• Journal for History of Mathematics
• /
• v.23 no.1
• /
• pp.53-66
• /
• 2010

This study concerns with partial sum of Fourier series, Fourier coefficients and the $L^1$-convergence of Fourier series. First, we introduce the $L^1$-convergence results. We consider equivalence relations of the partial sum of Fourier series from the early 20th century until the middle of. Second, we investigate the minor lineage of $L^1$-convergence theorem from W. H. Young to G. A. Fomin. Finally, we compare and reinterpret the $L^1$-convergence theorems.

• Communications for Statistical Applications and Methods
• /
• v.10 no.3
• /
• pp.955-969
• /
• 2003

The classical Fourier analysis, which is the typical frequency domain approach, is used to detect periodic trends that are of the sinusoidal shape in time series data. In this article, using a sequence of periodic step functions, describes an adaptive Fourier series where the patterns may take general periodic shapes that include sinusoidal as a special case. The results, which extend both Fourier analysis and Walsh-Fourier analysis, are applies to investigate the shape of the periodic component. Through the real data, compare the goodness-of-fit of the model using two methods, the adaptive Fourier method which is proposed method in this paper and classical Fourier method.

• Journal of the Korean Institute of Telematics and Electronics B
• /
• v.31B no.5
• /
• pp.26-33
• /
• 1994

Phase retrieval is concerned with the reconstruction of a signal from its Fourier transform magnitude (or intensity), which arises in many areas such as X-ray crystallography, optics, astronomy, or digital signal processing. In such areas, the Fourier transform phase of the desired signal is lost while measuring Fourier transform magnitude (F.T.M.). However, if a reference 'signal is added to the desired signal, then, in the Fourier trans form magnitude of the added signal, the Fourier transform phase of the desired signal is encoded. This paper addresses uniqueness and retrieval of the encoded Fourier phase of a multidimensional signal from the Fourier transform magnitude of the added signal along with the Fourier transform magnitude of the desired signal and the information of the additive reference signal. In Part I, several conditions under which the desired signal can be uniquely specified from the two Fourier transform magnitudes and the additive reference signal are presented. In Part II, the development of non-iterative algorithms and an iterative algorithm that may be used to reconstruct the desired signal(s) is considered.

• Journal of the Korean Institute of Telematics and Electronics B
• /
• v.31B no.5
• /
• pp.34-41
• /
• 1994

Phase retrieval is concerned with the reconstruction of a signal from its Fourier transform magnitude (or intensity), which arises in many areas such as X-ray crystallography, optics, astronomy, or digital signal processing In such areas, the Fourier transform phase of the desired signal is lost while measuring Fourier transform magnitude (F.T.M.). However, if a reference 'signal is added to the desired signal, then, in the Fourier trans form magnitude of the added signal, the Fourier transform phase of the desired signal is encoded This paper addresses uniqueness and retrieval of the encoded Fourier phase of a multidimensional signal from the Fourier transform magnitude of the added signal along with Fourier transform magnitude of the desired signal and the information of the additive reference signal In Part I, several conditions under which the desired signal can be uniquely specified from the two Fourier transform magnitudes and the additive reference signal are presented In Part II, the development of non-iterative algorithms and an iterative algorithm that may be used to reconstruct the desired signal (s) is considered

• The Journal of Korean Medicine
• /
• v.32 no.4
• /
• pp.139-148
• /
• 2011

Objective: The purpose of this study was to investigate the effect of food intake on the Fourier components of radial pulse wave. Methods: Thirty-one healthy male subjects participated in this study. Radial pulse was measured using 3 dimensional pulse imaging system (DMP-3000) before, right after, 40 minutes after, 80 minutes after and 120 minutes after food intake. Fourier transform was performed and the frequency and amplitude of Fourier components were analyzed. Results: 1. The frequency and the amplitude of Fourier components of radial pulse wave increased significantly after food intake. 2. The frequency of Fourier components increased right after food intake and then gradually decreased as time passed, however the amplitude of Fourier components increased and maintained certain levels and patterns throughout the experimental period of 120 minutes. 3. The change ratios of the frequency and the amplitude of Fourier components after food intake varied with the pulse measuring locations. Conclusions: Food intake exerts an influence on radial pulse wave, resulting in increase of frequency and amplitude of Fourier components. The change ratios of the frequency and the amplitude of Fourier components after food intake varied with the pulse measuring locations.

• Journal of the Chungcheong Mathematical Society
• /
• v.23 no.1
• /
• pp.1-10
• /
• 2010

In this note, we investigate the infinitesimal generators of the transformation groups induced by the Fourier-Gauss and Fourier-Mehler transforms on white noise functionals.

• Journal of the Korea Institute of Information and Communication Engineering
• /
• v.17 no.3
• /
• pp.540-545
• /
• 2013

The Fractional Fourier transform, as a generalization of the classical Fourier Transform, was first introduced in quantum mechanics. Because of its simple and useful properties of Fractional Fourier transform in time-frequency plane, various research results in sonar and radar signal processing have been introduced and shown superior results to conventional method utilizing Fourier transform until now. In this paper, we applied Fractional Fourier transform to sonar signal processing to detect and separate the overlapping linear frequency modulated signals. Experimental results show that received overlapping LFM(Linear Frequency Modulation) signals can be detected and separated effectively in Fractional Fourier transform domain.

• Bulletin of the Korean Mathematical Society
• /
• v.41 no.4
• /
• pp.785-803
• /
• 2004

We research properties of analytic functions which are exponentially decreasing or increasing. Also we show that the space of test functions is dense in the space of extended Fourier hyper-functions, and that the Fourier transform of the space of extended Fourier hyperfunctions into itself is an isomorphism and Parseval's inequality holds.

• Korean Journal of Mathematics
• /
• v.6 no.1
• /
• pp.47-66
• /
• 1998

In this paper, we introduce an $L_p$ analytic Fourier-Feynman transformation, show the existence of the $L_p$ analytic Fourier-Feynman transforms for a certain class of cylinder functionals on an abstract Wiener space, and investigate its interesting properties. Moreover, we define a convolution product for two functionals on the abstract Wiener space and establish the relationships between the Fourier-Feynman transform for the convolution product of two cylinder functionals and the Fourier-Feynman transform for each functional.

• Kyungpook Mathematical Journal
• /
• v.54 no.1
• /
• pp.43-63
• /
• 2014

The range spaces of Fourier cosine and sine transforms on $L^1$([0, ${\infty}$)) are characterized. Using Fourier cosine and sine type convolutions, Fourier cosine and sine transformable Boehmian spaces have been constructed, which properly contain $L^1$([0, ${\infty}$)). The Fourier cosine and sine transforms are extended to these Boehmian spaces consistently and their properties are established.