DOCSLIB.ORG

The Hilbert Transform

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

(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 {0} 9. Appendix: the snake lemma

Formulaically, the Cauchy principal-value functional η is

  • Z
  • Z

∞

  • f(x)
  • f(x)

  • ηf = principal-value functional of f = P.V.
  • dx = lim

dx x

ε→0+

x

−∞

|x|>ε

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
  • Z

  • ∞
  • ∞

  • f(x)
  • f(x)

lim

  • dx = −iπ f(0) + P.V.
  • dx

ε→0+

  • x + iε
  • x

  • −∞
  • −∞

See also [Gelfand-Silov 1964]. Other plausible identities are best certified using uniqueness, such as

Z

f(x) − f(−x)

12

  • ηf =
  • dx

(for f ∈ C1(R) ∩ L1(R))

x

R

f(x)−f(−x)

  • using the canonical continuous extension of
  • at 0. Also,

x

Z

2

f(x) − f(0) · e−x

  • ηf =
  • dx

(for f ∈ Co(R) ∩ L1(R))

x

R

The Hilbert transform of a function f on R is awkwardly described as a principal-value integral

  • Z
  • Z

∞

1
(Hf)(x) = P.V.

π

  • f(t)
  • 1
  • f(t)

dt =

lim

dt x − t π ε→0+ x − t

|t−x|>ε

−∞

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). The formulaic presentation of H makes it appear to be a convolution with (a constant multiple of) the principal-value functional. The fragility of these presentations make existence, continuity properties, and range of applicability of H less clear than one would like.

1

Paul Garrett: The Hilbert transform (February 14, 2017)

1. The principal-value functional

The principal-value functional η is better characterized as the unique (up to a constant multiple) odd distribution on R, positive-homogeneous of degree 0 as a distribution (see below). This characterization allows unambiguous comparison of various limiting expressions closely related to the principal-value functional. Further, the characterization of the principal-value functional makes discussion of the Hilbert transform less fragile and more convincing.

Again, let

  • Z
  • Z

∞

  • f(x)
  • f(x)

  • ηf = principal-value functional of f = P.V.
  • dx = lim

dx x

ε→0+

x

−∞

|x|>ε

We prove that η is a tempered distribution: [1.1] Claim: For f ∈ S , for every h > 0,

|η(f)| ≤

Z
ꢀ

ꢀꢀ
ꢀꢀꢀ

f(x) dx + 2h sup |f0(x)|

x

  • |x|≤h
  • |x|≥h

Thus, u is a tempered distribution:

  • |η(f)| ꢀ sup(1 + x2)|f(x)| + sup |f0(x)|
  • (implied constant independent of f)

  • x
  • x

Proof: Certainly

  • Z
  • Z

ꢀ

ꢀꢀ
ꢀꢀꢀ
ꢀꢀꢀ
ꢀꢀꢀ

  • f(x)
  • f(x)

  • |η(f)| =
  • dx + lim

dx x

ε→0+

x

|x|≥h

ε<|x|≤h

By the Mean Value Theorem, f(x) = f(0) + x f0(ξx) for some ξx between 0 and x. Thus,

  • Z
  • Z
  • Z
  • Z

  • f(x)
  • f(0)

  • dx =
  • dx +

f(ξx) dx = 0 +

f(ξx) dx

  • x
  • x

  • ε<|x|≤h
  • ε<|x|≤h
  • ε<|x|≤h
  • ε<|x|≤h

because 1/x is odd and x → f(0) is even. Thus,

  • Z
  • Z
  • Z

ꢀ

ꢀꢀ
ꢀꢀꢀ
ꢀꢀꢀ
ꢀꢀꢀ

f(x)

dx

=

f(ξx) dx

≤

|f(ξx)| dx x

  • ε<|x|≤h
  • ε<|x|≤h
  • ε<|x|≤h

≤ 2h · sup |f0(x)| ≤ 2h · sup |f0(x)|

ε<|x|≤h

|x|≤h

The latter is independent of ε. With 0 < h ≤ 1,

  • Z
  • Z
  • Z

ꢀ

ꢀꢀ
ꢀꢀꢀ
ꢀꢀꢀ
ꢀꢀꢀ
ꢀꢀꢀ
ꢀ

  • f(x)
  • f(x)
  • f(x)

