Power-type distributions x_+^lambda and (x+i0)^lambda

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Problem 16.7 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 16.1 "Main concepts"

Define the generalized function (x+i0)λ,Re⁡λ>−1(x+i 0)^{\lambda }, \operatorname {Re} \lambda > -1, as the limit of the regular generalized functions

(x+iε)λ=eλ(ln⁡∣x+iε∣+iarg⁡(x+iε)) (x+i \varepsilon )^{\lambda } = e^{\lambda (\ln \left|x+i \varepsilon \right|+i \arg (x+i \varepsilon ))}

(where arg⁡z∈[0,2π)\arg z \in [0,2 \pi )) as ε→+0\varepsilon \rightarrow +0. Prove that

(x+i0)λ=x+λ+eiπλx−λ. (x+i 0)^{\lambda } = x_{+}^{\lambda }+e^{i \pi \lambda } x_{-}^{\lambda }.
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Problem 16.8 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 16.1 "Main concepts"

Let Re⁡λ⩽−1,λ≠−1,−2,…\operatorname {Re} \lambda \leqslant -1, \lambda \neq -1,-2, \ldots, and let nn be the integer part of the number −Re⁡λ-\operatorname {Re} \lambda. For each function φ∈D\varphi \in \mathcal{D} set

⟨x+λ,φ⟩:=∫01xλ[φ(x)−φ(0)−xφ′(0)−…−xn−1(n−1)!φ(n−1)(0)]dx++∫1∞xλφ(x)dx+∑k=1nφ(k−1)(0)(k−1)!(λ+k)⟨x−λ,φ⟩:=∫01xλ[φ(−x)−φ(0)+xφ′(0)−…−(−x)n−1(n−1)!φ(n−1)(0)]dx++∫1∞xλφ(−x)dx+∑k=1n(−1)k−1φ(k−1)(0)(k−1)!(λ+k) \begin{aligned} \left\langle x_{+}^{\lambda }, \varphi \right\rangle : = & \int _{0}^{1} x^{\lambda }\left[\varphi (x)-\varphi (0)-x \varphi ^{\prime }(0)-\ldots -\frac{x^{n-1}}{(n-1)!} \varphi ^{(n-1)}(0)\right] d x+ \\ & +\int _{1}^{\infty } x^{\lambda } \varphi (x) d x+\sum _{k = 1}^{n} \frac{\varphi ^{(k-1)}(0)}{(k-1)!(\lambda +k)} \\ \left\langle x_{-}^{\lambda }, \varphi \right\rangle : = & \int _{0}^{1} x^{\lambda }\left[\varphi (-x)-\varphi (0)+x \varphi ^{\prime }(0)-\ldots -\frac{(-x)^{n-1}}{(n-1)!} \varphi ^{(n-1)}(0)\right] d x+ \\ & +\int _{1}^{\infty } x^{\lambda } \varphi (-x) d x+\sum _{k = 1}^{n} \frac{(-1)^{k-1} \varphi ^{(k-1)}(0)}{(k-1)!(\lambda +k)} \end{aligned}

Prove that these formulas define generalized functions on D. Prove that in every strip −n−1<Re⁡λ<−n-n-1 < \operatorname {Re} \lambda < -n these functionals can be given by the formulas

⟨x+λ,φ⟩:=∫0∞xλ[φ(x)−φ(0)−xφ′(0)−…−xn−1(n−1)!φ(n−1)(0)]dx⟨x−λ,φ⟩:=∫0∞xλ[φ(−x)−φ(0)+xφ′(0)−…−(−x)n−1(n−1)!φ(n−1)(0)]dx \begin{aligned} \left\langle x_{+}^{\lambda }, \varphi \right\rangle & : = \int _{0}^{\infty } x^{\lambda }\left[\varphi (x)-\varphi (0)-x \varphi ^{\prime }(0)-\ldots -\frac{x^{n-1}}{(n-1)!} \varphi ^{(n-1)}(0)\right] d x \\ \left\langle x_{-}^{\lambda }, \varphi \right\rangle & : = \int _{0}^{\infty } x^{\lambda }\left[\varphi (-x)-\varphi (0)+x \varphi ^{\prime }(0)-\ldots -\frac{(-x)^{n-1}}{(n-1)!} \varphi ^{(n-1)}(0)\right] d x \end{aligned}
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Problem 16.9 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 16.1 "Main concepts"

Prove that for fixed φ\varphi the function g(λ):=⟨x+λ,φ⟩g(\lambda ): = \left\langle x_{+}^{\lambda }, \varphi \right\rangle is holomorphic on the domain C\{−1,−2,…}\mathbb {C} \backslash \left\{ -1,-2, \ldots \right\} and, in general, has simple poles at the points −1,−2,…-1,-2, \ldots Find the residues at these points.

