Fourier transform

Let the function $f \in L_{1}(\mathbb {R})$. Prove that:

Find the Fourier transform of the characteristic function $\chi_{[a, b]}(x)$ of the segment $[a, b]$.

Prove property (1) of the Fourier transform: $\hat{F}$ is a continuous operator from $L_1(\mathbb {R})$ into $C_0(\mathbb {R})$, with $\left\| \hat{F}\right\| = \frac{1}{\sqrt{2\pi }}$

Give an example of a function $f \in L_{1}(\mathbb {R})$ for which $\hat{f} \notin L_{1}(\mathbb {R})$.

Prove that if $f \in L_{1}(\mathbb {R})$ and $\hat{f} \equiv 0$, then $f = 0$.

Give an example of a sequence of functions $\left\{ f_{n}\right\}_{1}^{\infty } \subset L_{1}(\mathbb {R})$ with the properties $\left\| f_{n}\right\|_{L_{1}(\mathbb {R})} = 1, n = 1,2, \ldots$, and $\left\| \hat{f}_{n}\right\|_{C_{0}(\mathbb {R})} \rightarrow 0$ as $n \rightarrow \infty$.

Give an example of a function $g \in C_{0}(\mathbb {R})$ which is not the Fourier transform of any function from $L_{1}(\mathbb {R})$.

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 $\hat{F} D^{n} = (-1)^{n} M^{n} \hat{F}$.

Prove property (5) of the Fourier transform: $\widehat{f(x-a)} = e^{-i a y} \hat{f}$, where $\hat{f}(y) = \frac{1}{\sqrt{2 \pi }} \int_{-\infty }^{+\infty } f(x) e^{-i x y} \, d x$ is the Fourier transform of a function $f \in L_{1}(\mathbb {R})$.

Prove property (6) of the Fourier transform: if a function $f \in L_{1}(\mathbb {R})$ satisfies the Dini condition at a point $x_{0}$,

then

where $\hat{f}(y) = \frac{1}{\sqrt{2 \pi }} \int_{-\infty }^{+\infty } f(x) e^{-i x y} \, d x$ is the Fourier transform of $f$.

Prove property (2) of the Fourier transform: $\hat{F}$ is an isomorphism of the space $\mathcal{S}$ onto itself, with $\hat{F}^{-1}\hat{F} = I$.

Suppose there exists a positive number $a$ such that $e^{a\left|x\right|}. f \in L_{1}(\mathbb {R})$. Prove that $\hat{f}(y)$ extends to a function $\tilde{f}(z)$, analytic in the strip $\left|\operatorname {Im} z\right| < a$.

Let a function $\varphi \in \mathcal{D}$ have support $\operatorname {supp} \varphi \subset [-a, a]$. Prove that its Fourier transform $\hat{\varphi }$ extends to an entire (analytic on the whole complex plane) function of the complex variable $z = s+i t$, satisfying, for each $n = 0,1,2, \ldots$, the inequality $\left|z\right|^{n}\left|\hat{\varphi }(z)\right| \leqslant C(\varphi , n) e^{a\left|t\right|}$.

Let an entire function $\psi$ of the complex variable $z = s+$ it satisfy, for each $n = 0,1,2, \ldots$, the inequality

Then there exists a function $\varphi \in \mathcal{D}$ such that $\operatorname {supp} \varphi \subset [-a, a]$ and $\hat{\varphi } = \psi$.

Prove that $\psi_{k} \rightarrow 0$ in $\mathcal{Z}$ if and only if two properties hold:

1. $\left|z\right|^{n}\left|\psi_{k}(z)\right| \leqslant C(n) e^{a\left|\operatorname {Im} z\right|}$ with constants $C(n)$ and $a > 0$, not depending on $k$;

2. $\psi_{n} \rightrightarrows 0$ on every interval $[\alpha , \beta ] \in \mathbb {R}$.

Suppose that all the power moments of a function $f$ are zero:

Does it follow that $f \equiv 0$, if:

Prove that for every set $A \subset \mathbb {R}$ of positive measure there exists a function $f \in L_{1}(\mathbb {R})$ with support on $A$ such that the function $\hat{f}$ has no zeros on the real axis.