ꢀ

  • dx =
  • dx +
  • dx

ꢀ

  • x
  • x
  • x

  • |x|≥h
  • |x|≥1
  • h≤|x|≤1

  • Z
  • Z
  • Z

  • 1
  • 1

≤

|f(x)| dx +

|f(x)| dx ≤

  • (1 + x2)|f(x)| ·
  • dx + 2h sup |f(x)|

h

1 + x2

x

  • |x|≥1
  • h≤|x|≤1
  • |x|≥1

Z

1

  • ≤ sup(1 + x2)|f(x)| ·
  • dx + 2h sup |f(x)| ꢀ sup(1 + x2)|f(x)|

1 + x2

  • x
  • x
  • x

|x|≥1

Of course, sup|x|≤h |f0(x)| ≤ supx |f0(x)|, giving the corollary.

///

2

Paul Garrett: The Hilbert transform (February 14, 2017)

To make the degree of a positive-homogeneous function ϕ agree with the degree of the integration-against-ϕ distribution uϕ, due to change-of-measure, we find that

  • Z
  • Z

(uϕ ◦ t)(f) = uϕ(f ◦ t−1) =

uϕ(x) f(t−1x) dx =

uϕ(tx) f(x) d(tx) = t · uϕ◦t(f)

  • R
  • R

Thus, for agreement of the notion of homogeneity for distributions and (integrate-against) functions, the dilation action u → u ◦ t on distributions should be

1
(u ◦ t)(f) = u(f ◦ t−1

  • )
  • (for t > 0, test function f, and distribution u)

t

Parity is as expected: u is odd when u(x → f(−x)) = −u(x → f(x)) for all test functions f, and is even when u(x → f(−x)) = +u(x → f(x)) for all test functions f.

[1.2] Claim: The principal-value distribution η is positive-homogeneous of degree −1, and is odd. Proof: The εth integral in the limit definition of η is itself odd, by changing variables:

  • Z
  • Z

  • f(−x)
  • f(x)

η(f ◦ (−1)) = lim dx = lim d(−x)

ε→0+

x

ε→0+

−x

  • |x|≥ε
  • |x|≥ε

  • Z
  • Z

  • f(x)
  • f(x)

=

  • lim −
  • dx = − lim

dx = −η(f)

ε→0+

x

ε→0+

x

  • |x|≥ε
  • |x|≥ε

as claimed. For the degree of homogeneity, for t > 0 and test function f,

  • Z
  • Z
  • Z

  • f(xt )
  • 1
  • 1
  • 1
  • 1
  • f(x)
  • 1
  • f(x)

(η ◦t)(f) = η(f ◦ ) = lim

dx =

Z

  • lim
  • d(tx) =
  • lim
  • d(x)

  • t
  • t
  • t ε→0+
  • x
  • t ε→0+
  • tx
  • t ε→0+
  • x

|x|≥ε

  • |tx|≥ε
  • |x|≥ε/t

  • 1
  • f(x)
  • 1

  • =
  • lim
  • d(x) = η(f)

  • t ε→0+
  • x
  • t

|x|≥ε

That is, η is homogeneous of degree −1.