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Problem 16.10 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 16.1 "Main concepts"

Prove that for fixed φ\varphi the function g(λ):=⟨∣x∣λ,φ⟩g(\lambda ): = \left\langle \left|x\right|^{\lambda }, \varphi \right\rangle is holomorphic on the domain C\{−1,−3,−5,…}\mathbb {C} \backslash \left\{ -1,-3,-5, \ldots \right\} and, in general, has simple poles at the points −1,−3,−5,…-1,-3,-5, \ldots Find the residues at these points.

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Problem 16.11 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 16.1 "Main concepts"

Prove that for fixed φ\varphi the functions g+(λ):=:=⟨(x+i0)λ,φ⟩g_{+}(\lambda ): = : = \left\langle (x+i 0)^{\lambda }, \varphi \right\rangle and g−(λ):=⟨(x−i0)λ,φ⟩g_{-}(\lambda ): = \left\langle (x-i 0)^{\lambda }, \varphi \right\rangle are holomorphic on C\mathbb {C}.

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Problem 16.29 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 16.2 "Operations on generalized functions"

Prove that xmx+λ=x+m+λ,λ≠−1,−2,…,m∈Nx^{m} x_{+}^{\lambda } = x_{+}^{m+\lambda }, \lambda \neq -1,-2, \ldots , m \in \mathbb {N}.

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Problem 16.30 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 16.2 "Operations on generalized functions"

Prove that x2m∣x∣λ=∣x∣2m+λx^{2 m}\left|x\right|^{\lambda } = \left|x\right|^{2 m+\lambda }, and x2m+1∣x∣λ=∣x∣2m+1+λsign⁡xx^{2 m+1}\left|x\right|^{\lambda } = \left|x\right|^{2 m+1+\lambda } \operatorname {sign} x, λ≠−1,−2,…,m∈N\lambda \neq -1,-2, \ldots , m \in \mathbb {N}.

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Problem 16.34 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 16.2 "Operations on generalized functions"

Compute the derivative of the following generalized functions:

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(a)

x+λx_{+}^{\lambda } for −1<Re⁡λ<0-1 < \operatorname {Re} \lambda < 0;

(b)

P1∣x∣\mathcal{P} \frac{1}{\left|x\right|};

(c)

ln⁡x+\ln x_{+};

(d)

ln⁡∣x∣\ln \left|x\right|;

(e)

P1x\mathcal{P} \frac{1}{x};

(f)

ln⁡(x+i0)\ln (x+i 0);

(g)

x+λx_{+}^{\lambda } for Re⁡λ⩽−1,λ≠−1,−2,…\operatorname {Re} \lambda \leqslant -1, \lambda \neq -1,-2, \ldots;

(h)

(x+i0)λ,λ∈C(x+i 0)^{\lambda }, \lambda \in \mathbb {C};

(i)

xδ0′x \delta_{0}^{\prime }.

Problem 16.35 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 16.2 "Operations on generalized functions"

Compute the second derivative of the following generalized functions:

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(a)

∣x∣\left|x\right|;

(b)

∣sin⁡x∣\left|\sin x\right|;

(c)

x+λx_{+}^{\lambda } for −1<Re⁡λ<0-1 < \operatorname {Re} \lambda < 0;

(d)

ln⁡x+\ln x_{+};

(e)

ln⁡∣x∣\ln \left|x\right|;

(f)

ln⁡(x+i0)\ln (x+i 0);

(g)

P1x\mathcal{P} \frac{1}{x}.

Problem 16.36 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 16.2 "Operations on generalized functions"

Compute the nn-th derivative of x+λx_{+}^{\lambda } for −1<Re⁡λ<0-1 < \operatorname {Re} \lambda < 0 (for the definition of x+λx_{+}^{\lambda }, see problem 16.8).

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Problem 16.44 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 16.2 "Operations on generalized functions"

Solve in D′\mathcal{D}^{\prime } the differential equations:

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(a)

F(n)=0F^{(n)} = 0;

(b)
  • xF′=λFx F^{\prime } = \lambda F, where λ∈C\lambda \in \mathbb {C};
(c)

F′+F=δ0F^{\prime }+F = \delta_{0};

(d)

F′′+F=δ0F^{\prime \prime }+F = \delta_{0};

(e)

xF′=1x F^{\prime } = 1;

(f)

F′′−F=δ0F^{\prime \prime }-F = \delta_{0}.

