Math Problems Vault

Problem 17.34

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

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Problem 17.35

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 $\hat{g}$ .

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Problem 17.36

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

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Problem 17.37

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?

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Problem 17.38

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

f(z+h) = \sum _{k = 0}^{\infty } f^{(k)}(z) \frac{h^{k}}{k!}, \quad z, h \in \mathbb {C}

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

\langle f(z+h), \psi (z)\rangle = \langle f(z), \psi (z-h)\rangle

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

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Problem 17.39

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

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(a)

$\mathcal{P} \frac{1}{x}$ ;

(b)

$\delta (x-a)$ ;

(c)

$\delta^{(n)}(x-a)$ ;

(d)

$\frac{1}{x \pm i 0}$ ;

(e)

$\operatorname {sign}(x)$ ;

(f)

$\theta (x)$ ;

(g)

$\operatorname {arctg} x$ ;

(h)

$x \operatorname {arctg} x$ ;

(i)

v. p. $\frac{\cos x}{x}$ ;

(j)

$\theta (x) e^{a x}, a > 0$ ;

(k)

$x_{+}^{\lambda }$ ;

(l)

$\ln x_{+}$ ;

(m)

$\cos (a x), a > 0$ ;

(n)

$\sin (a x), a > 0$ ;

(o)

$e^{i x^{2}}$ ;

(p)

$\mathcal{P} \frac{1}{\left|x\right|}$ .

Problem 17.40

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

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(a)

$x^{n} f = 0$ ;

(b)

$x f^{\prime } = 0$ ;

(c)

$x f = \operatorname {sign} x$ ;

(d)

$f^{\prime }+f = \delta_{0}$ ;

(e)

$f^{\prime }+f = \delta_{0}^{\prime }$ ;

(f)

$-f^{\prime \prime }+f = i \delta_{0}$ .