///

2. Other descriptions of the principal-value functional

[2.1] Claim: Let ϕ be any even function in L1(R) ∩ C1(R), with ϕ(0) = 1. Then

  • Z
  • Z

  • f(x)
  • f(x) − f(0) · ϕ(x)

  • P.V.
  • dx =
  • dx

(for f ∈ Co(R) ∩ L1(R), for example)

  • x
  • x

  • R
  • R

  • ꢁ
  • ꢂ

where f(x) − f(0) · ϕ(x) /x is extended by continuity at x = 0. Proof: By the demonstrated odd-ness of the principal-value integral, it is 0 on the even function ϕ.

f(x)−f(0)·ϕ(x)

  • Extending
  • by continuity at 0,

x

  • Z
  • Z

  • f(x) − f(0) · ϕ(x)
  • f(x) − f(0) · ϕ(x)

dx = lim

dx x

ε→0+

x

R

|x|≥ε

  • Z
  • Z
  • Z

  • f(x)
  • ϕ(x)
  • f(x)

  • =
  • lim
  • dx + f(0) lim

dx = P.V.

dx + 0

ε→0+

x

ε→0+

  • x
  • x

  • |x|≥ε
  • |x|≥ε

R

as claimed.

///

3

Paul Garrett: The Hilbert transform (February 14, 2017)

The leading term in the following might be unexpected:

[2.2] Claim: (Sokhotski-Plemelj)

  • Z
  • Z

  • ∞
  • ∞

  • f(x)
  • f(x)

lim

  • dx = −iπ f(0) + P.V.
  • dx

ε→0+

  • x + iε
  • x

  • −∞
  • −∞

Proof: Aiming to apply the previous claim,

  • Z
  • Z
  • Z

f(0) 1+x2

f(x) −

  • f(x)
  • 1/(1 + x2)

  • lim
  • dx = lim
  • dx + f(0) lim

dx

ε→0+

x + iε

ε→0+

x + iε

ε→0+

x + iε

  • R
  • R
  • R

For 0 < ε < 1, the right-most integral is evaluated by residues, moving the contour through the upper half-plane:

Z

  • 1/(1 + x2)
  • 1
  • 1

πidx = 2πi Resx=i

= 2πi

−→

= −iπ x + iε

  • (1 + x2)(x + iε)
  • (i + i)(i + iε)

R

as claimed. [2.3] Claim: With f continuously differentiable at 0, thereby extending

///

f(x)−f(−x)

by continuity at 0,

x

  • Z
  • Z

∞

  • f(x) − f(−x)
  • f(x)

12

  • dx = P.V.
  • dx

(for f ∈ C1(R) ∩ L1(R))

  • x
  • x

R

−∞

f(x)−f(−x)

Proof: For test function f, since functional, is continuous as 0, using the odd-ness of the principal-value

x

  • Z
  • Z
  • Z

  • f(x) − f(−x)
  • f(x) − f(−x)

f(x)

12
12

dx =

  • lim
  • dx = lim

dx x

R

ε→0+

x

ε→0+

x

  • |x|≥ε
  • |x|≥ε

as claimed.

///

3. Uniqueness of odd/even homogeneous distributions

After some clarification of notions of parity and positive-homogeneity, we show that there is a unique distribution of a given parity and positive-homogeneity degree.

d

[3.1] Claim: For a positive-homogeneous distribution u of degree s, xdx u is again positive-homogeneous of degree s, and with the same parity.

Proof: For test function f,

d

1

d

1

d

1

d

  • ((x u) ◦ t)(f) = (x u)(f ◦ t−1) =
  • (
  • u)(x · f ◦ t−1) = − u( (x · (f ◦ t−1)))

  • dx
  • t
  • dx
  • t dx
  • t
  • dx

  • 1
  • 1
  • 1

  • = − u(f ◦ t−1 + x · t−1 · (f0 ◦ t−1)) = − u(f ◦ t−1 + (x · f0) ◦ t−1)) = − u((f + xf0) ◦ t−1
  • )

  • t
  • t
  • t

  • ꢁ
  • ꢂ

  • d
  • d
  • d

= −(u ◦ t)(f + xf0) = −(u ◦ t)( (xf)) = ts · − u( (xf)) = ts · (x u)(f)

