scholarly journals The Abel summation of the inverse Fourier transform of the homogeneous functions in $R^n$

2019 ◽  
pp. 98-107
Author(s):  
Sergey Viktorovich Arkhipov ◽  
Keyword(s):  
Fourier Transform ◽  
Inverse Fourier Transform ◽  
Homogeneous Functions ◽  
Abel Summation

scholarly journals Smooth Wavelet Approximations of Truncated Legendre Polynomials via the Jacobi Theta Function

2014 ◽  
Vol 2014 ◽  
pp. 1-24 ◽  
Author(s):  
David W. Pravica ◽  
Njinasoa Randriampiry ◽  
Michael J. Spurr
Keyword(s):  
Fourier Transform ◽  
Theta Function ◽  
Legendre Polynomial ◽  
Legendre Polynomials ◽  
Inverse Fourier Transform ◽  
Recursion Relations ◽  
Jacobi Theta Function ◽  
Convergence Results ◽  
The Family ◽  
Reciprocal Symmetry

The family ofnth orderq-Legendre polynomials are introduced. They are shown to be obtainable from the Jacobi theta function and to satisfy recursion relations and multiplicatively advanced differential equations (MADEs) that are analogues of the recursion relations and ODEs satisfied by thenth degree Legendre polynomials. Thenth orderq-Legendre polynomials are shown to have vanishingkth moments for0≤k<n, as does thenth degree truncated Legendre polynomial. Convergence results are obtained, approximations are given, a reciprocal symmetry is shown, and nearly orthonormal frames are constructed. Conditions are given under which a MADE remains a MADE under inverse Fourier transform. This is used to construct new wavelets as solutions of MADEs.

Effect of the Condition Number of the Inverse Fourier Transform Matrix on the Solution Behavior of the Time Spectral Equation System

2020 ◽  
Author(s):  
Sen Zhang ◽  
Dingxi Wang ◽  
Yi Li ◽  
Hangkong Wu ◽  
Xiuquan Huang
Keyword(s):  
Fourier Transform ◽  
Stability Analysis ◽  
Condition Number ◽  
Inverse Fourier Transform ◽  
Real Life ◽  
Equation System ◽  
Time Instant ◽  
Transform Matrix ◽  
Spectral Equation ◽  
The Impact

Abstract The time spectral method is a very popular reduced order frequency method for analyzing unsteady flow due to its advantage of being easily extended from an existing steady flow solver. Condition number of the inverse Fourier transform matrix used in the method can affect the solution convergence and stability of the time spectral equation system. This paper aims at evaluating the effect of the condition number of the inverse Fourier transform matrix on the solution stability and convergence of the time spectral method from two aspects. The first aspect is to assess the impact of condition number using a matrix stability analysis based upon the time spectral form of the scalar advection equation. The relationship between the maximum allowable Courant number and the condition number will be derived. Different time instant groups which lead to the same condition number are also considered. Three numerical discretization schemes are provided for the stability analysis. The second aspect is to assess the impact of condition number for real life applications. Two case studies will be provided: one is a flutter case, NASA rotor 67, and the other is a blade row interaction case, NASA stage 35. A series of numerical analyses will be performed for each case using different time instant groups corresponding to different condition numbers. The conclusion drawn from the two real life case studies will corroborate the relationship derived from the matrix stability analysis.

scholarly journals The Bernstein Projector Determined by a Weak Associate Class of Good Cosets

2021 ◽  
Author(s):  
Yeansu Kim ◽  
Loren Spice ◽  
Sandeep Varma
Keyword(s):  
Fourier Transform ◽  
Characteristic Function ◽  
Lie Algebra ◽  
Equivalence Relation ◽  
Characteristic Zero ◽  
Closed Subset ◽  
Inverse Fourier Transform ◽  
Reductive Group ◽  
Associate Class ◽  
Weak Associate

