Integral transform used in various branches of mathematics
For summation transformation, see summation by parts.
In mathematics, the Abel transform,[1] named for Niels Henrik Abel, is an integral transform often used in the analysis of spherically symmetric or axially symmetric functions. The Abel transform of a function f(r) is given by
Assuming that f(r) drops to zero more quickly than 1/r, the inverse Abel transform is given by
In image analysis, the forward Abel transform is used to project an optically thin, axially symmetric emission function onto a plane, and the inverse Abel transform is used to calculate the emission function given a projection (i.e. a scan or a photograph) of that emission function.
In absorption spectroscopy of cylindrical flames or plumes, the forward Abel transform is the integrated absorbance along a ray with closest distance y from the center of the flame, while the inverse Abel transform gives the local absorption coefficient at a distance r from the center. Abel transform is limited to applications with axially symmetric geometries. For more general asymmetrical cases, more general-oriented reconstruction algorithms such as algebraic reconstruction technique (ART), maximum likelihood expectation maximization (MLEM), filtered back-projection (FBP) algorithms should be employed.
In recent years, the inverse Abel transform (and its variants) has become the cornerstone of data analysis in photofragment-ion imaging and photoelectron imaging. Among recent most notable extensions of inverse Abel transform are the "onion peeling" and "basis set expansion" (BASEX) methods of photoelectron and photoion image analysis.
^N. H. Abel, Journal für die reine und angewandte Mathematik, 1, pp. 153–157 (1826).
In mathematics, the Abeltransform, named for Niels Henrik Abel, is an integral transform often used in the analysis of spherically symmetric or axially...
even n − m ≥ 0. Fourier transform Integral transformAbeltransform Fourier–Bessel series Neumann polynomial Y and H transforms Louis de Branges (1968)...
In mathematics, an integral transform is a type of transform that maps a function from its original function space into another function space via integration...
Niels Henrik Abel (/ˈɑːbəl/ AH-bəl, Norwegian: [ˌnɪls ˈhɛ̀nːɾɪk ˈɑ̀ːbl̩]; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering...
In mathematics, the Laplace transform, named after Pierre-Simon Laplace (/ləˈplɑːs/), is an integral transform that converts a function of a real variable...
explanation needed] Taking the inverse Fourier transform of the resulting product {r|k|fk} gives the Abel means Arf of f: A r f ( e 2 π i x ) = ∑ k ∈ Z...
certain types of sums. It is also called Abel's lemma or Abel transformation, named after Niels Henrik Abel who introduced it in 1826. Suppose { f k }...
theorem Abel polynomials Abel's summation formula Abelian means Abel's test Abel's theorem Abelian theorem Abel–Ruffini theorem AbeltransformAbel transformation...
amount of bending can be related to the refractive index by using an Abeltransform on the formula relating bending angle to refractivity. In the case of...
The Abel Prize (/ˈɑːbəl/ AH-bəl; Norwegian: Abelprisen [ˈɑ̀ːbl̩ˌpriːsn̩]) is awarded annually by the King of Norway to one or more outstanding mathematicians...
Beerends, R. J. (1987), "The Fourier transform of Harish-Chandra's c-function and inversion of the Abeltransform", Math. Ann., 277: 1–23, doi:10.1007/BF01457275...
Niels Henrik Abel and Alfred Tauber. The original examples are Abel's theorem showing that if a series converges to some limit then its Abel sum is the...
translated as "Divergent series are an invention of the devil …") N. H. Abel, letter to Holmboe, January 1826, reprinted in volume 2 of his collected...
compute its Laplace transform, calculate the Laplace transform of d ℓ / d y {\displaystyle {d\ell }/{dy}} and then take the inverse transform (or try to) to...
the GNSS satellite rise or set behind the Earth. Using an Inverse Abel'stransform, a radial profile of refractivity at that tangent point on earth can...
Abel Azcona (born 1 April 1988) is a Spanish artist, specializing in performance art. His work includes installations, sculptures, and video art. He is...
series. The solution of the Jacobi inversion problem for the hyperelliptic Abel map by Weierstrass in 1854 required the introduction of the hyperelliptic...
product of their derivative and antiderivative. It is frequently used to transform the antiderivative of a product of functions into an antiderivative for...
laboratory of a mad scientist named Dr. Abel in his attempt to complete his project of forming a cyborg army. Abel thought that a perfect cyborg must have...
include data processing (e.g., batch processing, stream processing, extract-transform-load (ETL)), Internet of things (IoT) services for Internet-connected...
Howard Wang (English) Abel Walker (アベル・ウォーカー, Aberu Wōkā) is the prefect of Lang House and the leader of the Magia Lupus. Abel is a noble who believes...
However, they also highlight how the birth of Abel and the discovery of his father's memoirs transformed him as a person. The Chicago Tribune remarked...
Other Boleyn Girl (2008). He attracted attention for his portrayal of Rudolf Abel in Steven Spielberg's Bridge of Spies (2015), for which he won the Academy...
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