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One can construct various "exotic" classes of distributions by playing with spaces of test functions. For example, if we choose a suitable subspace space of real analytic functions as test functions, the dual space will include "distributions" which are not finite sums of derivatives of measures.

Let $$T(x)=\sum\limits_{k=0}^{\infty}c_k \delta^{(k)}(x),\quad c_k\in \mathbb C, \quad \limsup\limits_{k\to\infty}\left(k!|c_k|\right)^{1/k}=0.\qquad\qquad(*)$$ $T(.)$ is not a distribution in the classical sense. Its support is localized at $x=0$ but $T(.)$ cannot be written as a finite linear combination of $\delta^{(k)}(x)$. Yet $(*)$ defines a continuous linear functional on a subspace of the space of entire analytic functions on $\mathbb C$. This is a simple example of a hyperfunction (a.k.a. analytic functional).

The analogue of the structure theorem for hyperfunctions basically says that every hyperfunction can be written as a convolution $T*\mu$, where $T$ is of the form $(*)$ and $\mu$ is a Borel measure on $\mathbb R$.
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One can construct various exotic "exotic" classes of distributions by playing with spaces of test functions. Choosing For example, if we choose a smaller space $E\subset C_0^{\infty}(\mathbb R^n)$ suitable subspace of real analytic functions as test functionsgives rise to a larger , the dual space $E^*$ will include "distributions" which are not finite sums of continuous functionals on $E$ (e.g. we may use a space derivatives of holomorphic functions as test functions)measures.

Let's define formally the series

Let $$T(x)=\sum\limits_{k=0}^{\infty}c_k \delta^{(k)}(x),\quad c_k\in \mathbb C, \quad \limsup\limits_{k\to\infty}\left(k!|c_k|\right)^{1/k}=0.\qquad\qquad(*)$$ $T(.)$ is not a distribution in the classical sense. Its support is localized at $x=0$ but $T(.)$ cannot be written as a finite linear combination of $\delta^{(k)}(x)$. Yet (*) $(*)$ defines a continuous linear functional on a subspace of the space of entire analytic functions on $\mathbb C$.

Monsters This is a simple example of the form (*) are known as hyperfunctionsa hyperfunction (a.k.a a.k.a. analytic functionals, a.k.afunctional).distributions

The analogue of infinite order). They have quite the structure theorem for hyperfunctions basically says that every hyperfunction can be written as a number convolution $T*\mu$, where $T$ is of applications in complex analysis, PDEs and mathematical physics (e.g. in the theory of S-matrices)form $(*)$ and $\mu$ is a Borel measure on $\mathbb R$.
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One can construct various "exotic" exotic classes of distributions by playing with spaces of test functions.

Let Choosing a smaller space $E\subset C_0^{\infty}(\mathbb R^n)$ of test functions gives rise to a larger dual space $E^*$ of continuous functionals on $E$ (e.g. we may use a space of holomorphic functions as test functions).

Let's define formally the series $$T(x)=\sum\limits_{k=0}^{\infty}c_k \delta^{(k)}(x),\quad c_k\in \mathbb C, \quad \limsup\limits_{k\to\infty}\left(k!|c_k|\right)^{1/k}=0.\qquad\qquad(*)$$ $T(.)$ is not a distribution in the classical sense(every distribution with . Its support is localized at $x=0$ can but $T(.)$ cannot be written as a finite linear combination of $\delta^{(k)}(x)$). \delta^{(k)}(x)$. Yet (*) defines a continuous linear functional on a suitable subspace of the space of entire analytic functions on $\mathbb C$.

Monsters of the form (*) are known as hyperfunctions (a.k.a analytic functionals, a.k.a. distributions of infinite order). They have quite a number of applications in complex analysis(the theory of sheaves) , PDEs and mathematical physics (e.g. in the theory of S-matrices).
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