scholarly journals Lattice Tilings by Cubes: Whole, Notched and Extended

10.37236/1352 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Mihail Kolountzakis
Keyword(s):  
Fourier Transform ◽  
Harmonic Analysis ◽  
Equivalent Form ◽  
Unit Cube ◽  
Indicator Function ◽  
Geometric Method ◽  
Zero Set ◽  
New Class ◽  
Lattice Tiling ◽  
The Fourier Transform

We discuss some problems of lattice tiling via Harmonic Analysis methods. We consider lattice tilings of ${\bf R}^d$ by the unit cube in relation to the Minkowski Conjecture (now a theorem of Hajós) and give a new equivalent form of Hajós's theorem. We also consider "notched cubes" (a cube from which a rectangle has been removed from one of the corners) and show that they admit lattice tilings. This has also been been proved by S. Stein by a direct geometric method. Finally, we exhibit a new class of simple shapes that admit lattice tilings, the "extended cubes", which are unions of two axis-aligned rectangles that share a vertex and have intersection of odd codimension. In our approach we consider the Fourier Transform of the indicator function of the tile and try to exhibit a lattice of appropriate volume in its zero-set.

Translation uniqueness of phase retrieval and magnitude retrieval of band-limited signals

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Lung-Hui Chen
Keyword(s):  
Fourier Transform ◽  
Entire Function ◽  
Convex Hull ◽  
Scattering Theory ◽  
Phase Retrieval ◽  
Indicator Function ◽  
Band Limited ◽  
The Fourier Transform ◽  
Complex Valued ◽  
Spectral Invariant

Abstract In this paper, we discuss how to partially determine the Fourier transform F ⁢ ( z ) = ∫ - 1 1 f ⁢ ( t ) ⁢ e i ⁢ z ⁢ t ⁢ 𝑑 t , z ∈ ℂ , F(z)=\int_{-1}^{1}f(t)e^{izt}\,dt,\quad z\in\mathbb{C}, given the data | F ⁢ ( z ) | {\lvert F(z)\rvert} or arg ⁡ F ⁢ ( z ) {\arg F(z)} for z ∈ ℝ {z\in\mathbb{R}} . Initially, we assume [ - 1 , 1 ] {[-1,1]} to be the convex hull of the support of the signal f. We start with reviewing the computation of the indicator function and indicator diagram of a finite-typed complex-valued entire function, and then connect to the spectral invariant of F ⁢ ( z ) {F(z)} . Then we focus to derive the unimodular part of the entire function up to certain non-uniqueness. We elaborate on the translation of the signal including the non-uniqueness associates of the Fourier transform. We show that the phase retrieval and magnitude retrieval are conjugate problems in the scattering theory of waves.

scholarly journals Beyond Random Causes: Harmonic Analysis of Business Cycles at the Moscow Conjuncture Institute

2021 ◽  
Author(s):  
Marco Paulo Vianna Franco ◽  
Leonardo Costa Ribeiro ◽  
Eduardo da Motta e Albuquerque
Keyword(s):  
Fourier Transform ◽  
Business Cycles ◽  
Harmonic Analysis ◽  
Research Agenda ◽  
Random Events ◽  
Cyclic Processes ◽  
Push Forward ◽  
The Fourier Transform

This article proposes a historical assessment of harmonic analysis of business cycles and its ability to both decompose and build cycles, as received at the Moscow Conjuncture Institute. It traces how the Fourier transform arrived at the Institute, mediated by Henry L. Moore, in the works and actions of Albert Vainshtein, Nikolai Chetverikov, and Nikolai Kondratiev, ultimately leading to Eugen Slutsky’s well-known 1927 article The Summation of Random Causes as the Source of Cyclic Processes. Although the evidence does not warrant the assumption that there was an orchestrated effort at the Institute to push forward a research agenda on harmonic analysis of business cycles, it certainly unfolded as more than the summation of random events and individual incursions. Moreover, the Institute as a whole could have produced much more on this matter if it had escaped Stalinist oppression for at least a few more years.

scholarly journals On translation-bounded measures

1984 ◽  
Vol 37 (1) ◽  
pp. 139-142
Author(s):  
A. P. Robertson ◽  
M. L. Thornett
Keyword(s):  
Fourier Transform ◽  
Positive Measure ◽  
Borel Subset ◽  
Indicator Function ◽  
The Fourier Transform

AbstractIt is shown that a positive measure μ on the Borel subsets of Rk is translation-bounded if and only if the Fourier transform of the indicator function of every bounded Borel subset of Rk belongs to L2(μ).

scholarly journals Inversion formulas for Riemann-Liouville transform and its dual associated with singular partial differential operators

2006 ◽  
Vol 2006 ◽  
pp. 1-26 ◽  
Author(s):  
C. Baccar ◽  
N. B. Hamadi ◽  
L. T. Rachdi
Keyword(s):  
Fourier Transform ◽  
Harmonic Analysis ◽  
Differential Operators ◽  
Partial Differential ◽  
Inversion Formulas ◽  
Plancherel Theorem ◽  
Partial Differential Operators ◽  
The Fourier Transform

We define Riemann-Liouville transformℛαand its dualtℛαassociated with two singular partial differential operators. We establish some results of harmonic analysis for the Fourier transform connected withℛα. Next, we prove inversion formulas for the operatorsℛα,tℛαand a Plancherel theorem fortℛα.

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