All Questions

Suppose we have some Schwartz function $h$. Denote its Fourier transform $\widehat{h}$. Let $\xi_0$, $a$, $\Delta$ be positive and fixed. I have a function $\Omega: \mathbb{R}\times \mathbb{R}^+ \to \...

Let $v$ be a vector $v \in \mathbb{R}^p$, with non-negative entries and $p$ prime. The Hausdorff-Young inequality gives bounds of the form: $$\|\mathcal{F}v\|_a \le C_{a,b} \|v\|_b$$ where the ...

I am looking for $\delta>0$, such that $$ \delta \int_{-\infty}^{\infty} \exp(its) { \Gamma\{2(it+1)/3\}\over \Gamma\{(it+1)/2\} }dt \le \\ \int_{-\infty}^{\infty} \exp(its) { \Gamma (it+1)\over \...

I know that the Fourier transform of a compactly support distribution $u\in \mathscr{E}'(\mathbb{R}^{n})$ is smooth and also satisfies $$ |\hat{u}(\xi)|\leqslant C_{N}(1+|\xi|)^N,\label{1}\tag{1} $$ ...

Well I have a function parametrized by $z$ $$g_z(x,y) = \frac{z}{i \lambda r^2} e^{i k r}, \quad r = \sqrt{x^2+y^2+z^2},$$ where $\lambda > 0$ is real constant and $k = \frac{2\pi}{\lambda}$. This ...