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))).

scholarly journals Convolutors on $${\mathcal S}_{\omega }({\mathbb R}^N)$$

2021 ◽  
Vol 115 (4) ◽  
Author(s):  
Angela A. Albanese ◽  
Claudio Mele
Keyword(s):  
Fourier Transform ◽  
Dual Space ◽  
Strong Operator ◽  
Dual Spaces ◽  
Structure Theorems ◽  
The Fourier Transform ◽  
Decreasing Functions

AbstractIn this paper we continue the study of the spaces $${\mathcal O}_{M,\omega }({\mathbb R}^N)$$ O M , ω ( R N ) and $${\mathcal O}_{C,\omega }({\mathbb R}^N)$$ O C , ω ( R N ) undertaken in Albanese and Mele (J Pseudo-Differ Oper Appl, 2021). We determine new representations of such spaces and we give some structure theorems for their dual spaces. Furthermore, we show that $${\mathcal O}'_{C,\omega }({\mathbb R}^N)$$ O C , ω ′ ( R N ) is the space of convolutors of the space $${\mathcal S}_\omega ({\mathbb R}^N)$$ S ω ( R N ) of the $$\omega $$ ω -ultradifferentiable rapidly decreasing functions of Beurling type (in the sense of Braun, Meise and Taylor) and of its dual space $${\mathcal S}'_\omega ({\mathbb R}^N)$$ S ω ′ ( R N ) . We also establish that the Fourier transform is an isomorphism from $${\mathcal O}'_{C,\omega }({\mathbb R}^N)$$ O C , ω ′ ( R N ) onto $${\mathcal O}_{M,\omega }({\mathbb R}^N)$$ O M , ω ( R N ) . In particular, we prove that this isomorphism is topological when the former space is endowed with the strong operator lc-topology induced by $${\mathcal L}_b({\mathcal S}_\omega ({\mathbb R}^N))$$ L b ( S ω ( R N ) ) and the last space is endowed with its natural lc-topology.

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 The Fourier transform of thick distributions

2020 ◽  
pp. 1-26
Author(s):  
Ricardo Estrada ◽  
Jasson Vindas ◽  
Yunyun Yang
Keyword(s):  
Fourier Transform ◽  
Dual Space ◽  
Canonical Projection ◽  
Test Functions ◽  
Tempered Distributions ◽  
Delta Functions ◽  
The Fourier Transform ◽  
The One ◽  
One Point Compactification ◽  
Logarithmic Type

We first construct a space [Formula: see text] whose elements are test functions defined in [Formula: see text] the one point compactification of [Formula: see text] that have a thick expansion at infinity of special logarithmic type, and its dual space [Formula: see text] the space of sl-thick distributions. We show that there is a canonical projection of [Formula: see text] onto [Formula: see text] We study several sl-thick distributions and consider operations in [Formula: see text] We define and study the Fourier transform of thick test functions of [Formula: see text] and thick tempered distributions of [Formula: see text] We construct isomorphisms [Formula: see text] [Formula: see text] that extend the Fourier transform of tempered distributions, namely, [Formula: see text] and [Formula: see text] where [Formula: see text] are the canonical projections of [Formula: see text] or [Formula: see text] onto [Formula: see text] We determine the Fourier transform of several finite part regularizations and of general thick delta functions.

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.

scholarly journals A New Application Methodology of the Fourier Transform for Rational Approximation of the Complex Error Function

2016 ◽  
Vol 8 (1) ◽  
pp. 14 ◽  
Author(s):  
S. M. Abrarov ◽  
B. M. Quine
Keyword(s):  
Fourier Transform ◽  
Rational Approximation ◽  
Error Function ◽  
Practical Importance ◽  
Exponential Functions ◽  
New Approach ◽  
Practical Applications ◽  
Input Parameters ◽  
The Fourier Transform

