Quantum extensions of Fourier-Gauss and Fourier-Mehler transforms

지운식 대한수학회 2008 대한수학회지 Vol.45 No.6

Noncommutative extensions of the Gross and Beltrami Laplacians, called the quantum Gross Laplacian and the quantum Beltrami Laplacian, resp., are introduced and their basic properties are studied. As noncommutative extensions of the Fourier-Gauss and Fourier-Mehler transforms, we introduce the quantum Fourier-Gauss and quantum Fourier-Mehler transforms. The infinitesimal generators of all differentiable one parameter groups induced by the quantum Fourier-Gauss transform are linear combinations of the quantum Gross Laplacian and quantum Beltrami Laplacian. A characterization of the quantum Fourier-Mehler transform is studied. Noncommutative extensions of the Gross and Beltrami Laplacians, called the quantum Gross Laplacian and the quantum Beltrami Laplacian, resp., are introduced and their basic properties are studied. As noncommutative extensions of the Fourier-Gauss and Fourier-Mehler transforms, we introduce the quantum Fourier-Gauss and quantum Fourier-Mehler transforms. The infinitesimal generators of all differentiable one parameter groups induced by the quantum Fourier-Gauss transform are linear combinations of the quantum Gross Laplacian and quantum Beltrami Laplacian. A characterization of the quantum Fourier-Mehler transform is studied.

Genetic Analysis of Generalized S-Transform

Yun Lin,Xiaowan Yu,Chunguang Ma,Zheng Dou,Zhiqiang Wu,Zhiping Zhang 보안공학연구지원센터 2016 International Journal of u- and e- Service, Scienc Vol.9 No.4

This text starts with the short time Fourier transform and continuous wavelet transform to deduce the generalized S-transformation. From the point of generation views, we analyzed a relative relationship between generalized S-transformation and the short time Fourier transform, and the other relative relationship between generalized S transform and continuous wavelet transform. The article gives the definition of “the gene mutation of formula” and “the genetic restructuring of formula”, and introduces the deriving process of the two core concept. Theoretical analyses show that generalized S-transformation inherited the desirable characteristics in short time Fourier transform which use the window function to select suitable signal. Through genome sequencing of specific parameters, generalized S-transformation has a stronger adaptation that the time-frequency window could make real-time adjusting of frequency. Moreover, generalized S-transformation breaks out limitation that the wavelet function has to content the admissible conditions. From the point of gene mutation, we give the definition of “the gene mutation of formula”. Based on the structure form of wavelet functions, we define the generalized S-transformation with a wider domain of definition. Generalized S-transformation inherited the desirable characteristics of the short time Fourier transform and continuous wavelet transform. It has great utility and flexibility in analyzing non-stationary signals.

FOURIER-FEYNMAN TRANSFORM AND CONVOLUTION OF FOURIER-TYPE FUNCTIONALS ON WIENER SPACE

Kim, Byoung Soo The Youngnam Mathematical Society 2013 East Asian mathematical journal Vol.29 No.5

We develop a Fourier-Feynman theory for Fourier-type functionals ${\Delta}^kF$ and $\widehat{{\Delta}^kF}$ on Wiener space. We show that Fourier-Feynman transform and convolution of Fourier-type functionals exist. We also show that the Fourier-Feynman transform of the convolution product of Fourier-type functionals is a product of Fourier-Feynman transforms of each functionals.

FOURIER-FEYNMAN TRANSFORM AND CONVOLUTION OF FOURIER-TYPE FUNCTIONALS ON WIENER SPACE

김병수 영남수학회 2013 East Asian mathematical journal Vol.29 No.5

We develop a Fourier-Feynman theory for Fourier-type functionals ΔkF and ΔkF on Wiener space. We show that Fourier-Feynman transform and convolution of Fourier-type functionals exist. We also show that the Fourier-Feynman transform of the convolution product of Fourier-type functionals is a product of Fourier-Feynman transforms of each functionals.

GENERALIZED FOURIER-FEYNMAN TRANSFORM AND SEQUENTIAL TRANSFORMS ON FUNCTION SPACE

Choi, Jae-Gil,Chang, Seung-Jun Korean Mathematical Society 2012 대한수학회지 Vol.49 No.5

In this paper we first investigate the existence of the generalized Fourier-Feynman transform of the functional F given by $$F(x)={\hat{\nu}}((e_1,x)^{\sim},{\ldots},(e_n,x)^{\sim})$$, where $(e,x)^{\sim}$ denotes the Paley-Wiener-Zygmund stochastic integral with $x$ in a very general function space $C_{a,b}[0,T]$ and $\hat{\nu}$ is the Fourier transform of complex measure ${\nu}$ on $B({\mathbb{R}}^n)$ with finite total variation. We then define two sequential transforms. Finally, we establish that the one is to identify the generalized Fourier-Feynman transform and the another transform acts like an inverse generalized Fourier-Feynman transform.

Generalized Fourier-Feynman transform and sequential transforms on function space

최재길,장승준 대한수학회 2012 대한수학회지 Vol.49 No.5

In this paper we rst investigate the existence of the gener-alized Fourier-Feynman transform of the functional F given by F(x) = ^((e1; x)~,..., (en, x)~);where (e, x) denotes the Paley-Wiener-Zygmund stochastic integral with x in a very general function space Ca;b[0; T] and ^ is the Fourier transform of complex measure on B(Rn) with nite total variation. We then dene two sequential transforms. Finally, we establish that the one is to identify the generalized Fourier-Feynman transform and the another transform acts like an inverse generalized Fourier-Feynman transform.

