Estimates of averages of Fourier transforms with respect to general measures

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.

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

Geophysics ◽

10.1190/1.1442017 ◽

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.

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FOURIER TRANSFORMS OF FINITE LENGTH THEORETICAL GRAVITY ANOMALIES

Geophysics ◽

10.1190/1.1440804 ◽

1977 ◽

Vol 42 (7) ◽

pp. 1450-1457 ◽

Cited By ~ 5

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.

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Two notes on spectral synthesis for discrete Abelian groups

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100038950 ◽

1965 ◽

Vol 61 (3) ◽

pp. 617-620 ◽

Cited By ~ 17

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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Optimization of Stepsize in X-Ray Powder Diffractogram Collection

Powder Diffraction ◽

10.1017/s0885715600013099 ◽

1988 ◽

Vol 3 (1) ◽

pp. 32-38 ◽

Cited By ~ 4

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.

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Inversion of Fourier Transforms by Means of Scale-Frequency Series

International Journal of Engineering Mathematics ◽

10.1155/2014/686785 ◽

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.

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Uncertainty principles invariant under the fractional Fourier transform

The Journal of the Australian Mathematical Society Series B Applied Mathematics ◽

10.1017/s0334270000006986 ◽

1991 ◽

Vol 33 (2) ◽

pp. 180-191 ◽

Cited By ~ 39

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.

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Fourier transforms in generalized Fock spaces

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171290000655 ◽

1990 ◽

Vol 13 (3) ◽

pp. 431-441

Author(s):

John Schmeelk

Keyword(s):

Fourier Transform ◽

Taylor Series ◽

Fourier Transforms ◽

Fock Space ◽

Natural Generalization ◽

Fock Spaces ◽

Test Functions ◽

Tempered Distributions ◽

Tempered Distribution ◽

The Fourier Transform

A classical Fock space consists of functions of the form,Φ↔(ϕ0,ϕ1,…,ϕq,…),whereϕ0∈Candϕq∈L2(R3q),q≥1. We will replace theϕq,q≥1withq-symmetric rapid descent test functions within tempered distribution theory. This space is a natural generalization of a classical Fock space as seen by expanding functionals having generalized Taylor series. The particular coefficients of such series are multilinear functionals having tempered distributions as their domain. The Fourier transform will be introduced into this setting. A theorem will be proven relating the convergence of the transform to the parameter,s, which sweeps out a scale of generalized Fock spaces.

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The practice of using the fourier transform in processing the results of mathematical modeling of the hydrodynamics of getaways

Journal of Physics Conference Series ◽

10.1088/1742-6596/2131/3/032029 ◽

2021 ◽

Vol 2131 (3) ◽

pp. 032029

Author(s):

I Lipatov ◽

M Molchanova ◽

O Lebedev

Keyword(s):

Mathematical Modeling ◽

Fourier Transform ◽

Cross Sections ◽

Reference Data ◽

Fourier Transforms ◽

Scientific Analysis ◽

Engineering Measures ◽

The Fourier Transform ◽

Lock In

Abstract The article deals with the actual aspects of practical mathematical modeling of hydro-dynamic processes in the chambers of navigational locks. The use of direct and inverse Fourier transforms has been tested to obtain the representations of non-stationary graphs acceptable for analysis. Cross-sections of the water flow filling the chamber of a typical lock in the Volga-Don shipping channel were used as reference data (VDSC). The control sections in the flow were selected with a qualitatively different hydrodynamic nature of motion. A two-dimensional array of non-stationary data results were decomposed into Fourier series. The resulting graph of the amplitude-frequency spectrum was analyzed by the harmonics forming it. Its amplitude was taken as the criterion for the harmonics’ selection. After zeroing the insignificant harmonics, the inverse Fourier transform was performed. The quality of the data array approximation was controlled by visual overlay of the original graphs on the processed one. In all cases, it was possible to obtain the acceptable approximation results. This created a reliable basis for the scientific analysis and development of engineering measures for the implementation of safe ship passage through gateways. At the end of the article, a number of the data processing specific features are presented, caused by a variety of hydrodynamic features of the flow in various sections.

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On the detection of stellar differential rotation based on the Fourier transform of spectral line profiles

Publications of the Astronomical Society of Japan ◽

10.1093/pasj/psz129 ◽

2019 ◽

Vol 72 (1) ◽

Author(s):

Yoichi Takeda

Keyword(s):

Fourier Transform ◽

Spectral Line ◽

Differential Rotation ◽

Fourier Transforms ◽

Rotational Velocity ◽

Model Atmosphere ◽

Strict Sense ◽

Line Profiles ◽

Classical Formulation ◽

The Fourier Transform

Abstract It is known that stellar differential rotation can be detected by analyzing the Fourier transform of spectral line profiles, since the ratio of the first and second zero frequencies is a useful indicator. This approach essentially relies on the conventional formulation that the observed flux profile is expressible as a convolution of the rotational broadening function and the intrinsic profile, which implicitly assumes that the local intensity profile does not change over the disk. Although this postulation is unrealistic in the strict sense, how the result is affected by this approximation is still unclear. With the aim of examining this problem, flux profiles of several test lines (showing different center-to-limb variations) were simulated using a model atmosphere corresponding to a mid-F dwarf by integrating the intensity profiles for various combinations of vesin i (projected rotational velocity), α (differential degree), and i (inclination angle), and their Fourier transforms were computed to check whether the zeros are detected at the predicted positions or not. For this comparison a large grid of standard rotational broadening functions and their transforms/zeros were also calculated. It turned out that the situation depends critically on vesin i: In the case of vesin i ≳ 20 km s−1, where rotational broadening is predominant over other line broadening velocities (typically several km s−1), the first/second zeros of the transform are confirmed almost at the expected positions. In contrast, deviations begin to appear as vesin i is lowered, and the zero features of the transform are totally different from those expected at vesin i as low as ∼10 km s−1, which means that the classical formulation is no longer valid. Accordingly, while the zero-frequency approach is safely applicable to studying differential rotation in the former broader-line case, it would be difficult to practice for the latter sharp-line case.

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