A Remark on Fourier Transforms

1936 ◽  
Vol 32 (2) ◽  
pp. 321-327 ◽  
Author(s):  
A. Zygmund
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Complex Function ◽  
The Fourier Transform ◽  
Image Position ◽  
The Right

1. Let f(x) be a complex function belonging to LP (−∞, ∞); i.e. let f(x) be measurable, and |f(x)|p integrable, over (−∞, ∞). The functionis called the Fourier transform of f(x), if the integral on the right exists, in some sense, for almost every value of y. It is well known that, if 1 ≤ p ≤ 2, the integral (1) converges in mean, with index p′ = p/(p – l)† i.e. thatwhere

The a-Local Operator Problem

1959 ◽  
Vol 11 ◽  
pp. 583-592 ◽  
Author(s):  
Louis de Branges
Keyword(s):  
Fourier Transform ◽  
Total Variation ◽  
Convergence Theorem ◽  
Borel Measure ◽  
Local Operator ◽  
Operator Problem ◽  
Dominated Convergence ◽  
The Fourier Transform ◽  
Image Position ◽  
The Right

Let be the Fourier transform of a Borel measure of finite total variation. The formulacan be justified if the integral on the right converges absolutely. ForwhereNow let h → 0 in both sides of this equation and use the Lebesgue dominated convergence theorem.

scholarly journals On fourier transforms of radial functions

1965 ◽  
Vol 5 (3) ◽  
pp. 289-298 ◽  
Author(s):  
James L. Griffith
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Radial Functions ◽  
Cartesian Space ◽  
The Fourier Transform ◽  
Image Position

The Fourier transformF(y) of a functionf(t) inL1(Ek) whereEkis thek-dimensional cartesian space will be defined by.

Fourier Transforms of Unbounded Measures

1979 ◽  
Vol 31 (6) ◽  
pp. 1281-1292 ◽  
Author(s):  
James Stewart
Keyword(s):  
Fourier Transform ◽  
Real Line ◽  
Fourier Transforms ◽  
Dual Group ◽  
Locally Compact Abelian Group ◽  
Stieltjes Transform ◽  
The Real ◽  
Bounded Complex ◽  
The Fourier Transform ◽  
Image Position

1. Introduction. One of the basic objects of study in harmonic analysis is the Fourier transform (or Fourier-Stieltjes transform) μ of a bounded (complex) measure μ on the real line R:(1.1)More generally, if μ is a bounded measure on a locally compact abelian group G, then its Fourier transform is the function(1.2)where Ĝ is the dual group of G and One answer to the question “Which functions can be represented as Fourier transforms of bounded measures?” was given by the following criterion due to Schoenberg [11] for the real line and Eberlein [5] in general: f is a Fourier transform of a bounded measure if and only if there is a constant M such that(1.3)for all ϕ ∈ L1(G) where

The Fourier Transforms of Smooth Measures on Hypersurfaces of Rn + 1

1986 ◽  
Vol 38 (2) ◽  
pp. 328-359 ◽  
Author(s):  
Bernard Marshall
Keyword(s):  
Fourier Transform ◽  
Asymptotic Expansion ◽  
Unit Sphere ◽  
Gaussian Curvature ◽  
Fourier Transforms ◽  
Surface Measure ◽  
Positive Gaussian Curvature ◽  
Smooth Measures ◽  
The Fourier Transform ◽  
Image Position

The Fourier transform of the surface measure on the unit sphere in Rn + 1, as is well-known, equals the Bessel functionIts behaviour at infinity is described by an asymptotic expansionThe purpose of this paper is to obtain such an expression for surfaces Σ other than the unit sphere. If the surface Σ is a sufficiently smooth compact n-surface in Rn + 1 with strictly positive Gaussian curvature everywhere then with only minor changes in the main term, such an asymptotic expansion exists. This result was proved by E. Hlawka in [3]. A similar result concerned with the minimal smoothness of Σ was later obtained by C. Herz [2].

Notes on special systems of orthogonal functions (IV): the orthogonal functions of whittaker's cardinal series

1941 ◽  
Vol 37 (4) ◽  
pp. 331-348 ◽  
Author(s):  
G. H. Hardy
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Orthogonal System ◽  
Orthogonal Functions ◽  
The Fourier Transform ◽  
Image Position ◽  
Parseval’S Theorem

Suppose that n runs through all integral values, that (øn) is a system of normal orthogonal functions for the interval (−∞, ∞), and that ψn is the Fourier transform of øn. Then, by Parseval's theorem for Fourier transforms,and (ψn) is also a normal orthogonal system.

Wavelet packets associated with linear canonical transform on spectrum

2021 ◽  
pp. 2150030
Author(s):  
M. Younus Bhat ◽  
Aamir H. Dar
Keyword(s):  
Fourier Transform ◽  
Multiresolution Analysis ◽  
Fourier Transforms ◽  
Fractional Fourier Transform ◽  
Integral Transforms ◽  
Wavelet Packets ◽  
Linear Canonical Transform ◽  
Fresnel Transform ◽  
The Fourier Transform ◽  
Vector Valued

The linear canonical transform (LCT) provides a unified treatment of the generalized Fourier transforms in the sense that it is an embodiment of several well-known integral transforms including the Fourier transform, fractional Fourier transform, Fresnel transform. Using this fascinating property of LCT, we, in this paper, constructed associated wavelet packets. First, we construct wavelet packets corresponding to nonuniform Multiresolution analysis (MRA) associated with LCT and then those corresponding to vector-valued nonuniform MRA associated with LCT. We investigate their various properties by means of LCT.

