Fourier Transforms

Suppose $f \in L^{1}, f > 0$. Prove that $\left|\hat{f}(y)\right| < \hat{f}(0)$ for every $y \neq 0$.

Compute the Fourier transform of the characteristic function of an interval. For $n = 1,2,3, \ldots$, let $g_{n}$ be the characteristic function of $[-n, n]$, let $h$ be the characteristic function of $[-1,1]$, and compute $g_{n} * h$ explicitly. (The graph is piecewise linear). Show that $g_{n} * h$ is the Fourier transform of a function $f_{n} \in L^{\mathbf{1}}$; except for a multiplicative constant,

Show that $\left\| f_{n}\right\|_{1} \rightarrow \infty$ and conclude that the mapping $f \rightarrow \hat{f}$ maps $L^{1}$ into a proper subset of $C_{0}$. Show, however, that the range of this mapping is dense in $C_{0}$.

Find

where $\lambda$ is a positive constant.

Give examples of $f \in L^{2}$ such that $f \notin L^{1}$ but $\hat{f} \in L^{1}$. Under what circumstances can this happen?

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 $i t \hat{f}(t)$?

Let $S$ be the class of all functions $f$ on $\mathbb {R}^{1}$ which have the following property: $f$ is infinitely differentiable, and there are numbers $A_{m n}(f) < \infty$, for $m$ and $n = 0,1,2, \ldots$, such that

Here $D$ is the ordinary differentiation operator. Prove that the Fourier transform maps $S$ onto $S$. Find examples of members of $S$.

If $p$ and $q$ are conjugate exponents, $f \in L^{p}, g \in L^{q}$, and $h = f * g$, prove that $h$ is uniformly continuous. If also $1 < p < \infty$, then $h \in C_{0}$; show that this fails for some $f \in L^{1}, g \in L^{\infty }$.

Suppose $1 \leq p < \infty , f \in L^{p}$, and

Prove that $g \in C_{0}$. What can you say about $g$ if $f \in L^{\infty }$?

Let $C^{\infty }$ be the class of all infinitely differentiable complex functions on $\mathbb {R}^{1}$, and let $C_{c}^{\infty }$ consist of all $g \in C^{\infty }$ whose supports are compact. Show that $C_{c}^{\infty }$ does not consist of 0 alone.

Let $L_{\text{loc }}^{1}$ be the class of all $f$ which belong to $L^{1}$ locally; that is, $f \in L_{\text{loc }}^{1}$ provided that $f$ is measurable and $\int_{I}\left|f\right| < \infty$ for every bounded interval $I$.

If $f \in L_{\mathrm{loc}}^{1}$ and $g \in C_{c}^{\infty }$, prove that $f * g \in C^{\infty }$. Prove that there are sequences $\left\{ g_{n}\right\}$ in $C_{c}^{\infty }$ such that

as $n \rightarrow \infty$, for every $f \in L^{1}$. (Compare Theorem 9.10). Prove that $\left\{ g_{n}\right\}$ can also be so chosen that $\left(f * g_{n}\right)(x) \rightarrow f(x)$ a.e., for every $f \in L_{\mathrm{loc}}^{1}$; in fact, for suitable $\left\{ g_{n}\right\}$ the convergence occurs at every point $x$ at which $f$ is the derivative of its indefinite integral.

Prove that $\left(f * h_{\lambda }\right)(x) \rightarrow f(x)$ a.e. if $f \in L^{\text{1 }}$, as $\lambda \rightarrow 0$, and that $f * h_{\lambda } \in C^{\infty }$, although $h_{\lambda }$ does not have compact support. ($h_{\lambda }$ is defined in Sec. 9.7).

Find conditions on $f$ and/or $f$ which ensure the correctness of the following formal argument: If

and

then $F$ is periodic, with period $2 \pi$, the $n$th Fourier coefficient of $F$ is $\varphi (n)$, hence $F(x) = \sum \varphi (n) e^{i n x}$. In particular,

More generally,

What does () say about the limit, as $\alpha \rightarrow 0$, of the right-hand side (for " nice " functions, of course)? Is this in agreement with the inversion theorem? [() is known as the Poisson summation formula].

Take $f(x) = e^{-\left|x\right|}$ in Exercise 11 and derive the identity

If $0 < c < \infty$, define $f_{c}(x) = \exp \left(-c x^{2}\right)$.

The Fourier transform can be defined for $f \in L^{1}\left(\mathbb {R}^{k}\right)$ by

where $x \cdot y = \sum \xi_{i} \eta_{i}$ if $x = \left(\xi_{1}, \ldots , \xi_{k}\right), y = \left(\eta_{1}, \ldots , \eta_{k}\right)$, and $m_{k}$ is Lebesgue measure on $\mathbb {R}^{k}$, divided by $(2 \pi )^{k / 2}$ for convenience. Prove the inversion theorem and the Plancherel theorem in this context, as well as the analogue of Theorem 9.23.

If $f \in L^{1}\left(\mathbb {R}^{k}\right), A$ is a linear operator on $\mathbb {R}^{k}$, and $g(x) = f(A x)$, how is $\hat{g}$ related to $\hat{f}$? If $f$ is invariant under rotations, i.e., if $f(x)$ depends only on the euclidean distance of $x$ from the origin, prove that the same is true of $\hat{f}$.

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).

Show that every Lebesgue measurable character of $\mathbb {R}^{1}$ is continuous. Do the same for $\mathbb {R}^{k}$. (Adapt part of the proof of Theorem 9.23). Compare with Exercise 18.

Show (with the aid of the Hausdorff maximality theorem) that there exist real discontinuous functions $f$ on $\mathbb {R}^{1}$ such that

for all $x$ and $y \in \mathbb {R}^{1}$. Show that if (1) holds and $f$ is Lebesgue measurable, then $f$ is continuous. Show that if (1) holds and the graph of $f$ is not dense in the plane, then $f$ is continuous. Find all continuous functions which satisfy (1).

Suppose $A$ and $B$ are measurable subsets of $\mathbb {R}^{1}$, having finite positive measure. Show that the convolution $\chi_{A} * \chi_{B}$ is continuous and not identically 0. Use this to prove that $A+B$ contains a segment. (A different proof was suggested in Exercise 5, Chap. 7).