Mollifier

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. They are also known as Friedrichs mollifiers after Kurt Otto Friedrichs, who introduced them.

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. They are also known as Friedrichs mollifiers after Kurt Otto Friedrichs, who introduced them. Let \varphi be a smooth function on \R^n , n \ge 1 , and put \varphi_\epsilon(x) := \epsilon^{-n}\varphi(x / \epsilon) for \epsilon > 0 \in\R . Then \varphi is a mollifier if it satisfies the following three requirements: it is compactly supported, \int_{\R^n}\!\varphi(x)\mathrm{d}x=1 , \lim_{\epsilon\to 0}\varphi_\epsilon(x) = \lim_{\epsilon\to 0}\epsilon^{-n}\varphi(x / \epsilon)=\delta(x) , where \delta(x) is the Dirac delta function, and the limit must be understood as taking place in the space of Schwartz distributions. The function \varphi may also satisfy further conditions of interest; for example, if it satisfies \varphi(x)\ge 0 for all x \in \R^n , then it is called a positive mollifier, and if it satisfies \varphi(x)=\mu(|x|) for some infinitely differentiable function \mu:\R^+\to\R , then it is called a symmetric mollifier.

Notes on Friedrichs' definition

Note 1. When the theory of distributions was still not widely known nor used, property above was formulated by saying that the convolution of the function \scriptstyle\varphi_\epsilon with a given function belonging to a proper Hilbert or Banach space will converge, in the metric of that space, as ε → 0 to that function: this is exactly what Friedrichs did. This also clarifies why mollifiers are related to approximate identities. Note 2. As briefly pointed out in the "Historical notes" section of this entry, originally, the term "mollifier" identified the following convolution operator : \Phi_\epsilon(f)(x)=\int_{\mathbb{R}^n}\varphi_\epsilon(x-y) f(y)\mathrm{d}y where \varphi_\epsilon(x)=\epsilon^{-n}\varphi(x/\epsilon) and \varphi is a smooth function satisfying the first three conditions stated above and one or more supplementary conditions as positivity and symmetry. \varphi(x) = \begin{cases} e^{-1/(1-|x|^2)}/I_n& \text{ if } |x| where the numerical constant I_n ensures normalization. This function is infinitely differentiable, non analytic with vanishing derivative for . \varphi can be therefore used as mollifier as described above: one can see that \varphi (x) defines a positive and symmetric mollifier. one]]

Smoothing property

For any distribution T , the following family of convolutions indexed by the real number \epsilon T_\epsilon = T\ast\varphi_\epsilon where \ast denotes convolution, is a family of smooth functions.

Approximation of identity

For any distribution T , the following family of convolutions indexed by the real number \epsilon converges to T \lim_{\epsilon\to 0}T_\epsilon = \lim_{\epsilon\to 0}T\ast\varphi_\epsilon=T\in D^\prime(\mathbb{R}^n)

Support of convolution

For any distribution T , \operatorname{supp}T_\epsilon=\operatorname{supp}(T\ast\varphi_\epsilon)\subset\operatorname{supp}T+\operatorname{supp}\varphi_\epsilon , where \operatorname{supp} indicates the support in the sense of distributions, and + indicates their Minkowski addition. S\cdot T := \lim_{\epsilon\to 0}S_\epsilon\cdot T=\lim_{\epsilon\to 0}S\cdot T_\epsilon . === "Weak=Strong" theorems === Mollifiers are used to prove the identity of two different kind of extension of differential operators: the strong extension and the weak extension. The paper by Friedrichs which introduces mollifiers illustrates this approach. By convolution of the characteristic function of the unit ball B_1 = \{x : |x| with the smooth function \varphi_{1/2} (defined as in with \epsilon = 1/2 ), one obtains the function \begin{align} \chi_{B_1,1/2}(x) &=\chi_{B_1}\ast\varphi_{1/2}(x) \\ &=\int_{\mathbb{R}^n}\!\!\!\chi_{B_1}(x-y)\varphi_{1/2}(y)\mathrm{d}y \\ &=\int_{B_{1/2}}\!\!\! \chi_{B_1}(x-y) \varphi_{1/2}(y)\mathrm{d}y \ \ \ (\because\ \mathrm{supp}(\varphi_{1/2})=B_{1/2}) \end{align} which is a smooth function equal to 1 on B_{1/2} = \{ x: |x| , with support contained in B_{3/2}=\{ x: |x| . This can be seen easily by observing that if |x| \le 1/2 and |y| \le 1/2 then |x-y| \le 1 . Hence for |x| \le 1/2 , \int_{B_{1/2}}\!\!\!\chi_{B_1}(x-y) \varphi_{1/2}(y)\mathrm{d}y= \int_{B_{1/2}}\!\!\! \varphi_{1/2}(y)\mathrm{d}y=1 . One can see how this construction can be generalized to obtain a smooth function identical to one on a neighbourhood of a given compact set, and equal to zero in every point whose distance from this set is greater than a given \epsilon . Such a function is called a (smooth) cutoff function; these are used to eliminate singularities of a given (generalized) function via multiplication. They leave unchanged the value of the multiplicand on a given set, but modify its support. Cutoff functions are used to construct smooth partitions of unity.

Notes

Text derived from https://en.wikipedia.org/w/index.php?oldid=1373372189 under CC BY-SA 4.0; modifications have been made.

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