Introduction to Hyperfunctions and Their Integral Transforms
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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). -
Dimension Formulas for the Hyperfunction Solutions to Holonomic D-Modules
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector ARTICLE IN PRESS Advances in Mathematics 180 (2003) 134–145 http://www.elsevier.com/locate/aim Dimension formulas for the hyperfunction solutions to holonomic D-modules Kiyoshi Takeuchi Institute of Mathematics, University of Tsukuba, 1-1-1, Tennodai, Tsukuba, Ibaraki 305-8571, Japan Received 15 July 2002; accepted 19 November 2002 Communicated by Takahiro Kawai Dedicated to Professor Mitsuo Morimoto on the Occasion of his 60th Birthday Abstract In this paper, we will prove an explicit dimension formula for the hyperfunction solutions to a class of holonomic D-modules. This dimension formula can be considered as a higher dimensional analogue of a beautiful theorem on ordinary differential equations due to Kashiwara (Master’s Thesis, University of Tokyo, 1970) and Komatsu (J. Fac. Sci. Univ. Tokyo Math. Sect. IA 18 (1971) 379). In the course of the proof, we will make use of a recent innovation of Schmid–Vilonen (Invent. Math. 124 (1996) 451) in the representation theory (in the theory of index theorems for constructible sheaves). r 2003 Elsevier Science (USA). All rights reserved. MSC: 32C38; 35A27 Keywords: D-module; Holonomic system; Index theorem 1. Introduction The aim of this paper is to prove a useful formula to compute the exact dimensions of the solutions to holonomic D-modules. The study of the dimension formula for the generalized function solutions to ordinary differential equations started only recently. In the 1970s, a beautiful theorem was obtained by Kashiwara [7] and Komatsu [12] using the hyperfunction theory. -
Design and Application of a Hilbert Transformer in a Digital Receiver
Proceedings of the SDR 11 Technical Conference and Product Exposition, Copyright © 2011 Wireless Innovation Forum All Rights Reserved DESIGN AND APPLICATION OF A HILBERT TRANSFORMER IN A DIGITAL RECEIVER Matt Carrick (Northrop Grumman, Chantilly, VA, USA; [email protected]); Doug Jaeger (Northrop Grumman, Chantilly, VA, USA; [email protected]); fred harris (San Diego State University, San Diego, CA, USA; [email protected]) ABSTRACT A common method of down converting a signal from an intermediate frequency (IF) to baseband is using a quadrature down-converter. One problem with the quadrature down-converter is it requires two low pass filters; one for the real branch and one for the imaginary branch. A more efficient way is to transform the real signal to a complex signal and then complex heterodyne the resultant signal to baseband. The transformation of a real signal to a complex signal can be done using a Hilbert transform. Building a Hilbert transform directly from its sampled data sequence produces suboptimal results due to time series truncation; another method is building a Hilbert transformer by synthesizing the filter coefficients from half Figure 1: A quadrature down-converter band filter coefficients. Designing the Hilbert transform filter using a half band filter allows for a much more Another way of viewing the problem is that the structured design process as well as greatly improved quadrature down-converter not only extracts the desired results. segment of the spectrum it rejects the undesired spectral image, the spectral replica present in the Hermetian 1. INTRODUCTION symmetric spectra of a real signal. Removing this image would result in a single sided spectrum which being non- The digital portion of a receiver is typically designed to Hermetian symmetric is the transform of a complex signal. -
Communication Theory II Slides 04