  • dx
  • dx
  • dx

4

Paul Garrett: The Hilbert transform (February 14, 2017)

as asserted. Preservation of parity is similar: writing f−(x) = f(−x) for functions f, and u−(f) = u(f−) for distributions u,

  • d
  • d
  • d
  • d
  • d

(x u)−(f) = (x u)(f−) = ( u)(x · f−) = −u( (x · f−)) = u( ((x · f)−))

  • dx
  • dx
  • dx
  • dx
  • dx

ꢁ

ꢂ−

  • d
  • d
  • d

  • = u(−
  • (x · f) ) = −u−( (x · f)) = ((x )u−)(f)

  • dx
  • dx
  • dx

as claimed.

///

d

The Euler operator xdx is the infinitesimal form of dilation f(x) → f(tx). This is manifest in the following:

Recommended publications
  • Design and Application of a Hilbert Transformer in a Digital Receiver

    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 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

    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.
  • Arxiv:1611.05269V3 [Cs.IT] 29 Jan 2018 Graph Analytic Signal, and Associated Amplitude and Frequency Modulations Reveal Com

    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

    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

    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).
  • Discrete Hilbert Transform. Numeric Algorithms

    Discrete Hilbert Transform. Numeric Algorithms

    Volume 49, Number 4, 2008 485 Discrete Hilbert Transform. Numeric Algorithms Gheorghe TODORAN, Rodica HOLONEC and Ciprian IAKAB Abstract - The Hilbert and Fourier transforms are tools used for signal analysis in the time/frequency domains. The Hilbert transform is applied to casual continuous signals. The majority of the practical signals are discrete signals and they are limited in time. It appeared therefore the need to create numeric algorithms for the Hilbert transform. Such an algorithm is a numeric operator, named the Discrete Hilbert Transform. This paper makes a brief presentation of known algorithms and proposes an algorithm derived from the properties of the analytic complex signal. The methods for time and frequency calculus are also presented. 1. INTRODUCTION spectrum in order to avoid the aliasing process – the Nyquist condition. Signals can be classified into two classes: The discrete signal will be analyzed on a analytic signals (for instance = sin)( ωtAtx ), computer system, which implies its and experimental signals (measured signals). digitization (the digital signal is the discrete The last category represents real signals and is signal converted in binary format, accordingly of great importance in applications. to the adopted analog/numeric conversion; in An experimental signal represents a signal most of the cases, the signal acquisition observed during a limited interval of time. It is hardware also does the digitization of the a sample of the original signal, which signal samples). The resulted digital signal has characterizes a physical process of interest. the greatest importance in numeric analysis The experimental signal can be a operations. continuous time signal (analogical), or a Some other remarks need to be made.
  • Spectral Decompositions in Banach Spaces and the Hilbert Transform

    Spectral Decompositions in Banach Spaces and the Hilbert Transform

    LINEAR OPERATORS BANACH CENTER PUBLICATIONS, VOLUME 38 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 1997 SPECTRAL DECOMPOSITIONS IN BANACH SPACES AND THE HILBERT TRANSFORM T. A. GILLESPIE Department of Mathematics and Statistics University of Edinburgh, James Clerk Maxwell Building Edinburgh EH9 3JZ, Scotland E-mail: [email protected] Abstract. This paper gives a survey of some recent developments in the spectral theory of linear operators on Banach spaces in which the Hilbert transform and its abstract analogues play a fundamental role. 1. Introduction. Various notions of self-adjointness have been developed for oper- ators acting on Banach spaces, each reflecting some aspect of the Hilbert space theory. One such notion is that of well-boundedness, a concept introduced by Smart [23] and first studied by Smart and Ringrose [21–23]. An operator is (by definition) well-bounded if it has a functional calculus based on the Banach algebra of absolutely continuous func- tions on a compact real interval. This functional calculus gives rise to a form of spectral diagonalization, as will be described in more detail below. Initially, there were relatively few examples of well-bounded operators, other than rather obvious ones, until Dowson and Spain [17] gave an interesting example of a well- bounded operator A acting on Lp(Z) for p in the range 1 <p< ∞. It can be shown that the operator they considered has the property that eiA is the bilateral shift on Lp(Z) (see [18, p. 1044]) and this observation illustrates the more general fact [19] that every translation operator on Lp(G), where G is a locally compact abelian group and 1 <p< ∞, is of the form eiA for some well-bounded operator A.
  • An Extension of Hewitt's Inversion Formula and Its Application to Fluctuation Theory