An analysis of the spectrum of a gravity anomaly over an inclined dike

Geophysics ◽  
1985 ◽  
Vol 50 (9) ◽  
pp. 1500-1501
Author(s):  
B. N. P. Agarwal ◽  
D. Sita Ramaiah
Keyword(s):  
Fourier Transform ◽  
Gravity Anomaly ◽  
Fourier Spectrum ◽  
Fourier Transforms ◽  
Weighted Average ◽  
Window Function ◽  
Average Spectrum ◽  
Amplitude Spectra ◽  
Distance Data ◽  
The Fourier Transform

Bhimasankaram et al. (1977) used Fourier spectrum analysis for a direct approach to the interpretation of gravity anomaly over a finite inclined dike. They derived several equations from the real and imaginary components and from the amplitude and phase spectra to relate various parameters of the dike. Because the width 2b of the dike (Figure 1) appears only in sin (ωb) term—ω being the angular frequency—they determined its value from the minima/zeroes of the amplitude spectra. The theoretical Fourier spectrum uses gravity field data over an infinite distance (length), whereas field observations are available only for a limited distance. Thus, a set of observational data is viewed as a product of infinite‐distance data with an appropriate window function. Usually, a rectangular window of appropriate distance (width) and of unit magnitude is chosen for this purpose. The Fourier transform of the finite‐distance and discrete data is thus represented by convolution operations between Fourier transforms of the infinite‐distance data, the window function, and the comb function. The combined effect gives a smooth, weighted average spectrum. Thus, the Fourier transform of actual observed data may differ substantially from theoretic data. The differences are apparent for low‐ and high‐frequency ranges. As a result, the minima of the amplitude spectra may change considerably, thereby rendering the estimate of the width of the dike unreliable from the roots of the equation sin (ωb) = 0.

scholarly journals On dilation equations and the Hölder continuity of the de Rham functions

1994 ◽  
Vol 36 (3) ◽  
pp. 309-311
Author(s):  
Yibiao Pan
Keyword(s):  
Fourier Transform ◽  
Approximation Method ◽  
Hölder Continuity ◽  
Simple Approximation ◽  
Holder Continuity ◽  
Integrable Solution ◽  
De Rham ◽  
The Fourier Transform ◽  
Image Position ◽  
Dilation Equations

AbstractWe use a simple approximation method to prove the Holder continuity of the generalized de Rham functions.1. Consider the following dilatation equationwhere |α|<l/2. Suppose that f is an integrable solution of (1); then f must satisfywhere is the Fourier transform of f, andwhich immediately leads to

FOURIER TRANSFORMS OF FINITE LENGTH THEORETICAL GRAVITY ANOMALIES

Geophysics ◽  
1977 ◽  
Vol 42 (7) ◽  
pp. 1450-1457 ◽  
Author(s):  
Robert D. Regan ◽  
William J. Hinze
Keyword(s):  
Fourier Transform ◽  
Analytical Solution ◽  
Finite Length ◽  
Fourier Transforms ◽  
Gravity Anomalies ◽  
Mathematical Structure ◽  
Practical Application ◽  
Convolution Integrals ◽  
The Fourier Transform ◽  
Interpretation Method

The mathematical structure of the Fourier transformations of theoretical gravity anomalies of several geometrically simple bodies appears to have distinct advantages in the interpretation of these anomalies. However, the practical application of this technique is dependent upon the transformation of an observed gravity anomaly of finite length. Ideally, interpretation methods similar to those for the transformations of the theoretical gravity anomalies should be developed for anomalies of a finite length. However, the mathematical complexity of the convolution integrals in the transform calculations of theoretical anomaly segments indicate that no general closed analytical solution useful for interpretation is available. Thus, in order to utilize the Fourier transform interpretation method, the data must be of sufficient length for the finite transform to closely approximate the theoretical transforms.

Two notes on spectral synthesis for discrete Abelian groups

1965 ◽  
Vol 61 (3) ◽  
pp. 617-620 ◽  
Author(s):  
R. J. Elliott
Keyword(s):  
Fourier Transform ◽  
Dual Space ◽  
Fourier Transforms ◽  
Spectral Synthesis ◽  
Division Problem ◽  
Exponential Functions ◽  
Translation Invariant ◽  
Annihilator Ideal ◽  
The Fourier Transform ◽  
Translation Invariant Subspace

Introduction. Spectral synthesis is the study of whether functions in a certain set, usually a translation invariant subspace (a variety), can be synthesized from certain simple functions, exponential monomials, which are contained in the set. This problem is transformed by considering the annihilator ideal in the dual space, and after taking the Fourier transform the problem becomes one of deciding whether a function is in a certain ideal, that is, we have a ‘division problem’. Because of this we must take into consideration the possibility of the Fourier transforms of functions having zeros of order greater than or equal to 1. This is why, in the original situation, we study whether varieties are generated by their exponential monomials, rather than just their exponential functions. This viewpoint of the problem as a division question, of course, perhaps throws light on why Wiener's Tauberian theorem works, and is implicit in the construction of Schwartz's and Malliavin's counter examples to spectral synthesis in L1(G) (cf. Rudin ((4))).

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