Communication Theory II Lecture 4: Review on Fourier analysis and probabilty theory Ahmed Elnakib, PhD Assistant Professor, Mansoura University, Egypt Febraury 19th, 2015 1 Course Website o http://lms.mans.edu.eg/eng/ o The site contains the lectures, quizzes, homework, and open forums for feedback and questions o Log in using your name and password o Password for quizzes: third o One page quiz: for less download time 2 Lecture Outlines o Review on Fourier analysis of signals and systems . The Dirac delta function . Fourier transform of periodic signals . Transmission of signals through LTI systems . Hilbert transform o Review on probability theory . Deterministic vs. probabilistic mathematical models . Probability theory, random variables, and the distribution functions . The concept of expectation and second order statistics . Characteristic function, the center limit theory and the Bayesian interface 3 The Dirac Delta Function (Unit Impulse) δ(t) t 0 o An even function of time t, centered at the origin t = 0 o Sifting property: sifts out the value g(t0) of the function g(t) at time t = t0, where o Replication property: convolution of any function with the delta function leaves that function unchanged 4 The Dirac Delta Function (cont’d) W=1 Amplitude W=2 Amplitude W=5 f(t)=δ(t) F(f)=1 1 t Amplitude 0 0 f5 The Dirac Delta Function (cont’d) T→0 The delta function may be viewed Rectangular impulse as the limiting form of a pulse of unit area (symmetric with respect to the origin) as the duration of the pulse approaches zero Sinc impulse T=1/2W 6 Existence of Fourier Transform o Physical realizability is a sufficient condition for the existence of a Fourier transform (e.g., all energy signals are Fourier transformable ). -
On Some Integral Operators Appearing in Scattering Theory, And
On some integral operators appearing in scattering theory, and their resolutions S. Richard,∗ T. Umeda† Graduate school of mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan Department of Mathematical Sciences, University of Hyogo, Shosha, Himeji 671-2201, Japan E-mail: [email protected], [email protected] Abstract We discuss a few integral operators and provide expressions for them in terms of smooth functions of some natural self-adjoint operators. These operators appear in the context of scattering theory, but are independent of any perturbation theory. The Hilbert transform, the Hankel transform, and the finite interval Hilbert transform are among the operators considered. 2010 Mathematics Subject Classification: 47G10 Keywords: integral operators, Hilbert transform, Hankel transform, dilation operator, scattering theory 1 Introduction Investigations on the wave operators in the context of scattering theory have a long history, and several powerful technics have been developed for the proof of their existence and of their completeness. More recently, properties of these operators in various spaces have been studied, and the importance of these operators for non-linear problems has also been acknowledged. A quick search on MathSciNet shows that the terms wave operator(s) appear in the title of numerous papers, confirming their importance in various fields of mathematics. For the last decade, the wave operators have also played a key role for the arXiv:1909.01712v1 [math-ph] 4 Sep 2019 search of index theorems in scattering theory, as a tool linking the scattering part of a physical system to its bound states. For such investigations, a very detailed ∗Supported by the grantTopological invariants through scattering theory and noncommuta- tive geometry from Nagoya University, and by JSPS Grant-in-Aid for scientific research (C) no 18K03328, and on leave of absence from Univ. -
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Arxiv:1611.05269V3 [Cs.IT] 29 Jan 2018 Graph Analytic Signal, and Associated Amplitude and Frequency Modulations Reveal Com