고속 푸리에 변환(fast Fourier transform, FFT)을 이용한 겹친지문 분리의 효과와 한계

김채원,김채린,이한나,유제설,장윤식 한국경호경비학회 2019 시큐리티연구 Vol.- No.61

Photography is the most commonly used method of documenting the crime and incident scene as it helps maintaining chain of custody (COC) and prove integrity of the physical evidence. It can also capture phenomena as they are. However, digital images can be manipulated and lose their authenticity as admissible evidence. Thus only limited techniques can be used to enhance images, and one of them is Fourier transform. Fourier transform refers to transformation of images into frequency signals. Fast Fourier transform (FFT) is used in this study. In this experiment, we overlapped fingerprints with graph paper or other fingerprints and separated the fingerprints. Then we evaluated and compared quality of the separated fingerprints to the original fingerprints, and examined whether the two fingerprints can be identified as same fingerprints. In the case of the fingerprints on graph paper and general pattern-overlapping fingerprints, fingerprint ridges are enhanced. On the other hand, in case of separating complicated fingerprints such as core-to-core overlapping and delta-to-delta overlapping fingerprints, quality of fingerprints can be deteriorated. Quality of fingerprints is known to possibly bring negative effects on the credibility of examiners. The result of this study may be applicable to other areas using digital imaging enhancement technology. 사진은 가장 일반적으로 사용되는 현장 기록 방법으로, 증거물 관리의 연속성을 유지하고 현상을 있는 그대로 담아낼 수 있기 때문에 범죄수사 및 각종 사고조사에 널리 이용된다. 최근에는 디지털 카메라가 이용되기 때문에 이미지 조작에 따른 진정성(authenticity) 훼손위험이 있어 이미지 증강에 사용할 수 있는 방법이 제한적인데, 그 중 널리 활용되는 하나가 푸리에 변환 (Fourier transform)이다. 푸리에 변환은 이미지를 주파수 신호로 변환하는 것으로, 본 연구에서는 fast Fourier transform (FFT)을 사용하였다. 본 실험에서는 지문을 규칙적인 패턴을 가진 모눈종이나 또 다른 지문과 겹치게 한 뒤 FFT로 분리하였다. 그리고 원래의 지문과 분리한 지문의 품질을 각각 평가하고 비교하였으며, 두 지문이 같은 지문으로 판정될 수 있는지 검사하였다. 규칙적인 패턴을 가진 배경에 찍힌 지문이나 general pattern끼리 겹쳐진 지문을 분리한 경우 지문 융선이 증강되어 식별에 용이하였다. 하지만 core나 delta와 같이 복잡한 패턴끼리 겹쳐진 지문의 경우 원래의 지문보다 FFT로 분리하여 얻은 지문의 품질이 낮고 식별오류가 많았다. 지문 품질의 저하는 감정관의 신뢰성에 부정적 영향을 미칠 수 있다. 연구의 주요 논점은 지문감정 이외에도 디지털이미지 개선기술이 활용되는 여러 상황에서 참고할 수 있을 것이다.

QUANTUM EXTENSIONS OF FOURIER-GAUSS AND FOURIER-MEHLER TRANSFORMS

Ji, Un-Cig Korean Mathematical Society 2008 대한수학회지 Vol.45 No.6

Noncommutative extensions of the Gross and Beltrami Laplacians, called the quantum Gross Laplacian and the quantum Beltrami Laplacian, resp., are introduced and their basic properties are studied. As noncommutative extensions of the Fourier-Gauss and Fourier-Mehler transforms, we introduce the quantum Fourier-Gauss and quantum Fourier- Mehler transforms. The infinitesimal generators of all differentiable one parameter groups induced by the quantum Fourier-Gauss transform are linear combinations of the quantum Gross Laplacian and quantum Beltrami Laplacian. A characterization of the quantum Fourier-Mehler transform is studied.

FOURIER'S TRANSFORM OF FRACTIONAL ORDER VIA MITTAG-LEFFLER FUNCTION AND MODIFIED RIEMANN-LIOUVILLE DERIVATIVE

Jumarie, Guy Korean Society of Computational and Applied Mathem 2008 Journal of applied mathematics & informatics Vol.26 No.5

One proposes an approach to fractional Fourier's transform, or Fourier's transform of fractional order, which applies to functions which are fractional differentiable but are not necessarily differentiable, in such a manner that they cannot be analyzed by using the so-called Caputo-Djrbashian fractional derivative. Firstly, as a preliminary, one defines fractional sine and cosine functions, therefore one obtains Fourier's series of fractional order. Then one defines the fractional Fourier's transform. The main properties of this fractal transformation are exhibited, the Parseval equation is obtained as well as the fractional Fourier inversion theorem. The prospect of application for this new tool is the spectral density analysis of signals, in signal processing, and the analysis of some partial differential equations of fractional order.

CONDITIONAL FOURIER-FEYNMAN TRANSFORMS OF VARIATIONS OVER WIENER PATHS IN ABSTRACT WIENER SPACE

Cho, Dong-Hyun Korean Mathematical Society 2006 대한수학회지 Vol.43 No.5

In this paper, we evaluate first variations, conditional first variations and conditional Fourier-Feynman transforms of cylinder type functions over Wiener paths in abstract Wiener space and then, investigate relationships among first variation, conditional first variation, Fourier-Feynman transform and conditional Fourier-Feynman transform of those functions. Finally, we derive the conditional Fourier-Feynman transform for the product of cylinder type function which defines the functions in a Banach algebra introduced by Yoo, with n linear factors.