What is the meaning of $1^\lor=\delta$

Question

I tried to look up the definition of a $\lor$ and it does not seem to explain this particular usage

$$1^\lor=\delta$$

This is used in a proof that inverse fourier transform of $1$ is $\delta$, but i have no idea what the above usage of notation implies.

Thanks

Answer 1

The check accent is often used to indicate the inverse Fourier transform. Since the difference between direct and inverse transform is a sign, it is a question of taste if this notation is needed. $$ \check{\hat\phi}=\hat{\check\phi}=\phi $$

Answer 2

Given a distribution $D$, we use $\hat D$ to denote the Fourier transform, $$ \left< \hat D, \varphi \right> = \left< D, \hat \varphi \right>$$ Where $\hat \varphi$ is the classical Fourier transform on Schwartz space.

We can define ${\varphi}^{\vee}$ as, $$ \int_{\mathbb{R}} \varphi(-x) e^{-i\omega x} ~ dx $$ So this is like a transform but with a negative.

So I am guessing $D^{\vee}$ is defined just like for Fourier transform.

It does explain why $1^{\vee} = \delta$.