On Hilbert Transform, Analytic Signal, and Modulation Analysis for Signals over Graphs Arun Venkitaraman, Saikat Chatterjee, Peter Handel¨ Department of Information Science and Engineering, School of Electrical Engineering and ACCESS Linnaeus Center KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden . Abstract We propose Hilbert transform and analytic signal construction for signals over graphs. This is motivated by the popularity of Hilbert transform, analytic signal, and mod- ulation analysis in conventional signal processing, and the observation that comple- mentary insight is often obtained by viewing conventional signals in the graph setting. Our definitions of Hilbert transform and analytic signal use a conjugate-symmetry-like property exhibited by the graph Fourier transform (GFT), resulting in a ’one-sided’ spectrum for the graph analytic signal. The resulting graph Hilbert transform is shown to possess many interesting mathematical properties and also exhibit the ability to high- light anomalies/discontinuities in the graph signal and the nodes across which signal discontinuities occur. Using the graph analytic signal, we further define amplitude, phase, and frequency modulations for a graph signal. We illustrate the proposed con- cepts by showing applications to synthesized and real-world signals. For example, we show that the graph Hilbert transform can indicate presence of anomalies and that arXiv:1611.05269v3 [cs.IT] 29 Jan 2018 graph analytic signal, and associated amplitude and frequency modulations reveal com- plementary information in speech signals. Keywords: Graph signal, analytic signal, Hilbert transform, demodulation, anomaly detection. Email addresses: [email protected] (Arun Venkitaraman), [email protected] (Saikat Chatterjee), [email protected] (Peter Handel)¨ Preprint submitted to Signal Processing 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 (a) (b) Figure 1: Anomaly highlighting behavior of the graph Hilbert transform for 2D image signal graph. -
Hilbert Transform and Singular Integrals on the Spaces of Tempered Ultradistributions
ALGEBRAIC ANALYSIS AND RELATED TOPICS BANACH CENTER PUBLICATIONS, VOLUME 53 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 2000 HILBERT TRANSFORM AND SINGULAR INTEGRALS ON THE SPACES OF TEMPERED ULTRADISTRIBUTIONS ANDRZEJKAMINSKI´ Institute of Mathematics, University of Rzesz´ow Rejtana 16 C, 35-310 Rzesz´ow,Poland E-mail: [email protected] DUSANKAPERIˇ SIˇ C´ Institute of Mathematics, University of Novi Sad Trg Dositeja Obradovi´ca4, 21000 Novi Sad, Yugoslavia E-mail: [email protected] STEVANPILIPOVIC´ Institute of Mathematics, University of Novi Sad Trg Dositeja Obradovi´ca4, 21000 Novi Sad, Yugoslavia E-mail: [email protected] Abstract. The Hilbert transform on the spaces S0∗(Rd) of tempered ultradistributions is defined, uniquely in the sense of hyperfunctions, as the composition of the classical Hilbert transform with the operators of multiplying and dividing a function by a certain elliptic ultra- polynomial. We show that the Hilbert transform of tempered ultradistributions defined in this way preserves important properties of the classical Hilbert transform. We also give definitions and prove properties of singular integral operators with odd and even kernels on the spaces S0∗(Rd), whose special cases are the Hilbert transform and Riesz operators. 1. Introduction. The Hilbert transform on distribution and ultradistribution spaces has been studied by many mathematicians, see e.g. Tillmann [17], Beltrami and Wohlers [1], Vladimirov [18], Singh and Pandey [15], Ishikawa [2], Ziemian [20] and Pilipovi´c[11]. In all these papers the Hilbert transform is defined by one of the two methods: by the method of adjoints or by considering a generalized function on the kernel which belongs to the corresponding test function space. -
The Hilbert Transform
The Hilbert transform 1 Definition and properties 1 Recall the distribution pv( x ), defined by Z '(x) pv(1=x)(') := lim dx: !0 jx|≥ x The Hilbert transform is defined via the convolution with pv(1=x), namely 1 Z f(x − t) (Hf)(x) := lim dt: π !0 jt|≥ t The main theorem we are going to prove in this note is the following. Theorem 1.1. For any p 2 (1; +1), there exists a constant Cp such that kHfkp < Cpkfkp (1.1) for all f 2 S(R). Thus, H extends to a bounded linear operator on all of Lp(R). The first remark we make is that the above theorem is false when p = 1. To see this, note that for smooth f with compact support, when x is very large, the