scholarly journals On Fourier transforms of functions supported on sets of finite Lebesgue measure

1985 ◽  
Vol 106 (1) ◽  
pp. 180-183 ◽  
Author(s):  
Michael Benedicks
Keyword(s):  
Lebesgue Measure ◽  
Fourier Transforms ◽  
Finite Lebesgue Measure

Osculatory Packings by Spheres

1970 ◽  
Vol 13 (1) ◽  
pp. 59-64 ◽  
Author(s):  
David W. Boyd
Keyword(s):  
Hausdorff Dimension ◽  
Lebesgue Measure ◽  
Simple Proof ◽  
Pairwise Disjoint ◽  
Exponent Of Convergence ◽  
Residual Set ◽  
Open Set ◽  
Finite Lebesgue Measure

If U is an open set in Euclidean N-space EN which has finite Lebesgue measure |U| then a complete packing of U by open spheres is a collection C={Sn} of pairwise disjoint open spheres contained in U and such that Σ∞n=1|Sn| = |U|. Such packings exist by Vitali's theorem. An osculatory packing is one in which the spheres Sn are chosen recursively so that from a certain point on Sn+1 is the largest possible sphere contained in (Here S- will denote the closure of a set S). We give here a simple proof of the "well-known" fact that an osculatory packing is a complete packing. Our method of proof shows also that for osculatory packings, the Hausdorff dimension of the residual set is dominated by the exponent of convergence of the radii of the Sn.

The lemma of the logarithmic derivative for subharmonic functions

1996 ◽  
Vol 120 (2) ◽  
pp. 347-354 ◽  
Author(s):  
Walter Rudin
Keyword(s):  
Lebesgue Measure ◽  
Meromorphic Functions ◽  
Logarithmic Derivative ◽  
Value Distribution ◽  
Nevanlinna Characteristic ◽  
Value Distribution Theory ◽  
Subharmonic Functions ◽  
Classical Statement ◽  
Image Position ◽  
Finite Lebesgue Measure

The classical statement of the lemma in question [7], [3] is about meromorphic functions f on ℂ and says thatfor all r > 0, with the possible exception of a set of finite Lebesgue measure. Here T(r, f) is the Nevanlinna characteristic of f. The lemma plays an important role in value distribution theory.

Measures with Fourier Transforms in L2 of a Half-space

2011 ◽  
Vol 54 (1) ◽  
pp. 172-179
Author(s):  
Bassam Shayya
Keyword(s):  
Fourier Transform ◽  
Lebesgue Measure ◽  
Half Space ◽  
Fourier Transforms ◽  
Geometric Information ◽  
Absolutely Continuous ◽  
Compactly Supported ◽  
The Fourier Transform

AbstractWe prove that if the Fourier transform of a compactly supported measure is in L2 of a half-space, then the measure is absolutely continuous to Lebesgue measure. We then show how this result can be used to translate information about the dimensionality of a measure and the decay of its Fourier transform into geometric information about its support.

scholarly journals On two functionals involving the maximum of the torsion function

2018 ◽  
Vol 24 (4) ◽  
pp. 1585-1604 ◽  
Author(s):  
Antoine Henrot ◽  
Ilaria Lucardesi ◽  
Gérard Philippin
Keyword(s):  
Lebesgue Measure ◽  
Lower Bounds ◽  
Convex Sets ◽  
Upper And Lower Bounds ◽  
Dirichlet Eigenvalue ◽  
Torsion Function ◽  
Open Set ◽  
First Dirichlet Eigenvalue ◽  
Finite Lebesgue Measure

In this paper we investigate upper and lower bounds of two shape functionals involving the maximum of the torsion function. More precisely, we consider T(Ω)∕(M(Ω)|Ω|) and M(Ω)λ1(Ω), where Ω is a bounded open set of ℝd with finite Lebesgue measure |Ω|, M(Ω) denotes the maximum of the torsion function, T(Ω) the torsion, and λ1(Ω) the first Dirichlet eigenvalue. Particular attention is devoted to the subclass of convex sets.

scholarly journals Exponential polynomials with Fatou and non-escaping sets of finite Lebesgue measure

2020 ◽  
pp. 1-20
Author(s):  
MAREIKE WOLFF
Keyword(s):  
Lebesgue Measure ◽  
Exponential Polynomial ◽  
Fatou Set ◽  
Exponential Polynomials ◽  
Finite Lebesgue Measure

