The Funk-Hecke formula, harmonic polynomials, and derivatives of radial distributions

Estrada, Ricardo

doi:10.5269/bspm.v37i3.34198

Cited by 7 publications

(9 citation statements)

References 20 publications

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“…Notice that this delta part is in fact a spherical delta part 9. Naturally, when k − n = 2m ≥ 0, the projection of the thick delta part is precisely the distributional delta part, and this agrees with[36, (7.7)].…”

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confidence: 80%

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The Fourier transform of thick distributions

2020

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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.

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confidence: 80%

“…Proof. Notice that for a general k, the Fourier transform of r λ Y k (w) , a homogeneous distribution of degree λ, is homogeneous of degree − (λ + n) , so that the Funk-Hecke formula [18,20] as presented in [9] yields that…”

Section: Some Fourier Transformsmentioning

confidence: 99%

The Fourier transform of thick distributions

2020

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“…The equations of the form (3.1) can be brought to diagonal form, by means of a transformation to spherical harmonics (essentially the so-called Funk-Hecke theorem [28]).…”

Section: Invertibility Of the Transformmentioning

confidence: 99%

Tools for visualizing and analyzing Fourier space sampling in Cryo-EM

Baldwin

Lyumkis

2021

Progress in Biophysics and Molecular Biology

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A complete understanding of how an orientation distribution contributes to a cryo-EM reconstruction remains lacking. It is necessary to begin critically assessing the set of views to gain an understanding of its effect on experimental reconstructions. Toward that end, we recently suggested that the type of orientation distribution may alter resolution measures in a systematic manner. We introduced the sampling compensation factor (SCF), which incorporates how the collection geometry might change the spectral signal-to-noise ratio (SSNR), irrespective of the other experimental aspects. We show here that knowledge of the sampling restricted to spherical surfaces of sufficiently large radii in Fourier space is equivalent to knowledge of the set of projection views. Moreover, the SCF geometrical factor may be calculated from one such surface. To aid cryo-EM researchers, we developed a graphical user interface (GUI) tool that evaluates experimental orientation distributions. The GUI returns plots of projection directions, sampling constrained to the surface of a sphere, the SCF value, the fraction of the empty region of Fourier space, and a histogram of the sampling values over the points on a sphere. Finally, a fixed tilt angle may be incorporated to determine how tilting the grid during collection may improve the distribution of views and Fourier space sampling. We advocate this simple conception of sampling and the use of such tools as a complement to the distribution of views to capture the different aspects of the effect of projection directions on cryo-EM reconstructions.

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“…Therefore, h Y = q Y ∇ δ(x) for some polynomial q Y . The Funk-Hecke formula [9,12], as presented in [3], shows that there are constants λ k , that depend on k but not otherwise on Y, such that…”

Section: Preliminaries and Notationmentioning

confidence: 99%

“…As we explain in the Section 6, for a general α the simplicity of (1.4) is just lost ¶ , but (1.5) remains almost the same for any α. Actually if we replace r 2−n by a more general function f (r) (1.5) remains basically the same but (1.4) is lost; this is a consequence of the Funk-Hecke formula [3,9,12].…”

Section: Introductionmentioning

confidence: 99%

Harmonic Polynomials Via Differentiation

Estrada¹

2018

ATA

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It is well-known that if p is a homogeneous polynomial of degree k in n variables, p ∈ P k , then the ordinary derivative p(∇) r 2−n has the form A n,k Y(x)r 2−n−2k where A n,k is a constant and where Y is a harmonic homogeneous polynomial of degree k, Y ∈ H k , actually the projection of p onto H k. Here we study the distributional derivative p ∇ r 2−n and show that the ordinary part is still a multiple of Y, but that the delta part is independent of Y, that is, it depends only on p−Y. We also show that the exponent 2−n is special in the sense that the corresponding results for p(∇)(r α) do not hold if α = 2−n. Furthermore, we establish that harmonic polynomials appear as multiples of r 2−n−2k−2k when p(∇) is applied to harmonic multipoles of the form Y (x)r 2−n−2k for some Y ∈ H k .

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The Funk-Hecke formula, harmonic polynomials, and derivatives of radial distributions

Cited by 7 publications

References 20 publications

The Fourier transform of thick distributions

The Fourier transform of thick distributions

Tools for visualizing and analyzing Fourier space sampling in Cryo-EM

Harmonic Polynomials Via Differentiation

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