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I'm learning a bit about the theory of distributions. What examples of distributions will help me develop good intuition?

Definitions: Let $U$ be an open subset of $\mathbb{R}^n$. Write $C_c^\infty(U)$ for the complex vector space of infinitely differentiable functions $U \to \mathbb{C}$ with compact support. A distribution on $U$ is a linear map $C^\infty_c(U) \to \mathbb{C}$, continuous with respect to a certain topology on $C^\infty_c(U)$.

Examples: If $\mu$ is a measure on $U$ then $f \mapsto \int_U f \mathrm{d}\mu$ is a distribution. (This covers, for instance, the Dirac distribution, $f \mapsto f(0)$.)

More generally, write $D_i = \partial/\partial x_i$. Then $$ f \mapsto \int_U D_{i_1} D_{i_2} \cdots D_{i_r} f \mathrm{d}\mu $$ is a distribution, for any indices $i_1, \ldots, i_r$ and measure $\mu$.

Any linear combination of such things is again a distribution, since distributions form a vector space. E.g. if $n \geq 3$ then there's a distribution $$ f \mapsto \int_U D_3 D_1 f \mathrm{d}\mu + \int_U D_2^2 D_3 f \mathrm{d}\nu $$ for any measures $\mu$ and $\nu$. I guess we can also take infinite linear combinations, subject to convergence conditions.

My question: Is it OK if I go round thinking of things like the last example as being a typical sort of distribution? Or is the concept of distribution much more general than I'm realizing? The texts I've seen are short on this kind of intuition.

I'm learning a bit about the theory of distributions. What examples of distributions will help me develop good intuition?

Definitions: Let $U$ be an open subset of $\mathbb{R}^n$. Write $C_c^\infty(U)$ for the complex vector space of infinitely differentiable functions $U \to \mathbb{C}$ with compact support. A distribution on $U$ is a linear map $C^\infty_c(U) \to \mathbb{C}$, continuous with respect to a certain topology on $C^\infty_c(U)$.

Examples: If $\mu$ is a signed measure on $U$, finite on compact subsets, then $f \mapsto \int_U f \mathrm{d}\mu$ is a distribution. (This covers, for instance, the Dirac distribution, $f \mapsto f(0)$.)

More generally, write $D_i = \partial/\partial x_i$. Then $$ f \mapsto \int_U D_{i_1} D_{i_2} \cdots D_{i_r} f \mathrm{d}\mu $$ is a distribution, for any indices $i_1, \ldots, i_r$ and measure $\mu$.

Any linear combination of such things is again a distribution, since distributions form a vector space. E.g. if $n \geq 3$ then there's a distribution $$ f \mapsto \int_U D_3 D_1 f \mathrm{d}\mu + \int_U D_2^2 D_3 f \mathrm{d}\nu $$ for any measures $\mu$ and $\nu$. I guess we can also take infinite linear combinations, subject to convergence conditions.

My question: Is it OK if I go round thinking of things like the last example as being a typical sort of distribution? Or is the concept of distribution much more general than I'm realizing? The texts I've seen are short on this kind of intuition.

I'm learning a bit about the theory of distributions. What examples of distributions will help me develop good intuition?

Definitions: Let $U$ be an open subset of $\mathbb{R}^n$. Write $C_c^\infty(U)$ for the complex vector space of infinitely differentiable functions $U \to \mathbb{C}$ with compact support. A distribution on $U$ is a linear map $C^\infty_c(U) \to \mathbb{C}$, continuous with respect to a certain topology on $C^\infty_c(U)$.

Examples: If $\mu$ is a measure on $U$ then $f \mapsto \int_U f \mathrm{d}\mu$ is a distribution. (This covers, for instance, the Dirac distribution, $f \mapsto f(0)$.)

More generally, write $D_i = \partial/\partial x_i$. Then $$ f \mapsto \int_U D_{i_1} D_{i_2} \cdots D_{i_r} f \mathrm{d}\mu $$ is a distribution, for any indices $i_1, \ldots, i_r$ and measure $\mu$.

Any linear combination of such things is again a distribution, since distributions form a vector space. E.g. if $n \geq 3$ then there's a distribution $$ f \mapsto \int_U D_3 D_1 f \mathrm{d}\mu + \int_U D_2^2 D_3 f \mathrm{d}\nu $$ for any measures $\mu$ and $\nu$. I guess we can also take infinite linear combinations, subject to convergence conditions.

My question: Is it OK if I go round thinking of things like the last example as being a typical sort of distribution? Or is the concept of distribution much more general than I'm realizing? The texts I've seen are short on this kind of intuition.re

I'm learning a bit about the theory of distributions. What examples of distributions will help me develop good intuition?

Definitions: Let $U$ be an open subset of $\mathbb{R}^n$. Write $C_c^\infty(U)$ for the complex vector space of infinitely differentiable functions $U \to \mathbb{C}$ with compact support. A distribution on $U$ is a linear map $C^\infty_c(U) \to \mathbb{C}$, continuous with respect to a certain topology on $C^\infty_c(U)$.

Examples: If $\mu$ is a measure on $U$ then $f \mapsto \int_U f \mathrm{d}\mu$ is a distribution. (This covers, for instance, the Dirac distribution, $f \mapsto f(0)$.)

More generally, write $D_i = \partial/\partial x_i$. Then $$ f \mapsto \int_U D_{i_1} D_{i_2} \cdots D_{i_r} f \mathrm{d}\mu $$ is a distribution, for any indices $i_1, \ldots, i_r$ and measure $\mu$.

Any linear combination of such things is again a distribution, since distributions form a vector space. E.g. if $n \geq 3$ then there's a distribution $$ f \mapsto \int_U D_3 D_1 f \mathrm{d}\mu + \int_U D_2^2 D_3 f \mathrm{d}\nu $$ for any measures $\mu$ and $\nu$. I guess we can also take infinite linear combinations, subject to convergence conditions.

My question: Is it OK if I go round thinking of things like the last example as being a typical sort of distribution? Or is the concept of distribution much more general than I'm realizing? The texts I've seen are short on this kind of intuition.