Abstract We give conditions ensuring that the Fatou set and the complement of the fast escaping set of an exponential polynomial f both have finite Lebesgue measure. Essentially, these conditions are designed such that $|f(z)|\ge \exp (|z|^\alpha )$ for some $\alpha>0$ and all z outside a set of finite Lebesgue measure.

scholarly journals Three Problems on Exponential Bases

2019 ◽  
Vol 62 (1) ◽  
pp. 55-70
Author(s):  
Laura De Carli ◽  
Alberto Mizrahi ◽  
Alexander Tepper
Keyword(s):  
Lebesgue Measure ◽  
Riesz Basis ◽  
Unit Cube ◽  
Exponential Basis ◽  
Exponential Bases ◽  
Finite Lebesgue Measure

AbstractWe consider three special and significant cases of the following problem. Let $D\subset \mathbb{R}^{d}$ be a (possibly unbounded) set of finite Lebesgue measure. Let $E(\mathbb{Z}^{d})=\{e^{2\unicode[STIX]{x1D70B}ix\cdot n}\}\text{}_{n\in \mathbb{Z}^{d}}$ be the standard exponential basis on the unit cube of $\mathbb{R}^{d}$. Find conditions on $D$ for which $E(\mathbb{Z}^{d})$ is a frame, a Riesz sequence, or a Riesz basis for $L^{2}(D)$.

scholarly journals On Efficiency and Localisation for the Torsion Function

2021 ◽  
Author(s):  
M. van den Berg ◽  
D. Bucur ◽  
T. Kappeler
Keyword(s):  
Lebesgue Measure ◽  
Schrödinger Operator ◽  
Schrodinger Operator ◽  
Dirichlet Laplacian ◽  
Torsion Function ◽  
Open Set ◽  
Finite Lebesgue Measure

AbstractWe consider the torsion function for the Dirichlet Laplacian −Δ, and for the Schrödinger operator −Δ + V on an open set ${\Omega }\subset \mathbb {R}^{m}$ Ω ⊂ ℝ m of finite Lebesgue measure $0<|{\Omega }|<\infty $ 0 < | Ω | < ∞ with a real-valued, non-negative, measurable potential V. We investigate the efficiency and the phenomenon of localisation for the torsion function, and their interplay with the geometry of the first Dirichlet eigenfunction.

Computer processing of high-resolution images of periodic specimens

1986 ◽  
Vol 44 ◽  
pp. 2-5
Author(s):  
W. Chiu ◽  
M.F. Schmid ◽  
T.-W. Jeng
Keyword(s):  
High Resolution ◽  
Fourier Transforms ◽  
Complex Structure ◽  
Distribution Functions ◽  
Multiple Image ◽  
Cryo Electron Microscopy ◽  
Structure Factors ◽  
Unit Cells ◽  
High Resolution Images ◽  
Phase Probability Distribution

Cryo-electron microscopy has been developed to the point where one can image thin protein crystals to 3.5 Å resolution. In our study of the crotoxin complex crystal, we can confirm this structural resolution from optical diffractograms of the low dose images. To retrieve high resolution phases from images, we have to include as many unit cells as possible in order to detect the weak signals in the Fourier transforms of the image. Hayward and Stroud proposed to superimpose multiple image areas by combining phase probability distribution functions for each reflection. The reliability of their phase determination was evaluated in terms of a crystallographic “figure of merit”. Grant and co-workers used a different procedure to enhance the signals from multiple image areas by vector summation of the complex structure factors in reciprocal space.

The extended Fourier algorithm: Application in discrete optics and electron holography

1994 ◽  
Vol 52 ◽  
pp. 918-919
Author(s):  
E. Voelkl ◽  
L. F. Allard
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Sampling Rate ◽  
Electron Holography ◽  
Optical Bench ◽  
Fourier Space ◽  
Ccd Cameras ◽  
Simulated Image ◽  
Initial Sampling ◽  
Fourier Algorithm

The conventional discrete Fourier transform can be extended to a discrete Extended Fourier transform (EFT). The EFT allows to work with discrete data in close analogy to the optical bench, where continuous data are processed. The EFT includes a capability to increase or decrease the resolution in Fourier space (thus the argument that CCD cameras with a higher number of pixels to increase the resolution in Fourier space is no longer valid). Fourier transforms may also be shifted with arbitrary increments, which is important in electron holography. Still, the analogy between the optical bench and discrete optics on a computer is limited by the Nyquist limit. In this abstract we discuss the capability with the EFT to change the initial sampling rate si of a recorded or simulated image to any other(final) sampling rate sf.

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