Modulation Spaces

Bényi, Árpád; Okoudjou, Kasso A.

doi:10.1007/978-1-0716-0332-1_2

Cited by 2 publications

(1 citation statement)

References 44 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Modulation spaces can be naturally introduced at this stage as spaces of distributions characterized by prescribed summability conditions for the corresponding phase-space representations [BO20]. For 1 ≤ p ≤ ∞ and a fixed g…”

Section: Introductionmentioning

confidence: 99%

On the convergence of a novel family of time slicing approximation operators for Feynman path integrals

Trapasso

2021

Preprint

View full text Add to dashboard Cite

In this note we study the properties of a sequence of approximate propagators for the Schrödinger equation, in the spirit of Feynman's path integrals. Precisely, we consider Hamiltonian operators arising as the Weyl quantization of a quadratic form in phase space, plus a bounded potential perturbation in the form of a pseudodifferential operator with a rough symbol. It is known that the corresponding Schrödinger propagator is a generalized metaplectic operator. This naturally motivates the introduction of a manageable time slicing approximation consisting of operators of the same type. By means of techniques and function spaces of time-frequency analysis it is possible to obtain several convergence results with precise rates in terms of the mesh size of the time slicing subdivision. In particular, we prove convergence in the norm operator topology in L 2 , as well as pointwise convergence of the corresponding integral kernels for non-exceptional times.

show abstract

Section: Introductionmentioning

confidence: 99%

On the convergence of a novel family of time slicing approximation operators for Feynman path integrals

Trapasso

2021

Preprint

View full text Add to dashboard Cite

show abstract

Completeness of shifted dilates in invariant Banach spaces of tempered distributions

Feichtinger

Gumber

2021

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

We show that well-established methods from the theory of Banach modules and time-frequency analysis allow to derive completeness results for the collection of shifted and dilated version of a given (test) function in a quite general setting. While the basic ideas show strong similarity to the arguments used in a recent paper by V. Katsnelson we extend his results in several directions, both relaxing the assumptions and widening the range of applications. There is no need for the Banach spaces considered to be embedded into ( L 2 ( R ) , ‖ ⋅ ‖ 2 ) \big ( {{{\boldsymbol {L}}^2}(\mathbb {R})}, \, \|\,\cdot \,\|_{2} \big ) , nor is the Hilbert space structure relevant. We choose to present the results in the setting of the Euclidean spaces, because then the Schwartz space S ′ ( R d ) {{{\boldsymbol {\mathcal {S}}}’}(\mathbb {R}^d)} ( d ≥ 1 d \geq 1 ) of tempered distributions provides a well-established environment for mathematical analysis. We also establish connections to modulation spaces and Shubin classes ( Q s ( R d ) , ‖ ⋅ ‖ Q s ) \big ( {{{\boldsymbol {Q}}_s}({{\mathbb {R}^{d}}})}, \, \|\,\cdot \,\|_{{{\boldsymbol {Q}}_s}} \big ) , showing that they are special cases of Katsnelson’s setting (only) for s ≥ 0 s \geq 0 .

show abstract

Modulation Spaces

Cited by 2 publications

References 44 publications

On the convergence of a novel family of time slicing approximation operators for Feynman path integrals

On the convergence of a novel family of time slicing approximation operators for Feynman path integrals

Completeness of shifted dilates in invariant Banach spaces of tempered distributions

Contact Info

Product

Resources

About