Abstract Let ${\text G}$ be a reductive group over a $p$-adic field $F$ of characteristic zero, with $p \gg 0$, and let $G={\text G}(F)$. In [ 15], J.-L. Kim studied an equivalence relation called weak associativity on the set of unrefined minimal $K$-types for ${\text G}$ in the sense of A. Moy and G. Prasad. Following [ 15], we attach to the set $\overline{\mathfrak{s}}$ of good $K$-types in a weak associate class of positive-depth unrefined minimal $K$-types a ${G}$-invariant open and closed subset $\mathfrak{g}_{\overline{\mathfrak{s}}}$ of the Lie algebra $\mathfrak{g} = {\operatorname{Lie}}({\text G})(F)$, and a subset $\tilde{{G}}_{\overline{\mathfrak{s}}}$ of the admissible dual $\tilde{{G}}$ of ${G}$ consisting of those representations containing an unrefined minimal $K$-type that belongs to $\overline{\mathfrak{s}}$. Then $\tilde{{G}}_{\overline{\mathfrak{s}}}$ is the union of finitely many Bernstein components of ${G}$, so that we can consider the Bernstein projector $E_{\overline{\mathfrak{s}}}$ that it determines. We show that $E_{\overline{\mathfrak{s}}}$ vanishes outside the Moy–Prasad ${G}$-domain ${G}_r \subset{G}$, and reformulate a result of Kim as saying that the restriction of $E_{\overline{\mathfrak{s}}}$ to ${G}_r\,$, pushed forward via the logarithm to the Moy–Prasad ${G}$-domain $\mathfrak{g}_r \subset \mathfrak{g}$, agrees on $\mathfrak{g}_r$ with the inverse Fourier transform of the characteristic function of $\mathfrak{g}_{\overline{\mathfrak{s}}}$. This is a variant of one of the descriptions given by R. Bezrukavnikov, D. Kazhdan, and Y. Varshavsky in [8] for the depth-$r$ Bernstein projector.

scholarly journals On the Inverse Fourier Transform of the Planck-Einstein law

2021 ◽  
Author(s):  
Alireza Jamali
Keyword(s):  
Fourier Transform ◽  
Inverse Fourier Transform ◽  
Maximum Frequency ◽  
Principle Of Maximum

After proposing a natural metric for the space in which particles spin which implements the principle of maximum frequency, E=hf is generalised and its inverse Fourier transform is calculated.

Quantitative characterization, classification and reconstruction of oocyst shapes of Eimeria species from cattle

Parasitology ◽  
1998 ◽  
Vol 116 (1) ◽  
pp. 21-28 ◽  
Author(s):  
C. SOMMER
Keyword(s):  
Fourier Transform ◽  
Quantitative Data ◽  
Inverse Fourier Transform ◽  
Morphological Variability ◽  
Eimeria Species ◽  
Equal Weight ◽  
Agglomerative Clustering ◽  
Multivariate Statistical ◽  
Mean Values ◽  
Size And Shape

This study reports on morphological variability of Eimeria species, which may be given either by drawings or as quantitative data. The drawings may be used to facilitate identification by eye of ‘unknown’ Eimeria specimens, whereas quantitative data may serve as a reference set for identification by multivariate statistical techniques. The morphology of 810 Eimeria specimens was defined in binary (b/w) digital images by pixels of their oocyst outline. A Fourier transform of pixel positions yielded size and shape features. To classify coccidia, the quantitative data were employed in an agglomerative clustering by average linkage algorithm with equal weight assigned to size and shape. An inverse Fourier transform served to reconstruct oocyst outlines, i.e. outlines of average shape and size, from mean values of features in resulting clusters. Clusters were subsequently identified based on their average morphology by comparison with drawings of species in an earlier taxonomical work. Five hundred oocyst outlines were simulated for each cluster representing a species, and shape/size variability was presented in contour diagrams. Differences in species shapes, and correspondence in length and width, were seen after reconstruction by inverse Fourier transform and comparison with earlier studies.

scholarly journals Inverse Fourier Transform in the Gamma Coordinate System

2011 ◽  
Vol 2011 ◽  
pp. 1-16
Author(s):  
Yuchuan Wei ◽  
Hengyong Yu ◽  
Ge Wang
Keyword(s):  
Fourier Transform ◽  
Coordinate System ◽  
Inverse Fourier Transform ◽  
Rotation Axis ◽  
General Scheme ◽  
Weight Functions ◽  
Projection Data ◽  
Parallel Beam ◽  
Beam Geometry ◽  
Parallel Beam Geometry

This paper provides auxiliary results for our general scheme of computed tomography. In 3D parallel-beam geometry, we first demonstrate that the inverse Fourier transform in different coordinate systems leads to different reconstruction formulas and explain why the Radon formula cannot directly work with truncated projection data. Also, we introduce a gamma coordinate system, analyze its properties, compute the Jacobian of the coordinate transform, and define weight functions for the inverse Fourier transform assuming a simple scanning model. Then, we generate Orlov's theorem and a weighted Radon formula from the inverse Fourier transform in the new system. Furthermore, we present the motion equation of the frequency plane and the conditions for sharp points of the instantaneous rotation axis. Our analysis on the motion of the frequency plane is related to the Frenet-Serret theorem in the differential geometry.

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