What does a hat or star means in math?

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What are the general uses of the hat and star symbol in math? Or could you please point me to a page that discusses this? Thanks.

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9 Answers

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There is a nice list for $*$ in this article

I guess another (more general) term for "hat" is Circumflex

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Different branches of mathematics may have varying conventional usages of these kind of "decorations". Typically they denote a transformed version of the base variable (e.g. $\hat{f}$ denoting the Fourier transform of $f$ as mentioned in another answer). Or, they may denote a special or specific value of a variable ($x^*$ giving the value of $x$ minimizing $f$ from another answer.) The $*$ symbol is often used for arbitrary associative binary operations. Etc. etc.

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how about omitted terms $$ \partial\langle x_0,...,x_n\rangle=\sum_{i=0}^n(-1)^i\langle x_0,...,\hat{x_i},...x_n\rangle $$

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I have seen the star used for multiplication, hermitian conjugate of a matrix, special values of a variable (given a function $f(x), x^*$ might be the value of $x$ that minimizes $f$), among others. In Conway's theory of games, * is the game that wins for the first player.

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$\hat{}$ can also be used to denote the Fourier transform $\hat{f}$ of an integrable function $f$.

$\ast$ can be used to denote the convolution product $f \ast g$ of two functions $f$ and $g$.

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According to ISO 31-11: $$\begin{align} \mathbb{N}^* = \mathbb{N}-\{0\} \\ \mathbb{Z}^* = \mathbb{Z}-\{0\} \end{align}$$ The same goes for $\mathbb{Q}, \mathbb{R}, \mathbb{C}$. $$\begin{align} z^* = \text{complex conjugate of } z. \end{align}$$

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The Hat symbol can be used to denote a vector. And the Star symbol may possibly used to denote a binary operation. For example a non empty set $G$ with a binary operation $\star$ is said to be a Group if.....

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Another possibility: if $V$ is a vector space over a field $\mathbb{F}$, its dual space, $V^*$, is the set of linear maps $V \to \mathbb{F}$. The dual space is also a vector space in its own right. The double dual of $V$ is $V^{**} = (V^*)^*$, and there's a nice correspondence between $V$ and $V^{**}$ such that given an element $v \in V$ we have a special corresponding element often called $\hat{v} \in V^{**}$.

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After looking through generating functions in this paper, on pp. $16$, a paragraph states the following: $$\begin{align} &\text{In general, we say a sequence $(c_n)_{n\geq 0}$ is the convolution of $(a_k)_{k\geq 0}$ and} \\ (b_m&)_{m\geq 0} \ \text{$($write $c=a\star b)$, if}\end{align}$$ $$c_n = \sum_{k=0}^n a_kb_{n-k}, \qquad n\geq 0, \tag{4.5}$$ I believe this provides the definition of the star $\star$ operation, precisley adding to what @RudytheReindeer mentioned.

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