    An Extension of Hewitt's Inversion Formula and Its Application to Fluctuation Theory

    AN EXTENSION OF HEWITT’S INVERSION FORMULA AND ITS APPLICATION TO FLUCTUATION THEORY S¸erban E. B˘adil˘a1 Abstract. We analyze fluctuations of random walks with generally distributed increments. Integral representations for key performance measures are obtained by extending an inversion theorem of Hewitt [11] for Laplace-Stieltjes transforms. Another important step in the analysis involves introducing the so-called harmonic measures associated to the walk. It is also pointed out that such representations can be explicitly calculated, if one assumes a form of rational structure for the increment transform. Applications include, but are not restricted to, queueing and insurance risk problems. Keywords: fluctuations of random walks, harmonic analysis, singular integrals 2010 Mathematics Subject Classification: Primary 60G50 Secondary 30E20 1. Introduction A standard assumption in the theory of fluctuations of random walks as they appear in, e.g., queueing or insurance applications, is that the increment of the random walk can be represented as the difference of two independent random variables. In the context of queueing theory, random walks with increments which do not have this property appear as embedded at arrival epochs of customers in a single server queue in which the service requirement of the current customer is correlated with the time until the next arrival. In risk reserve processes that appear in insurance, the independence assumption is violated when the current claim size depends on the time elapsed since the previous arrival, and hence on the premium gained meanwhile. The purpose of the present paper is to show how much can be done for random walks which do not satisfy this independence assumption, regarding their maxima (as in the waiting time/maximum aggregate loss), their minima (idle periods/deficit at ruin) and their excur- arXiv:1508.00751v1 [math.PR] 4 Aug 2015 sions which are related to busy periods or the time to ruin.
  • The Sy Llnetry of Lmaginary Unit and the Regularization Of

    The Sy Llnetry of Lmaginary Unit and the Regularization Of

    The Sy■llnetry of lmaginary Unit and the Regularization of lntegrals Shigeru OHKURO Abstract The ne、 、 integral apeared in our generalization of Hilbert transform is naturaHy and explicitly founded from the syHlrnetric consideration of complex number system. In this course 都/e obtained both parallel and antipara■ el integrals fo■ o、】、アing PARITY reasoning in quantum mechanics(the start of elementary quantum mathematics) The concept of the imaginary unit in information processing of formula manipulation by personalcomputer such as REDUCE and W【 ATHEふlIATICA wili be very helpful for every person to just catch this signiFicant discOvery The new branch of natural science,EXPERIMENTALふ /1ATHEMATICS,not as the artincial science starts from nO郡 ァon、、アith friendly relation to infOrmation processing Key wordsi Symmetry ofimaginary unit,Regularization of integrals,Generahzation of Hllbert transform,Para■ el integral,Experirnental lnathematics 1. INTRODUCTION In our recent paper we have considered a generalization (GHT)of r【 ildbert transform (HT).ヽ Ve have shown that using the new symbolfor our new type of complex integrals GHT can be expressed in the sirnilar form as the usual HT. ‐ The perpous of the present paper is to sho、 二、that the above integral is naturaHy and easily considered as the consequence of the symmetry of the imaginary unit in complex numbers, 2. THE SYMMETRY OF IMAGINARY UNIT 三 ユア Usually the imaginary unit ゲ==v「 =l haS been considered to be able to diline as one of t、 、o solutions of the eq, χ2+1=0 ′ “ he agreement of radical sign"does not apply in complex case. The concept of``sman(or 「 large)''does nOt exist.
  • Introduction to Hyperfunctions and Their Integral Transforms

