• Keine Ergebnisse gefunden

Functional Analysis II – Problem sheet 9

N/A
N/A
Protected

Academic year: 2022

Aktie "Functional Analysis II – Problem sheet 9"

Copied!
1
0
0

Wird geladen.... (Jetzt Volltext ansehen)

Volltext

(1)

Functional Analysis II – Problem sheet 9

Mathematisches Institut der LMU – SS2009 Prof. Dr. P. M¨uller, Dr. A. Michelangeli

Handout: 23.06.2009

Due: Tuesday 30.06.2009 by 1 p.m. in the “Funktionalanalysis II” box Questions and infos: Dr. A. Michelangeli, office B-334, michel@math.lmu.de Grader: Ms. S. Sonner – ¨Ubungen on Wednesdays, 4,30 - 6 p.m., room C-111

Exercise 24. Let ϕ, ψ ∈ S(Rd), the Schwartz space of smooth functions with rapid decrease in dimension d > 1, and let F be the Fourier transform1 on S(Rd). Prove Theorem 2.13, part (g), stated in the class. I.e., prove that F(ϕψ) = (2π)−d/2F(ϕ)∗ F(ψ), that is, the Fourier transform of the (pointwise) product of two Schwartz functions is, up to a pre-factor, the convolution of the Fourier transform of each function. (Compare this statement with the discussion of Exercise 22.)

Exercise 25. Let ϕ∈C0(R), ϕ≡/0. i.e., a non-zero compactly supported smooth function.

Prove that Fϕ, its Fourier transform, cannot have compact support. (Hint: assume by con- tradiction that ξ7→(Fϕ)(ξ) is compactly supported, extend it from ξ∈R toξ∈C and show that you get an holomorphic function. Then. . .)

Exercise 26. Let a > 0. Say in which sense (i.e., as elements of which space) the R R functions

x7→e−a|x|

x7→

r2 π

a a2 +x2

admit Fourier transform and compute it. Verify the Fourier inversion formula in this cases.

1remember that the convention adopted in the class is (Ff)(ξ) = (2π)−d/2R

Rdf(x)e−ixξdx

1

Referenzen

ÄHNLICHE DOKUMENTE

This characterisation of the spectrum of A usually goes under the name of Weyl’s criterion: it says that the points of spec(A) are “almost eigenvalues”.. of A (i.e., Aϕ n ≈ λϕ n

(Hint: the natural way to proceed is to construct explicitly the approximating step function. To this aim, observe that the range of f can be covered by a finite and conveniently

Prove that, due to (∗∗), the limit operator defined in (∗) is unique (i.e., independent of the approximating sequence of simple functions one considers) and (∗∗) holds for

10.4) Prove that the operator-valued function R 3 t 7→ e tA is norm-continuous on R and Lipschitz norm-continuous on any bounded subset of R. Estimate such a constant.. 10.5) Prove

Compare this result to the general statement of the Weyl’s criterion (→ Exercise 3): actually that statement does not exclude that the whole Spec(A), not only Spec ess (A), might

1 This goes beyond the standard example of the Dirac measure, which is singular with respect to the Lebesgue measure, but not continuous (it is a pure point

(Hint: connect this problem with Exercise 19 above and use the thesis stated there. To be sure to have fixed all the details, check the role played in this construction by

Applied Automata Theory (WS 2012/2013) Technische Universit¨ at Kaiserslautern.. Exercise