On the Bogolyubov endomorphisms of the renormalized square of white noise algebra

2021 ◽  
Author(s):  
Habib Rebei ◽  
Slaheddine Wannes
Keyword(s):  
White Noise ◽  
Function Space ◽  
The Other ◽  
Linear Transformations ◽  
Test Function ◽  
Canonical Commutation Relations ◽  
Arbitrary Complex ◽  
Borel Functions ◽  
Test Function Space ◽  
Complex Valued

We introduce the quadratic analogue of the Bogolyubov endomorphisms of the canonical commutation relations (CCR) associated with the re-normalized square of white noise algebra (RSWN-algebra). We focus on the structure of a subclass of these endomorphisms: each of them is uniquely determined by a quadruple [Formula: see text], where [Formula: see text] are linear transformations from a test-function space [Formula: see text] into itself, while [Formula: see text] is anti-linear on [Formula: see text] and [Formula: see text] is real. Precisely, we prove that [Formula: see text] and [Formula: see text] are uniquely determined by two arbitrary complex-valued Borel functions of modulus [Formula: see text] and two maps of [Formula: see text], into itself. Under some additional analytic conditions on [Formula: see text] and [Formula: see text], we discover that we have only two equivalent classes of Bogolyubov endomorphisms, one of them corresponds to the case [Formula: see text] and the other corresponds to the case [Formula: see text]. Finally, we close the paper by building some examples in one and multi-dimensional cases.

Tensor Bogolyubov representations of the renormalized square of white noise (RSWN) algebra

2019 ◽  
Vol 22 (04) ◽  
pp. 1950025
Author(s):  
Habib Rebei ◽  
Luigi Accardi ◽  
Hajer Taouil
Keyword(s):  
White Noise ◽  
Lebesgue Measure ◽  
Absolutely Continuous ◽  
First Order ◽  
Commutation Relations ◽  
Hyperbolic Sine ◽  
Arbitrary Complex ◽  
Borel Functions ◽  
Complex Valued

We introduce the quadratic analog of the tensor Bogolyubov representation of the CCR. Our main result is the determination of the structure of these maps: each of them is uniquely determined by two arbitrary complex-valued Borel functions of modulus [Formula: see text] and two maps of [Formula: see text] into itself whose inverses induce transformations that map the Lebesgue measure [Formula: see text] into measures [Formula: see text] absolutely continuous with respect to it. Furthermore, the Radon–Nikodyn derivatives [Formula: see text], of these measures with respect to [Formula: see text], must satisfy the relation [Formula: see text] for [Formula: see text]-almost every [Formula: see text]. This makes a surprising bridge with the hyperbolic sine and cosine defining the structure of usual (i.e. first-order) Bogolyubov transformations. The reason of the surprise is that the linear and quadratic commutation relations are completely different.

Continuous wavelet transform involving canonical convolution

2019 ◽  
Vol 13 (06) ◽  
pp. 2050104
Author(s):  
Zamir Ahmad Ansari
Keyword(s):  
Wavelet Transform ◽  
Function Space ◽  
Continuous Wavelet Transform ◽  
Convolution Operator ◽  
Linear Canonical Transform ◽  
Continuous Wavelet ◽  
Test Function ◽  
Test Function Space

The main objective of this paper is to study the continuous wavelet transform in terms of canonical convolution and its adjoint. A relation between the canonical convolution operator and inverse linear canonical transform is established. The continuity of continuous wavelet transform on test function space is discussed.

scholarly journals Periodic Tempered Distributions of Beurling Type and Periodic Ultradifferentiable Functions with Arbitrary Support

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Byung Keun Sohn
Keyword(s):  
Function Space ◽  
Dual Space ◽  
Ultradifferentiable Functions ◽  
Tempered Distributions ◽  
Test Function ◽  
Test Function Space

Let Sω′(R) be the space of tempered distributions of Beurling type with test function space Sω(R) and let Eω,p be the space of ultradifferentiable functions with arbitrary support having a period p. We show that Eω,p is generated by Sω(R). Also, we show that the mapping Sω(R)→Eω,p is linear, onto, and continuous and the mapping Sω,p′(R)→Eω,p′ is linear and onto where Sω,p′(R) is the subspace of Sω′(R) having a period p and Eω,p′ is the dual space of Eω,p.

scholarly journals Test function space for Wick power series

2000 ◽  
Vol 123 (3) ◽  
pp. 709-725 ◽  
Author(s):  
A. G. Smirnov ◽  
M. A. Solov'ev
Keyword(s):  
Power Series ◽  
Function Space ◽  
Test Function ◽  
Test Function Space

The n-Dimensional Distributional Hankel Transformation

1975 ◽  
Vol 27 (2) ◽  
pp. 423-433 ◽  
Author(s):  
E. L. Koh
Keyword(s):  
Function Space ◽  
Generalized Functions ◽  
One Dimension ◽  
Dimensional Case ◽  
Hankel Transformation ◽  
Test Function ◽  
Test Function Space

