Differentiation rules for the Fourier transform

Suppose $f \in L^{1}(\mathbb {R})$ and $n \in \mathbb {N}$. Define $g: \mathbb {R} \rightarrow \mathbb {C}$ by $g(x) = x^{n} f(x)$. Prove that if $g \in L^{1}(\mathbb {R})$, then $\widehat{f}$ is $n$ times continuously differentiable on $\mathbb {R}$ and

for all $t \in \mathbb {R}$.

Suppose $n \in \mathbb {N}$and $f \in L^{1}(\mathbb {R})$ is $n$ times continuously differentiable and $f^{(k)} \in L^{1}(\mathbb {R})$ for $k = 1, \ldots , n$. Prove that if $t \in \mathbb {R}$, then

Prove that for all $\varphi \in \mathcal{S}(\mathbb {R})$ the following equations hold

Prove property (3) of the Fourier transform. Prove that if $f(x), x f(x), \ldots , x^{n} f(x) \in L_{1}(\mathbb {R})$, then $\hat{f} \in C^{n}(\mathbb {R})$ and $\hat{F} M^{n} = D^{n} \hat{F}$.

Prove property (4) of the Fourier transform. Prove that if $f \in L_{1}(\mathbb {R}) \cap C^{n}(\mathbb {R})$ and $f^{(j)} \in L_{1}(\mathbb {R}), j = 1,2, \ldots , n$, then $\hat{f}(y) = o\left(\frac{1}{\left|y\right|^{k}}\right)$ as $\left|y\right| \rightarrow \infty$ and $\hat{F} D^{n} = (-1)^{n} M^{n} \hat{F}$.

Let the function $f$ be continuous on $R$, absolutely integrable on $R$, and satisfy the condition

Prove that $F[\varphi ] = -\frac{i}{y} \widehat{f}(y)$, where $\widehat{f}(y) = F[f]$.

If $f \in L^{1}$ and $\int \left|t \hat{f}(t)\right| d m(t) < \infty$, prove that $f$ coincides a.e. with a differentiable function whose derivative is

Suppose $f \in L^{1}, f$ is differentiable almost everywhere, and $f^{\prime } \in L^{1}$. Does it follow that the Fourier transform of $f^{\prime }$ is $i t \hat{f}(t)$?

The Laplacian of a function $f$ on $\mathbb {R}^{k}$ is

provided the partial derivatives exist. What is the relation between $\hat{f}$ and $\hat{g}$ if $g = \Delta f$ and all necessary integrability conditions are satisfied? It is clear that the Laplacian commutes with translations. Prove that it also commutes with rotations, i.e., that

whenever $f$ has continuous second derivatives and $A$ is a rotation of $\mathbb {R}^{k}$. (Show that it is enough to do this under the additional assumption that $f$ has compact support).

In the section we claimed that

The point of this exercise is to verify the middle equality, and hence all three equalities. Note that

Use integration by parts to show that this equals $i w \hat{f}(w)$, and hence that the equation holds.

Suppose is $f$ is both integrable with respect to Lebesgue measure $\mathfrak {m}$ on the real line and absolutely continuous, in the sense that $f(x) = \mathfrak {m}^{x}(\left\{ t \leq x\right\} \dot{f}(t))$, for all $x$, for some integrable function $\dot{f}$.