main contribution to the integral Z f(x − t) dt t comes from the values t which are not far away from x. This in general gives the decay 1 rate of order jxj , unless there is a magical cancellation due to the sign changes of f, in which case one can hope for a faster decay. Indeed, it is not hard to show that if f is continuous and has compact support with R f(t)dt = a, then a (Hf)(x) = + O(1=x2) πx 1 R for all large x. As a consequence, Hf 2 L (R) if and only if f = 0. The statement of 1 the above theorem is also not true for p = +1. The reason is that x is not integrable at infinity, so one can not bound on kHfk1 merely by using the maximum of f without any other information (e.g., the size of the support). -
Ballistic Electron Transport in Graphene Nanodevices and Billiards
Ballistic Electron Transport in Graphene Nanodevices and Billiards Dissertation (Cumulative Thesis) for the award of the degree Doctor rerum naturalium of the Georg-August-Universität Göttingen within the doctoral program Physics of Biological and Complex Systems of the Georg-August University School of Science submitted by George Datseris from Athens, Greece Göttingen 2019 Thesis advisory committee Prof. Dr. Theo Geisel Department of Nonlinear Dynamics Max Planck Institute for Dynamics and Self-Organization Prof. Dr. Stephan Herminghaus Department of Dynamics of Complex Fluids Max Planck Institute for Dynamics and Self-Organization Dr. Michael Wilczek Turbulence, Complex Flows & Active Matter Max Planck Institute for Dynamics and Self-Organization Examination board Prof. Dr. Theo Geisel (Reviewer) Department of Nonlinear Dynamics Max Planck Institute for Dynamics and Self-Organization Prof. Dr. Stephan Herminghaus (Second Reviewer) Department of Dynamics of Complex Fluids Max Planck Institute for Dynamics and Self-Organization Dr. Michael Wilczek Turbulence, Complex Flows & Active Matter Max Planck Institute for Dynamics and Self-Organization Prof. Dr. Ulrich Parlitz Biomedical Physics Max Planck Institute for Dynamics and Self-Organization Prof. Dr. Stefan Kehrein Condensed Matter Theory, Physics Department Georg-August-Universität Göttingen Prof. Dr. Jörg Enderlein Biophysics / Complex Systems, Physics Department Georg-August-Universität Göttingen Date of the oral examination: September 13th, 2019 Contents Abstract 3 1 Introduction 5 1.1 Thesis synopsis and outline.............................. 10 2 Fundamental Concepts 13 2.1 Nonlinear dynamics of antidot superlattices..................... 13 2.2 Lyapunov exponents in billiards............................ 20 2.3 Fundamental properties of graphene......................... 22 2.4 Why quantum?..................................... 31 2.5 Transport simulations and scattering wavefunctions................ -
An Elementary Theory of Hyperfunctions and Microfunctions
PARTIAL DIFFERENTIAL EQUATIONS BANACH CENTER PUBLICATIONS, VOLUME 27 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 1992 AN ELEMENTARY THEORY OF HYPERFUNCTIONS AND MICROFUNCTIONS HIKOSABURO KOMATSU Department of Mathematics, Faculty of Science, University of Tokyo Tokyo, 113 Japan 1. Sato’s definition of hyperfunctions. The hyperfunctions are a class of generalized functions introduced by M. Sato [34], [35], [36] in 1958–60, only ten years later than Schwartz’ distributions [40]. As we will see, hyperfunctions are natural and useful, but unfortunately they are not so commonly used as distributions. One reason seems to be that the mere definition of hyperfunctions needs a lot of preparations. In the one-dimensional case his definition is elementary. Let Ω be an open set in R. Then the space (Ω) of hyperfunctions on Ω is defined to be the quotient space B (1.1) (Ω)= (V Ω)/ (V ) , B O \ O where V is an open set in C containing Ω as a closed set, and (V Ω) (resp. (V )) is the space of all holomorphic functions on V Ω (resp. VO). The\ hyper- functionO f(x) represented by F (z) (V Ω) is written\ ∈ O \ (1.2) f(x)= F (x + i0) F (x i0) − − and has the intuitive meaning of the difference of the “boundary values”of F (z) on Ω from above and below. V Ω Fig. 1 [233] 234 H. KOMATSU The main properties of hyperfunctions are the following: (1) (Ω), Ω R, form a sheaf over R. B ⊂ Ω Namely, for all pairs Ω1 Ω of open sets the restriction mappings ̺Ω1 : (Ω) (Ω ) are defined, and⊂ for any open covering Ω = Ω they satisfy the B →B 1 α following conditions: S (S.1) If f (Ω) satisfies f = 0 for all α, then f = 0; ∈B |Ωα (S.2) If fα (Ωα) satisfy fα Ωα∩Ωβ = fβ Ωα∩Ωβ for all Ωα Ωβ = , then there is∈ Ban f (Ω) such| that f = f| . -