Does there exist a function $f$ with support on the segment

Prove that for any functions $f, g \in \mathcal{S}$ the Parseval equality holds: $(\hat{f}, \hat{g})_{L_{2}(\mathbb {R})} = (f, g)_{L_{2}(\mathbb {R})}$.

Find the Fourier transform of the following functions (the parameter $a$ is everywhere assumed positive):

For functions $\varphi \in \mathcal{D}$ prove the Poisson summation formula (cf. problem 16.48):

Prove that in the spaces $\mathcal{S}$ and $L_{2}(\mathbb {R})$ the identities hold:

Prove that the Chebyshev-Hermite functions $H_{n}(x) e^{-x^{2} / 2}$, obtained from the functions $x^{n} e^{-x^{2} / 2}, n = 0,1,2 \ldots$, by the process of orthogonalization in $L_{2}(\mathbb {R})$, are eigenfunctions of the Fourier transform operator in $L_{2}(\mathbb {R})$, and find the corresponding eigenvalues.

Find the spectrum of the Fourier transform operator in $L_{2}(\mathbb {R})$.

Let a function $f \in L_{2}(\mathbb {R})$ be almost everywhere nonzero and suppose there exists a positive number $a$ such that $e^{a\left|x\right|} \cdot f \in L_{2}(\mathbb {R})$. Prove that the system of functions $\left\{ x^{n} f\right\}_{n = 0}^{\infty }$ is complete in $L_{2}(\mathbb {R})$.

Prove that the Chebyshev-Hermite functions form a complete orthonormal system in $L_{2}(\mathbb {R})$ (cf. problem 3.55).

Prove the statement of problem 3.54 using the Fourier transform.

Prove that the Sobolev space $W_{2}^{n}(\mathbb {R})$ is isometrically isomorphic to the space $\left\{ f \in L_{2}(\mathbb {R}):|\lambda |^{n} \hat{f}(\lambda ) \in L_{2}(\mathbb {R})\right\}$.

Prove that every function from the space $W_{2}^{n}(\mathbb {R})$ is bounded.

(R. Paley, N. Wiener, 1934). Prove that a function $g: \mathbb {R} \rightarrow \mathbb {C}$ is the Fourier transform of some function $f \in L_{2}(\mathbb {R})$ with $\operatorname {supp} f \subseteq [-a, a]$ if and only if $g \in L_{2}(\mathbb {R})$ and $g$ is the restriction to $\mathbb {R}$ of an entire function $g(z)$ satisfying the inequality $\left|g(z)\right| \leqslant C e^{a\left|\operatorname {Im} z\right|}$ for all $z \in \mathbb {C}$.

Prove that for the functions $f$ and $g$ of problem 17.31 the equality holds

where the series converges in $L_{2}[-a, a]$.

(V. A. Kotelnikov, 1933). Prove that the function $g$ from problem 17.31 satisfies the equality

where the series on the right converges in $L_{2}(\mathbb {R}).$¹

Prove that formula (17.1) defines a continuous isomorphism of the space $\mathcal{S}^{\prime }$, and the inverse operator is given by the equality $\left\langle \hat{F}^{-1} f, \varphi \right\rangle = \left\langle f, \hat{F}^{-1} \varphi \right\rangle$.

Prove that if $f \in \mathcal{S}^{\prime }$ is a regular generalized function, generated by some function $g \in \mathcal{S}$, then the Fourier transform $\hat{F} f$, defined by formula (17.1), is a regular generalized function, generated by the function $\hat{g}$.

Prove properties (1)-(3) of the Fourier transform of generalized functions from $\mathcal{S}^{\prime }$.

Prove that formulas (17.2) and (17.3) define continuous operators from $\mathcal{D}^{\prime }$ into $\mathcal{Z}^{\prime }$. How do the Fourier transforms defined by these formulas differ?

Prove that a generalized function $f \in \mathcal{Z}^{\prime }$ is analytic in the following sense:

where the series on the right converges in the sense of $\mathcal{Z}^{\prime }$, and $f(z+h)$ is the generalized function in $\mathcal{Z}^{\prime }$, defined by the formula

for every $\psi \in \mathcal{Z}$.

Find the Fourier transform of the generalized functions in the corresponding spaces:

Find all functions $f \in \mathcal{S}^{\prime }$ satisfying the equation: