UNITARITY OF KUO'S FOURIER–MEHLER TRANSFORM

2004 ◽  
Vol 07 (01) ◽  
pp. 147-154 ◽  
Author(s):  
UN CIG JI ◽  
NOBUAKI OBATA
Keyword(s):  
Automorphism Group ◽  
White Noise ◽  
Gaussian Measure ◽  
Unitary Operators ◽  
Parameter Group

Kuo's Fourier–Mehler transforms {ℱΘ} form a one-parameter automorphism group of the space of white noise distributions [Formula: see text]. The adjoint transforms [Formula: see text] form a one-parameter automorphism group of the space of test white noise functions [Formula: see text] but do not admit unitary extensions with respect to the L2-norm induced from the Gaussian measure with variance 1. We prove that [Formula: see text] admits an extension to a one-parameter group of unitary operators on the L2-space with respect to the Gaussian measure with variance 1/2.

scholarly journals Spectral Type of One-Parameter Group of Unitary Operators with Transversal Group

1968 ◽  
Vol 32 ◽  
pp. 141-153 ◽  
Author(s):  
Masasi Kowada
Keyword(s):  
Hilbert Space ◽  
Probability Measure ◽  
Spectral Type ◽  
Measure Space ◽  
Unitary Operator ◽  
Separable Hilbert Space ◽  
Unitary Operators ◽  
Parameter Group ◽  
Probability Measure Space

It is an important problem to determine the spectral type of automorphisms or flows on a probability measure space. We shall deal with a unitary operator U and a 1-parameter group of unitary operators {Ut} on a separable Hilbert space H, and discuss their spectral types, although U and {Ut} are not necessarily supposed to be derived from an automorphism or a flow respectively.

scholarly journals Finite Dimensional Approximation to Band Limited White Noise

1967 ◽  
Vol 29 ◽  
pp. 211-216 ◽  
Author(s):  
Takeyuki Hida ◽  
Hisao Nomoto
Keyword(s):  
White Noise ◽  
Point Spectrum ◽  
Unitary Operators ◽  
Finite Dimensional Approximation ◽  
Finite Dimensional ◽  
Period 2 ◽  
Dimensional Approximation ◽  
Band Limited ◽  
Infinite Multiplicity ◽  
Countably Infinite

One of the authors discussed finite dimensional approximations to a white noise and a periodic Brownian motion with period 2 π on the projective limit space of spheres ([2]). The group of unitary operators derived from the periodic white noise has a pure point spectrum which consists of all integers with countably infinite multiplicity. We also have much interest in the investigation of a band limited white noise which is another typical example having quite different spectral type. Indeed, the corresponding group of unitary operators has a continuous spectrum with countably infinite multiplicity.

scholarly journals The automorphism group of the Gaussian measure cannot act pointwise

2005 ◽  
Vol 148 (1) ◽  
pp. 305-329 ◽  
Author(s):  
E. Glasner ◽  
B. Tsirelson ◽  
B. Weiss
Keyword(s):  
Automorphism Group ◽  
Gaussian Measure

scholarly journals Differential equations in spaces of Hilbert space valued distributions

2003 ◽  
Vol 68 (3) ◽  
pp. 491-500 ◽  
Author(s):  
Maxim Alshansky
Keyword(s):  
Hilbert Space ◽  
Differential Equations ◽  
White Noise ◽  
Wiener Process ◽  
Gaussian Measure ◽  
Additive Noise ◽  
Covariance Operator ◽  
Operator Coefficient ◽  
White Noise Process ◽  
Tempered Distributions

A Gaussian measure is introduced on the space of Hilbert space valued tempered distributions. It is used to define a Hilbert space valued Q-Wiener process and a white noise process with a nuclear covariance operator Q. The proposed construction is used for solving operator-differential equations with additive noise with the operator coefficient generating an n-times integrated exponentially bounded semigroup.

scholarly journals A remark on the space of testing random variables in the white noise calculus

1989 ◽  
Vol 115 ◽  
pp. 139-149 ◽  
Author(s):  
Izumi Kubo ◽  
Yoshitaka Yokoi
Keyword(s):  
White Noise ◽  
Random Variables ◽  
Analytic Functional ◽  
Continuous Functional ◽  
Basic Space ◽  
Parameter Group ◽  
Real Analytic ◽  
White Noise Calculus

The first author and S. Takenaka introduced the structure of a Gel’fand triplet ℋ ⊂ (L2) ⊂ ℋ* into Hida’s calculus on generalized Brownian functionals [4-7]. They showed that the space ℋ of testing random variables has nice properties. For example, ℋ is closed under multiplication of two elements in ℋ, each element of ℋ is a continuous functional on the basic space ℰ*, in addition it can be considered as an analytic functional, and moreover exp [tΔv] (Δv is Volterra’s Laplacian) is real analytic in t ∊ R as a one-parameter group of operators on ℋ, etc.

Integrated Fractional white Noise as an Alternative to Multifractional Brownian Motion

2007 ◽  
Vol 44 (02) ◽  
pp. 393-408 ◽  
Author(s):  
Allan Sly
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
White Noise ◽  
Gaussian Process ◽  
Discrete Data ◽  
Hölder Exponent ◽  
Scaling Properties ◽  
Multifractional Brownian Motion ◽  
Local Properties

Multifractional Brownian motion is a Gaussian process which has changing scaling properties generated by varying the local Hölder exponent. We show that multifractional Brownian motion is very sensitive to changes in the selected Hölder exponent and has extreme changes in magnitude. We suggest an alternative stochastic process, called integrated fractional white noise, which retains the important local properties but avoids the undesirable oscillations in magnitude. We also show how the Hölder exponent can be estimated locally from discrete data in this model.

Auditory Selective Attention in Cerebral-Palsied Individuals

1985 ◽  
Vol 16 (4) ◽  
pp. 260-266 ◽  
Author(s):  
Lee Ann Laraway
Keyword(s):  
Selective Attention ◽  
White Noise ◽  
Normal Subjects ◽  
Auditory Selective Attention ◽  
Normal Individuals ◽  
Significant Difference

The purpose of this study was to determine whether there is a statistically significant difference between the auditory selective attention abilities of normal and cerebral-palsied individuals. Twenty-three cerebral-palsied and 23 normal subjects between the ages of 5 and 21 were asked to repeat a series of 30 items consisting of from 2 to 4 digits in the presence of intermittent white noise. Results of the study indicate that cerebral-palsied individuals perform significantly poorer than normal individuals when the stimulus is accompanied by noise. Noise was not a significant factor in the performance of the normal subjects regardless of age.

White Noise and Stuttering

1961 ◽  
Vol 4 (1) ◽  
pp. 72-72 ◽  
Author(s):  
Samuel Sutton ◽  
Richard Allen Chase
Keyword(s):  
White Noise

Infant Responses to Recorded Sounds

1968 ◽  
Vol 11 (4) ◽  
pp. 811-816 ◽  
Author(s):  
Maurice I. Mendel
Keyword(s):  
White Noise ◽  
Narrow Band ◽  
Limited Bandwidth ◽  
Broadband Spectrum ◽  
Continuous Band

Thirty infants, ranging in age from 4 to 11 months, were tested with five different recorded sounds that varied in bandwidth and temporal configuration: a continuous band of white noise, the same band of noise interrupted twice per second, the crinkling of onionskin paper, a narrow band of noise centered at 3000 Hz, and a warbled 3000 Hz tone. With loudness and duration of the stimuli held constant, more responses occurred to sounds composed of a broadband spectrum than to those of a limited bandwidth. Temporal configuration of the sound had no effect on the number of responses elicited.

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