Integration kernels for smoothing out sharp features
A mollifier (top) in dimension one. At the bottom, in red is a function with a corner (left) and sharp jump (right), and in blue is its mollified version.
In mathematics, mollifiers (also known as approximations to the identity) are particular smooth functions, used for example in distribution theory to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution. Intuitively, given a (generalized) function, convolving it with a mollifier "mollifies" it, that is, its sharp features are smoothed, while still remaining close to the original.[1]
They are also known as Friedrichs mollifiers after Kurt Otto Friedrichs, who introduced them.[2]
^That is, the mollified function is close to the original with respect to the topology of the given space of generalized functions.
Intuitively, given a (generalized) function, convolving it with a mollifier "mollifies" it, that is, its sharp features are smoothed, while still remaining...
algebra. Specifically, it can mean: Nascent delta function, most commonly Mollifier, more narrowly Approximate identity, more abstractly This disambiguation...
Cambridge University Press. p. 349. ISBN 978-1-107-64695-7. See also: Mollifier Motulsky, Harvey; Christopoulos, Arthur (2004). Fitting Models to Biological...
gratify her if you yourself proposed the name Albert to her." The Queen was mollified by the proposal to name the new baby Albert, and wrote to the Duchess...
smooth and compactly supported then the sequence is called a mollifier. The standard mollifier is obtained by choosing η to be a suitably normalized bump...
laws, leading to complex patterns of heirship. Congress has attempted to mollify the impact of heirship by granting tribes the capacity to acquire fragmented...
concerned that everything now exists within the Momazon warehouse, they are mollified by the retail convenience. Bender's experience working with the Momazon...
The United Kingdom, France and Austria united in proposing amendments to mollify the Sultan, but the court of St. Petersburg ignored their suggestions....
Ceres's Aventine temple was most likely built at patrician expense, to mollify the plebs, the patricians brought the Magna Mater ("Great mother of the...
enforce Soviet domination of the lesser states of Central Europe and to mollify some states that had expressed interest in the Marshall Plan, and which...
reforms and the disasters of 1900 only exacerbated. Cixi in 1901 moved to mollify the foreign community, called for reform proposals, and initiated a set...
the nature of their relationship several times, but Wilde was able to mollify him. In June 1894, he called on Wilde at 16 Tite Street, without an appointment...
helped pass the Tariff of 1832, which lowered rates, but not enough to mollify the South Carolina nullifiers. The crisis ended when Clay and Calhoun agreed...
positions. However, the party bosses wanted to find a running mate who could mollify Taft's supporters, as the schism between the moderate and conservative...
Curve Fitting. By P. G. Guest, Philip George Guest. Page 349. See also: Mollifier Fitting Models to Biological Data Using Linear and Nonlinear Regression...
on Japanese language and ethics while Classical Chinese was included to mollify upper-class Taiwanese parents. These schools served a small percentage...
Tugh Temür gave lavish presents and an imperial seal to Eljigidey to mollify his anger. With the death of Ilkhan Abu Said Bahatur in 1335, Mongol rule...
against actions by state governments, but the passage of the document mollified some critics of the original constitution and shored up his support in...
Kaas expressed no preference or suggestion on the vote, but as a way of mollifying opposition by Centre members to the granting of further powers to Hitler...
"Narrow-Gauge" Platform winning the vote and being adopted. In an attempt to mollify suffragists who were incensed at the lack of a plank endorsing women's...
{\displaystyle \alpha } which are not smooth can be made smooth using a mollifier prior to discretization. As an example, discretization of the function...
Lighthill' explanation), which is more ad hoc. This now enters the theory as mollifier theory. This theory was very successful and is still widely used, but...
1014 that Rome followed suit (see Nicene Creed). Methodius' critics were mollified by Methodius having to accept the appointment of Wiching as his coadjutor...
Duchy of Warsaw, had "ill requited him for his loyalty", and he was only mollified for the time being by Napoleon's public declaration that he had no intention...
hope of an end to the war, but "an abiding fear of peace." He was only mollified after an 18 July meeting with Johnson in Honolulu, where Johnson affirmed...
convention voted on June 25 to ratify the Constitution, Henry was somewhat mollified by the fact that the convention had proposed about 40 amendments; some...
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