Erdélyi–Kober fractional integrals and radon transforms for mutually orthogonal affine planes

2020 ◽  
Vol 23 (4) ◽  
pp. 967-979
Author(s):  
Boris Rubin ◽  
Yingzhan Wang
Keyword(s):  
Fractional Integrals ◽  
Radon Transforms ◽  
Affine Planes ◽  
Inversion Formulas ◽  
Radial Functions ◽  
Compactly Supported ◽  
Radial Case ◽  
Compactly Supported Functions

AbstractWe apply Erdélyi–Kober fractional integrals to the study of Radon type transforms that take functions on the Grassmannian of j-dimensional affine planes in ℝn to functions on a similar manifold of k-dimensional planes by integration over the set of all j-planes that meet a given k-plane at a right angle. We obtain explicit inversion formulas for these transforms in the class of radial functions under minimal assumptions for all admissible dimensions. The general (not necessarily radial) case, but for j + k = n − 1, n odd, was studied by S. Helgason [8] and F. Gonzalez [4, 5] on smooth compactly supported functions.

scholarly journals On Radon transforms between lines and hyperplanes

2017 ◽  
Vol 28 (13) ◽  
pp. 1750093 ◽  
Author(s):  
Boris Rubin ◽  
Yingzhan Wang
Keyword(s):  
Radon Transform ◽  
Symmetric Case ◽  
Fractional Integrals ◽  
Radon Transforms ◽  
Full Generality ◽  
Inversion Formulas ◽  
Radial Functions ◽  
Kelvin Transform ◽  
Funk Transform ◽  
Dual Transform

We obtain new inversion formulas for the Radon transform and its dual between lines and hyperplanes in [Formula: see text]. The Radon transform in this setting is non-injective and the consideration is restricted to the so-called quasi-radial functions that are constant on symmetric clusters of lines. For the corresponding dual transform, which is injective, explicit inversion formulas are obtained both in the symmetric case and in full generality. The main tools are the Funk transform on the sphere, the Radon-John [Formula: see text]-plane transform in [Formula: see text], the Grassmannian modification of the Kelvin transform, and the Erdélyi–Kober fractional integrals.

Riesz potentials and orthogonal radon transforms on affine Grassmannians

2021 ◽  
Vol 24 (2) ◽  
pp. 376-392
Author(s):  
Boris Rubin ◽  
Yingzhan Wang
Keyword(s):  
Radon Transforms ◽  
Riesz Potentials ◽  
Fractional Powers ◽  
Affine Planes ◽  
Inversion Formulas ◽  
Affine Grassmannians ◽  
Existence Conditions

Abstract We establish intertwining relations between Riesz potentials associated with fractional powers of minus-Laplacian and orthogonal Radon transforms 𝓡 j,k of the Gonzalez-Strichartz type. The latter take functions on the Grassmannian of j-dimensional affine planes in ℝ n to functions on a similar manifold of k-dimensional planes by integration over the set of all j-planes that meet a given k-plane at a right angle. The main results include sharp existence conditions of 𝓡 j,k f on L p -functions, Fuglede type formulas connecting 𝓡 j,k with Radon-John k-plane transforms and Riesz potentials, and explicit inversion formulas for 𝓡 j,k f under the assumption that f belongs to the range of the j-plane transform. The method extends to another class of Radon transforms defined on affine Grassmannians by inclusion.

scholarly journals On weighted inequalities for fractional integrals of radial functions

2011 ◽  
Vol 55 (2) ◽  
pp. 575-587 ◽  
Author(s):  
Pablo L. De Nápoli ◽  
Irene Drelichman ◽  
Ricardo G. Durán
Keyword(s):  
Fractional Integrals ◽  
Weighted Inequalities ◽  
Radial Functions

METHOD OF MEAN VALUE OPERATORS FOR RADON TRANSFORMS IN THE SPACE OF MATRICES

2008 ◽  
Vol 19 (03) ◽  
pp. 245-283 ◽  
Author(s):  
E. OURNYCHEVA ◽  
B. RUBIN
Keyword(s):  
Radon Transform ◽  
Sufficient Conditions ◽  
Mean Value ◽  
Necessary And Sufficient Conditions ◽  
Inversion Method ◽  
Radon Transforms ◽  
Inversion Formulas ◽  
Higher Rank ◽  
Necessary And Sufficient

We extend the Funk–Radon–Helgason inversion method of mean value operators to the Radon transform [Formula: see text] of continuous and Lpfunctions which are integrated over matrix planes in the space of real rectangular matrices. Necessary and sufficient conditions of existence of [Formula: see text] for such f and explicit inversion formulas are obtained. New higher-rank phenomena related to this setting are investigated.

scholarly journals Fourier transforms of powers of well-behaved 2D real analytic functions

2018 ◽  
Vol 30 (3) ◽  
pp. 723-732
Author(s):  
Michael Greenblatt
Keyword(s):  
Analytic Functions ◽  
Fourier Transforms ◽  
Newton Polygon ◽  
Companion Paper ◽  
Two Dimensional ◽  
Compactly Supported ◽  
Sharp Estimates ◽  
Real Analytic Functions ◽  
Real Analytic ◽  
Compactly Supported Functions

AbstractThis paper is a companion paper to [6], where sharp estimates are proven for Fourier transforms of compactly supported functions built out of two-dimensional real-analytic functions. The theorems of [6] are stated in a rather general form. In this paper, we expand on the results of [6] and show that there is a class of “well-behaved” functions that contains a number of relevant examples for which such estimates can be explicitly described in terms of the Newton polygon of the function. We will further see that for a subclass of these functions, one can prove noticeably more precise estimates, again in an explicitly describable way.

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