Problem 17.39 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 17.2 "Fourier transform of generalized functions"

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

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(a)

P1x\mathcal{P} \frac{1}{x};

(b)

δ(x−a)\delta (x-a);

(c)

δ(n)(x−a)\delta^{(n)}(x-a);

(d)

1x±i0\frac{1}{x \pm i 0};

(e)

sign⁡(x)\operatorname {sign}(x);

(f)

θ(x)\theta (x);

(g)

arctg⁡x\operatorname {arctg} x;

(h)

xarctg⁡xx \operatorname {arctg} x;

(i)

v. p. cos⁡xx\frac{\cos x}{x};

(j)

θ(x)eax,a>0\theta (x) e^{a x}, a > 0;

(k)
  • x+λx_{+}^{\lambda };
(l)
  • ln⁡x+\ln x_{+};
(m)

cos⁡(ax),a>0\cos (a x), a > 0;

(n)

sin⁡(ax),a>0\sin (a x), a > 0;

(o)

eix2e^{i x^{2}};

(p)

P1∣x∣\mathcal{P} \frac{1}{\left|x\right|}.

Problem 441 from Kirillov A.A., Gvishiani A.D. "Теоремы и задачи функционального анализа" [KiGv88]More problems in Subsection III.4.4 "Generalized Functions"

Let φ∈D(R)\varphi \in \mathcal D(\mathbb {R}).

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(a)

Prove that the function

fφ(λ)=Γ(λ)−1∫0∞xλ−1φ(x) dx, f_\varphi (\lambda )=\Gamma (\lambda )^{-1}\int _0^\infty x^{\lambda -1}\varphi (x)\, dx,

which is defined for Re⁡λ>0\operatorname {Re}\lambda >0, admits an analytic continuation to the left half-plane.

(b)

Prove that for fixed λ∈C\lambda \in \mathbb {C} the correspondence φ↦fφ(λ)\varphi \mapsto f_\varphi (\lambda ) is a generalized function (usually denoted by x+λ−1/Γ(λ)x_+^{\lambda -1}/\Gamma (\lambda )).

(c)

Compute the generalized function defined above for the parameter values λ=−n\lambda =-n, n=0,1,2,…n=0,1,2,\ldots.

Problem 449 from Kirillov A.A., Gvishiani A.D. "Теоремы и задачи функционального анализа" [KiGv88]More problems in Subsection III.4.5 "Operations on Generalized Functions"

A generalized function FF on the line is said to be homogeneous of degree (λ,ε)(\lambda ,\varepsilon ), where λ∈C\lambda \in \mathbb {C} and ε=0\varepsilon =0 or 11, if F(tx)=∣t∣λ(sgn⁡t)εF(x)F(tx)=\left|t\right|^{\lambda }(\operatorname {sgn}t)^\varepsilon F(x) for t≠0t\ne 0.

Prove that a homogeneous function of degree (λ,ε)(\lambda ,\varepsilon ) on the line (see first-edition Problem 513) satisfies the differential equation xF′=λFxF'=\lambda F.

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Problem 450 from Kirillov A.A., Gvishiani A.D. "Теоремы и задачи функционального анализа" [KiGv88]More problems in Subsection III.4.5 "Operations on Generalized Functions"

Prove that for all λ∈C\lambda \in \mathbb {C} and ε=0,1\varepsilon =0,1 there exists a homogeneous generalized function on the line of degree (λ,ε)(\lambda ,\varepsilon ), and that it is unique to within a numerical factor.

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Problem 534 from Kirillov A.A., Gvishiani A.D. "Теоремы и задачи функционального анализа" [KiGv88]More problems in Subsection IV.1.2 "Convolutions of Generalized Functions"

Let fα(x)=[1/Γ(α)]xα−1θ(x)f_\alpha (x)=[1/\Gamma (\alpha )]x^{\alpha -1}\theta (x) for α>0\alpha >0.

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(a)

Verify that fα∈D+′(R)f_\alpha \in \mathcal{D}'_+(\mathbb {R}) for α>−1\alpha >-1.

(b)

Prove the identity fα∗fβ=fα+βf_\alpha *f_\beta =f_{\alpha +\beta }.

(c)

Prove the identity (d/dx)fα=fα−1(d/dx)f_\alpha =f_{\alpha -1} for α>1\alpha >1.

(d)

Find the limit of fαf_\alpha in D+′(R)\mathcal{D}'_+(\mathbb {R}) as α→0\alpha \to 0.

Problem 600 from Kirillov A.A., Gvishiani A.D. "Теоремы и задачи функционального анализа" [KiGv88]More problems in Subsection IV.2.4 "The Fourier Transform of Generalized Functions"

Let ff be a homogeneous generalized function of degree (λ,ε)(\lambda ,\varepsilon ). Prove that f~\tilde f is also homogeneous, and find its degree.

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References

[БСШ17] Бородин П.А., Савчук А.М., Шейпак И.А. Задачи по функциональному анализу. МЦНМО, 2017.
[KiGv88] Kirillov A.A., Gvishiani A.D. Теоремы и задачи функционального анализа. 2-е изд., Наука, 1988.