<p>This paper presents a new approach in application of the Fourier transform to the complex error function resulting in an efficient rational approximation. Specifically, the computational test shows that with only $17$ summation terms the obtained rational approximation of the complex error function provides accuracy ${10^{ - 15}}$ over the most domain of practical importance $0 \le x \le 40,000$ and ${10^{ - 4}} \le y \le {10^2}$ required for the HITRAN-based spectroscopic applications. Since the rational approximation does not contain trigonometric or exponential functions dependent upon the input parameters $x$ and $y$, it is rapid in computation. Such an example demonstrates that the considered methodology of the Fourier transform may be advantageous in practical applications.</p>

Optimization of Stepsize in X-Ray Powder Diffractogram Collection

1988 ◽  
Vol 3 (1) ◽  
pp. 32-38 ◽  
Author(s):  
David G. Cameron ◽  
Ernest E. Armstrong
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
X Ray Diffraction ◽  
Step Size ◽  
Reasonable Estimate ◽  
X Ray ◽  
Fourier Transform Methods ◽  
Step Interval ◽  
The Fourier Transform ◽  
Selection Of

AbstractFourier transform methods of smoothing and interpolation are applied to X-ray diffraction data. It is shown that, frequently, too small a step size is used. Major gains are to be expected by selection of the optimum step size and use of these methods.A comparison of Fourier transforms of diffractograms of quartz measured between 67 and 69° 2θ, collected at varying step intervals (0.1 to 0.01° 2θ) was used to illustrate these applications. By examining the Fourier transform of the diffractogram and noting where it decays to die baseline, a reasonable estimate of the optimal step interval can be obtained. In addition, Fourier interpolation can be used to enhance the appearance of the diffractogram, approximating a continuous plot.

scholarly journals Inversion of Fourier Transforms by Means of Scale-Frequency Series

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Nassar H. S. Haidar
Keyword(s):  
Fourier Transform ◽  
Nonlinear Programming ◽  
Fourier Transforms ◽  
Inverse Fourier Transform ◽  
Double Series ◽  
Nonlinear Programming Problem ◽  
Continuous Scale ◽  
Free Parameters ◽  
The Fourier Transform ◽  
Dual Scale

We report on inversion of the Fourier transform when the frequency variable can be scaled in a variety of different ways that improve the resolution of certain parts of the frequency domain. The corresponding inverse Fourier transform is shown to exist in the form of two dual scale-frequency series. Upon discretization of the continuous scale factor, this Fourier transform series inverse becomes a certain nonharmonic double series, a discretized scale-frequency (DSF) series. The DSF series is also demonstrated, theoretically and practically, to be rate-optimizable with respect to its two free parameters, when it satisfies, as an entropy maximizer, a pertaining recursive nonlinear programming problem incorporating the entropy-based uncertainty principle.

scholarly journals Uncertainty principles invariant under the fractional Fourier transform

1991 ◽  
Vol 33 (2) ◽  
pp. 180-191 ◽  
Author(s):  
David Mustard
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Fractional Fourier Transform ◽  
Intrinsic Property ◽  
Continuous Group ◽  
Uncertainty Principles ◽  
New Family ◽  
Special Cases ◽  
Fractional Fourier Transforms ◽  
The Fourier Transform

AbstractUncertainty principles like Heisenberg's assert an inequality obeyed by some measure of joint uncertainty associated with a function and its Fourier transform. The more groups under which that measure is invariant, the more that measure represents an intrinsic property of the underlying object represented by the given function. The Fourier transform is imbedded in a continuous group of operators, the fractional Fourier transforms, but the Heisenberg measure of overall spread turns out not to be invariant under that group. A new family is developed of measures that are invariant under the group of fractional Fourier transforms and that obey associated uncertainty principles. The first member corresponds to Heisenberg's measure but is generally smaller than his although equal to it in special cases.

Estimates of averages of Fourier transforms with respect to general measures

2003 ◽  
Vol 133 (4) ◽  
pp. 943-950 ◽  
Author(s):  
Per Sjölin ◽  
Fernando Soria
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
The Fourier Transform

We study a connection between the L2 average decay of the Fourier transform of functions with respect to a given measure and the Hausdorff behaviour of that measure.

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