have included several appendices. It is our intention that this appendix, and those that follow, perform the func tion of outlining our notational conventions, as well as providing a guide that the reader may follow in finding relevant topics to study to supple ment this text. As a result, some precision in language has been sacrificed in favor of understandability through heuristic descriptions. Our main goals in this appendix are to provide background material as well as mathematical justifications for our use of "singular functions" and "bandlimited delta functions," which are distributional objects not normally discussed in textbooks. A.l Introduction Distributions provide a mathematical framework that can be used to satisfy two important needs in the theory of partial differential equations. First, quantities that have point duration and/or act at a point location may be described through the use of the traditional "Dirac delta function" representation. Such quantities find a natural place in representing energy sources as the forcing functions of partial differential equations. In this usage, distributions can "localize" the values of functions to specific spatial 390 A. Distribution Theory and/or temporal values, representing the position and time at which the source acts in the domain of a problem under consideration. The second important role of distributions is to extend the process of differentiability to functions that fail to be differentiable in the classical sense at isolated points in their domains of definition. -
Schwartz Functions on Nash Manifolds and Applications to Representation Theory
SCHWARTZ FUNCTIONS ON NASH MANIFOLDS AND APPLICATIONS TO REPRESENTATION THEORY AVRAHAM AIZENBUD Joint with Dmitry Gourevitch and Eitan Sayag arXiv:0704.2891 [math.AG], arxiv:0709.1273 [math.RT] Let us start with the following motivating example. Consider the circle S1, let N ⊂ S1 be the north pole and denote U := S1 n N. Note that U is diffeomorphic to R via the stereographic projection. Consider the space D(S1) of distributions on S1, that is the space of continuous linear functionals on the 1 1 1 Fr´echet space C (S ). Consider the subspace DS1 (N) ⊂ D(S ) consisting of all distributions supported 1 at N. Then the quotient D(S )=DS1 (N) will not be the space of distributions on U. However, it will be the space S∗(U) of Schwartz distributions on U, that is continuous functionals on the Fr´echet space S(U) of Schwartz functions on U. In this case, S(U) can be identified with S(R) via the stereographic projection. The space of Schwartz functions on R is defined to be the space of all infinitely differentiable functions that rapidly decay at infinity together with all their derivatives, i.e. xnf (k) is bounded for any n; k. In this talk we extend the notions of Schwartz functions and Schwartz distributions to a larger geometric realm. As we can see, the definition is of algebraic nature. Hence it would not be reasonable to try to extend it to arbitrary smooth manifolds. However, it is reasonable to extend this notion to smooth algebraic varieties. -
Fundamental Theorems in Mathematics
SOME FUNDAMENTAL THEOREMS IN MATHEMATICS OLIVER KNILL Abstract. An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of 243 theorems are whether the result can be formulated elegantly, whether it is beautiful or useful and whether it could serve as a guide [6] without leading to panic. The order is not a ranking but ordered along a time-line when things were writ- ten down. Since [556] stated “a mathematical theorem only becomes beautiful if presented as a crown jewel within a context" we try sometimes to give some context. Of course, any such list of theorems is a matter of personal preferences, taste and limitations. The num- ber of theorems is arbitrary, the initial obvious goal was 42 but that number got eventually surpassed as it is hard to stop, once started. As a compensation, there are 42 “tweetable" theorems with included proofs. More comments on the choice of the theorems is included in an epilogue. For literature on general mathematics, see [193, 189, 29, 235, 254, 619, 412, 138], for history [217, 625, 376, 73, 46, 208, 379, 365, 690, 113, 618, 79, 259, 341], for popular, beautiful or elegant things [12, 529, 201, 182, 17, 672, 673, 44, 204, 190, 245, 446, 616, 303, 201, 2, 127, 146, 128, 502, 261, 172]. For comprehensive overviews in large parts of math- ematics, [74, 165, 166, 51, 593] or predictions on developments [47]. For reflections about mathematics in general [145, 455, 45, 306, 439, 99, 561]. Encyclopedic source examples are [188, 705, 670, 102, 192, 152, 221, 191, 111, 635]. -
Mathematical Problems on Generalized Functions and the Can–
Mathematical problems on generalized functions and the canonical Hamiltonian formalism J.F.Colombeau [email protected] Abstract. This text is addressed to mathematicians who are interested in generalized functions and unbounded operators on a Hilbert space. We expose in detail (in a “formal way”- as done by Heisenberg and Pauli - i.e. without mathematical definitions and then, of course, without mathematical rigour) the Heisenberg-Pauli calculations on the simplest model close to physics. The problem for mathematicians is to give a mathematical sense to these calculations, which is possible without any knowledge in physics, since they mimick exactly usual calculations on C ∞ functions and on bounded operators, and can be considered at a purely mathematical level, ignoring physics in a first step. The mathematical tools to be used are nonlinear generalized functions, unbounded operators on a Hilbert space and computer calculations. This text is the improved written version of a talk at the congress on linear and nonlinear generalized functions “Gf 07” held in Bedlewo-Poznan, Poland, 2-8 September 2007. 1-Introduction . The Heisenberg-Pauli calculations (1929) [We,p20] are a set of 3 or 4 pages calculations (in a simple yet fully representative model) that are formally quite easy and mimick calculations on C ∞ functions. They are explained in detail at the beginning of this text and in the appendices. The H-P calculations [We, p20,21, p293-336] are a basic formulation in Quantum Field Theory: “canonical Hamiltonian formalism”, see [We, p292] for their relevance. The canonical Hamiltonian formalism is considered as mainly equivalent to the more recent “(Feynman) path integral formalism”: see [We, p376,377] for the connections between the 2 formalisms that complement each other.