The Hankel transformation was extended to certain generalized functions of one dimension [1; 2; 3]. In this paper, we develop the n-dimensional case corresponding to [1]. The procedure in [1] is briefly as follows:A test function space Hμ is constructed on which the μth order Hankel transformation hμ defined byis an automorphism whenever μ ≧ —1/2. The generalized transformation hμ' is then defined on the dual Hμ' as the adjoint of hμ through a Parseval relation, i.e.

scholarly journals Paley–Wiener properties for spaces of entire functions

2020 ◽  
Vol 10 (2) ◽  
Author(s):  
Elmira Nabizadeh ◽  
Christine Pfeuffer ◽  
Joachim Toft
Keyword(s):  
Function Space ◽  
Entire Functions ◽  
Test Function ◽  
Test Function Space

Abstract We deduce Paley–Wiener results in the Bargmann setting. At the same time we deduce characterisations of Pilipović spaces of low orders. In particular we improve the characterisation of the Gröchenig test function space $$\mathcal {H}_{\flat _1}=\mathcal {S}_C$$ H ♭ 1 = S C , deduced in Toft (J Pseudo-Differ Oper Appl 8:83–139, 2017).

scholarly journals Conversion Between Cubic Bezier Curves and Catmull–Rom Splines

2021 ◽  
Vol 2 (5) ◽  
Author(s):  
Soroosh Tayebi Arasteh ◽  
Adam Kalisz
Keyword(s):  
Computer Graphics ◽  
Geometric Design ◽  
The Other ◽  
Control Points ◽  
Complex Surfaces ◽  
Linear Transformations ◽  
Computer Aided Geometric Design ◽  
Cubic Curves ◽  
Original Curve ◽  
Computer Aided

AbstractSplines are one of the main methods of mathematically representing complicated shapes, which have become the primary technique in the fields of Computer Graphics (CG) and Computer-Aided Geometric Design (CAGD) for modeling complex surfaces. Among all, Bézier and Catmull–Rom splines are the most common in the sub-fields of engineering. In this paper, we focus on conversion between cubic Bézier and Catmull–Rom curve segments, rather than going through their properties. By deriving the conversion equations, we aim at converting the original set of the control points of either of the Catmull–Rom or Bézier cubic curves to a new set of control points, which corresponds to approximately the same shape as the original curve, when considered as the set of the control points of the other curve. Due to providing simple linear transformations of control points, the method is very simple, efficient, and easy to implement, which is further validated in this paper using some numerical and visual examples.

Measurement of Wiener kernels

1984 ◽  
Vol 16 (1) ◽  
pp. 11-12
Author(s):  
Yoshifusa Ito
Keyword(s):  
Differential Operator ◽  
Mathematical Models ◽  
White Noise ◽  
Linear Systems ◽  
Main Part ◽  
The Other ◽  
Linear Functionals ◽  
Wiener Kernels ◽  
Non Linear ◽  
Non Linear Systems

Since the late 1960s Wiener's theory on the non-linear functionals of white noise has been widely applied to the construction of mathematical models of non-linear systems, especially in the field of biology. For such applications the main part is the measurement of Wiener's kernels, for which two methods have been proposed: one by Wiener himself and the other by Lee and Schetzen. The aim of this paper is to show that there is another method based on Hida's differential operator.

scholarly journals A MLMVN with Arbitrary Complex-Valued Inputs and a Hybrid Testability Approach for the Extraction of Lumped Models Using FRA

2019 ◽  
Vol 9 (1) ◽  
pp. 5-19 ◽  
Author(s):  
Igor Aizenberg ◽  
Antonio Luchetta ◽  
Stefano Manetti ◽  
Maria Cristina Piccirilli
Keyword(s):  
Neural Network ◽  
Frequency Response ◽  
Learning Algorithm ◽  
Repeated Measurements ◽  
Distributed Parameter ◽  
Response Analysis ◽  
Frequency Response Analysis ◽  
Arbitrary Complex ◽  
Lumped Models ◽  
Complex Valued

Abstract A procedure for the identification of lumped models of distributed parameter electromagnetic systems is presented in this paper. A Frequency Response Analysis (FRA) of the device to be modeled is performed, executing repeated measurements or intensive simulations. The method can be used to extract the values of the components. The fundamental brick of this architecture is a multi-valued neuron (MVN), used in a multilayer neural network (MLMVN); the neuron is modified in order to use arbitrary complex-valued inputs, which represent the frequency response of the device. It is shown that this modification requires just a slight change in the MLMVN learning algorithm. The method is tested over three completely different examples to clearly explain its generality.

Simple and Complex Reaction Time as a Function of Subliminal and Supraliminal Accessory Stimulation

1974 ◽  
Vol 38 (2) ◽  
pp. 417-418 ◽  
Author(s):  
Robert Zenhausern ◽  
Claude Pompo ◽  
Michael Ciaiola
Keyword(s):  
Reaction Time ◽  
White Noise ◽  
Visual Stimuli ◽  
The Other ◽  
Complex Reaction

Simple and complex reaction time to visual stimuli was tested under 7 levels of accessory stimulation (white noise). Only the highest level of stimulation (70 db above threshold) lowered reaction time. The other levels had no effect.

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