Verifying that an explicit functional is a distribution

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Problem 14.3 from Brayman V.B., Chaikovskyi A.V., Konstantinov O.Y., Kukush A.G., Mishura Y.S., Nesterenko O.N. "Functional Analysis and Operator Theory" [BCK+24]More problems in Section 14.1 "Examples of Problems with Solutions"

Using the definition, prove that the following functionals are the generalized functions:

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(1)

f(φ)=∫Rxφ(x)dxf(\varphi ) = \int_{\mathbb {R}} x \varphi (x) d x

(2)

g(φ)=∑n=1∞φ(n)(n),φ∈D(R)g(\varphi ) = \sum_{n = 1}^{\infty } \varphi^{(n)}(n), \varphi \in \mathcal{D}(\mathbb {R}).

Are they regular? Find supports of these generalized functions.

Problem 14.26 from Brayman V.B., Chaikovskyi A.V., Konstantinov O.Y., Kukush A.G., Mishura Y.S., Nesterenko O.N. "Functional Analysis and Operator Theory" [BCK+24]More problems in Section 14.2 "Problems to Solve"

Prove that the following functionals f:D(R)→Cf: \mathcal{D}(\mathbb {R}) \rightarrow \mathbb {C} are generalized function, i.e., f∈D′(R)f \in \mathcal{D}^{\prime }(\mathbb {R}).

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(1)

f(φ)=φ(0)+φ(1)f(\varphi ) = \varphi (0)+\varphi (1);

(2)

f(φ)=φ′(0)f(\varphi ) = \varphi^{\prime }(0);

(3)

f(φ)=∫Rφ(x)dxf(\varphi ) = \int_{\mathbb {R}} \varphi (x) d x;

(4)

f(φ)=∫−11∣x∣φ′(x)dx;f(\varphi ) = \int_{-1}^{1}\left|x\right| \varphi^{\prime }(x) d x ;

(5)

f(φ)=∫−22sign⁡xφ′(x)dxf(\varphi ) = \int_{-2}^{2} \operatorname {sign} x \varphi^{\prime }(x) d x;

(6)

f(φ)=∫Rln⁡∣x∣φ(x)dxf(\varphi ) = \int_{\mathbb {R}} \ln \left|x\right| \varphi (x) d x;

(7)

f(φ)=∫Rexφ′(x)dxf(\varphi ) = \int_{\mathbb {R}} e^{x} \varphi^{\prime }(x) d x;

(8)

f(φ)=∫Re−xφ(n)(x)dx,n∈Nf(\varphi ) = \int_{\mathbb {R}} e^{-x} \varphi^{(n)}(x) d x, n \in \mathbb {N};

(9)

f(φ)=φ′(1)+∫02πsin⁡xφ(x)dxf(\varphi ) = \varphi^{\prime }(1)+\int_{0}^{2 \pi } \sin x \varphi (x) d x;

(10)

f(φ)=φ(0)+∫13xφ′′(x)dxf(\varphi ) = \varphi (0)+\int_{1}^{3} x \varphi^{\prime \prime }(x) d x.

In which of these cases is the function regular? Find the support of ff.

Problem IV.5.10 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section IV.5 "Inductive Limits and the Space of Distributions"

Let ff be a Lebesgue measurable function on Ω\Omega (an open subset of Rd\mathbb {R}^d, d≥1d \ge 1) that is locally integrable (that is, ∫K∣f∣ dλ<∞\int_K \left|f\right| \, d\lambda < \infty for every compact subset KK of Ω\Omega, where λ\lambda is dd-dimensional Lebesgue measure). If Lf:D(Ω)→FL_f : \mathscr {D}(\Omega ) \to \mathbb {F} is defined by Lf(ϕ)=∫fϕ dλL_f(\phi ) = \int f\phi \, d\lambda, then LfL_f is a distribution (that is, a continuous linear functional on D(Ω)\mathscr {D}(\Omega )). Verify this statement.

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Problem IV.5.11 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section IV.5 "Inductive Limits and the Space of Distributions"

For the remaining exercises, Ω\Omega is always an open subset of Rd\mathbb {R}^d, d≥1d \ge 1. If μ\mu is a measure on Ω\Omega, show that ϕ↦∫ϕ dμ\phi \mapsto \int \phi \, d\mu is a distribution [on] Ω\Omega.

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References

[BCK+24] Brayman V.B., Chaikovskyi A.V., Konstantinov O.Y., Kukush A.G., Mishura Y.S., Nesterenko O.N. Functional Analysis and Operator Theory. Springer, 2024.
[Con07] Conway J.B. A Course in Functional Analysis. 2nd ed., Springer, 2007.