    Introduction to Hyperfunctions and Their Integral Transforms

    [chapter] [chapter] [chapter] 1 2 Introduction to Hyperfunctions and Their Integral Transforms Urs E. Graf 2009 ii Contents Preface ix 1 Introduction to Hyperfunctions 1 1.1 Generalized Functions . 1 1.2 The Concept of a Hyperfunction . 3 1.3 Properties of Hyperfunctions . 13 1.3.1 Linear Substitution . 13 1.3.2 Hyperfunctions of the Type f(φ(x)) . 15 1.3.3 Differentiation . 18 1.3.4 The Shift Operator as a Differential Operator . 25 1.3.5 Parity, Complex Conjugate and Realness . 25 1.3.6 The Equation φ(x)f(x) = h(x)............... 28 1.4 Finite Part Hyperfunctions . 33 1.5 Integrals . 37 1.5.1 Integrals with respect to the Independent Variable . 37 1.5.2 Integrals with respect to a Parameter . 43 1.6 More Familiar Hyperfunctions . 44 1.6.1 Unit-Step, Delta Impulses, Sign, Characteristic Hyper- functions . 44 1.6.2 Integral Powers . 45 1.6.3 Non-integral Powers . 49 1.6.4 Logarithms . 51 1.6.5 Upper and Lower Hyperfunctions . 55 α 1.6.6 The Normalized Power x+=Γ(α + 1) . 58 1.6.7 Hyperfunctions Concentrated at One Point . 61 2 Analytic Properties 63 2.1 Sequences, Series, Limits . 63 2.2 Cauchy-type Integrals . 71 2.3 Projections of Functions . 75 2.3.1 Functions satisfying the H¨olderCondition . 77 2.3.2 Projection Theorems . 78 2.3.3 Convergence Factors . 87 2.3.4 Homologous and Standard Hyperfunctions . 88 2.4 Projections of Hyperfunctions . 91 2.4.1 Holomorphic and Meromorphic Hyperfunctions . 91 2.4.2 Standard Defining Functions .
  • Proved That Every Real Separable Banach Space Contains a Separable Hilbert Space As a Dense Embedding, and This Space Is the Support of a Gaussian Measure

    Proved That Every Real Separable Banach Space Contains a Separable Hilbert Space As a Dense Embedding, and This Space Is the Support of a Gaussian Measure

    . SOME BANACH SPACES ARE ALMOST HILBERT TEPPER L. GILL1∗ AND MARZETT GOLDEN1 Abstract. The purpose of this note is to show that, if B is a uniformly convex Banach, then the dual space B0 has a \ Hilbert space representation" (defined in the paper), that makes B much closer to a Hilbert space then previously suspected. As an application, we prove that, if B also has a Schauder basis (S- basis), then for each A 2 C[B] (the closed and densely defined linear operators), there exists a closed densely defined linear operator A∗ 2 C[B] that has all the expected properties of an adjoint. Thus for example, the bounded linear operators, L[B], is a ∗algebra. This result allows us to give a natural definition to the Schatten class of operators on a uniformly convex Banach space with a S-basis. In particular, every theorem that is true for the Schatten class on a Hilbert space, is also true on such a space. The main tool we use is a special version of a result due to Kuelbs [K], which shows that every uniformly convex Banach space with a S-basis can be densely and continuously embedded into a Hilbert space which is unique up to a change of basis. 1. Introduction In 1965, Gross [G] proved that every real separable Banach space contains a separable Hilbert space as a dense embedding, and this space is the support of a Gaussian measure. This was a generalization of Wiener's theory, that was based on the use of the (densely embedded Hilbert) Sobolev space H1[0; 1] ⊂ C[0; 1].