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Phase space analysis of finite and infinite dimensional Fresnel integrals.

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Title: Phase space analysis of finite and infinite dimensional Fresnel integrals.
Authors: Mazzucchi, Sonia1 (AUTHOR), Nicola, Fabio2 (AUTHOR), Trapasso, S. Ivan1,2 (AUTHOR)
Source: Journal of Functional Analysis. Oct2025, Vol. 289 Issue 8, pN.PAG-N.PAG. 1p.
Subjects: Integrable functions, Fresnel function, Feynman integrals, Mathematical physics, Schrödinger operator
Abstract: The full characterization of the class of Fresnel integrable functions is an open problem in functional analysis, with significant applications to mathematical physics (Feynman path integrals) and the analysis of the Schrödinger equation. In finite dimension, we prove the Fresnel integrability of functions in the Sjöstrand class M ∞ , 1 — a family of continuous and bounded functions, locally enjoying the mild regularity of the Fourier transform of an integrable function. This result broadly extends the current knowledge on the Fresnel integrability of Fourier transforms of finite complex measures, and relies upon ideas and techniques of Gabor wave packet analysis. We also discuss infinite-dimensional extensions of this result. In this connection, we extend and make more concrete the general framework of projective functional extensions introduced by Albeverio and Mazzucchi. In particular, we obtain a concrete example of a continuous linear functional on an infinite-dimensional space beyond the class of Fresnel integrable functions. As an interesting byproduct, we obtain a sharp M ∞ , 1 → L ∞ operator norm bound for the free Schrödinger evolution operator. [ABSTRACT FROM AUTHOR]
Copyright of Journal of Functional Analysis is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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DbLabel: Engineering Source
An: 185654704
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  Data: Phase space analysis of finite and infinite dimensional Fresnel integrals.
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  Data: <searchLink fieldCode="AR" term="%22Mazzucchi%2C+Sonia%22">Mazzucchi, Sonia</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Nicola%2C+Fabio%22">Nicola, Fabio</searchLink><relatesTo>2</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Trapasso%2C+S%2E+Ivan%22">Trapasso, S. Ivan</searchLink><relatesTo>1,2</relatesTo> (AUTHOR)
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  Data: <searchLink fieldCode="JN" term="%22Journal+of+Functional+Analysis%22">Journal of Functional Analysis</searchLink>. Oct2025, Vol. 289 Issue 8, pN.PAG-N.PAG. 1p.
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  Data: <searchLink fieldCode="DE" term="%22Integrable+functions%22">Integrable functions</searchLink><br /><searchLink fieldCode="DE" term="%22Fresnel+function%22">Fresnel function</searchLink><br /><searchLink fieldCode="DE" term="%22Feynman+integrals%22">Feynman integrals</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+physics%22">Mathematical physics</searchLink><br /><searchLink fieldCode="DE" term="%22Schrödinger+operator%22">Schrödinger operator</searchLink>
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  Label: Abstract
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  Data: The full characterization of the class of Fresnel integrable functions is an open problem in functional analysis, with significant applications to mathematical physics (Feynman path integrals) and the analysis of the Schrödinger equation. In finite dimension, we prove the Fresnel integrability of functions in the Sjöstrand class M ∞ , 1 — a family of continuous and bounded functions, locally enjoying the mild regularity of the Fourier transform of an integrable function. This result broadly extends the current knowledge on the Fresnel integrability of Fourier transforms of finite complex measures, and relies upon ideas and techniques of Gabor wave packet analysis. We also discuss infinite-dimensional extensions of this result. In this connection, we extend and make more concrete the general framework of projective functional extensions introduced by Albeverio and Mazzucchi. In particular, we obtain a concrete example of a continuous linear functional on an infinite-dimensional space beyond the class of Fresnel integrable functions. As an interesting byproduct, we obtain a sharp M ∞ , 1 → L ∞ operator norm bound for the free Schrödinger evolution operator. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Journal of Functional Analysis is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1016/j.jfa.2025.111009
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      – Code: eng
        Text: English
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      – SubjectFull: Integrable functions
        Type: general
      – SubjectFull: Fresnel function
        Type: general
      – SubjectFull: Feynman integrals
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      – SubjectFull: Mathematical physics
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      – SubjectFull: Schrödinger operator
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      – TitleFull: Phase space analysis of finite and infinite dimensional Fresnel integrals.
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              Text: Oct2025
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