Weierstrass Transform on Howell's Space-A New Theory
IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 16, Issue 6 Ser. IV (Nov. – Dec. 2020), PP 57-62 www.iosrjournals.org Weierstrass Transform on Howell’s Space-A New Theory 1 2* 3 V. N. Mahalle , S.S. Mathurkar , R. D. Taywade 1(Associate Professor, Dept. of Mathematics, Bar. R.D.I.K.N.K.D. College, Badnera Railway (M.S), India.) 2*(Assistant Professor, Dept. of Mathematics, Govt. College of Engineering, Amravati, (M.S), India.) 3(Associate Professor, Dept. of Mathematics, Prof. Ram Meghe Institute of Technology & Research, Badnera, Amravati, (M.S), India.) Abstract: In this paper we have explained a new theory of Weierstrass transform defined on Howell’s space. The definition of the Weierstrass transform on this space of test functions is discussed. Some Fundamental theorems are proved and some results are developed. Weierstrass transform defined on Howell’s space is linear, continuous and one-one mapping from G to G c is also discussed. Key Word: Weierstrass transform, Howell’s Space, Fundamental theorem. --------------------------------------------------------------------------------------------------------------------------------------- Date of Submission: 10-12-2020 Date of Acceptance: 25-12-2020 --------------------------------------------------------------------------------------------------------------------------------------- I. Introduction A new theory of Fourier analysis developed by Howell, presented in an elegant series of papers [2-6] n supporting the Fourier transformation of the functions on R . This theory is similar to the theory for tempered distributions and to some extent, can be viewed as an extension of the Fourier theory. Bhosale [1] had extended Fractional Fourier transform on Howell’s space. Mahalle et. al. [7, 8] described a new theory of Mellin and Laplace transform defined on Howell’s space. -
Memristor-Based Approximation of Gaussian Filter
Memristor-based Approximation of Gaussian Filter Alex Pappachen James, Aidyn Zhambyl, Anju Nandakumar Electrical and Computer Engineering department Nazarbayev University School of Engineering Astana, Kazakhstan Email: [email protected], [email protected], [email protected] Abstract—Gaussian filter is a filter with impulse response of fourth section some mathematical analysis will be provided to Gaussian function. These filters are useful in image processing obtained results and finally the paper will be concluded by of 2D signals, as it removes unnecessary noise. Also, they could stating the result of the done work. be helpful for data transmission (e.g. GMSK modulation). In practice, the Gaussian filters could be approximately designed II. GAUSSIAN FILTER AND MEMRISTOR by several methods. One of these methods are to construct Gaussian-like filter with the help of memristors and RLC circuits. A. Gaussian Filter Therefore, the objective of this project is to find and design Gaussian filter is such filter which convolves the input signal appropriate model of Gaussian-like filter, by using mentioned devices. Finally, one possible model of Gaussian-like filter based with the impulse response of Gaussian function. In some on memristor designed and analysed in this paper. sources this process is also known as the Weierstrass transform [2]. The Gaussian function is given as in equation 1, where µ Index Terms—Gaussian filters, analog filter, memristor, func- is the time shift and σ is the scale. For input signal of x(t) tion approximation the output is the convolution of x(t) with Gaussian, as shown in equation 2.