Paley–Wiener theorem for functions in L_p(ℝⁿ)

ABSTRACT

In this paper we examine necessary and sufficient conditions on the sequences of norm of the derivatives of functions in $L_p\left({\mathbb{R}}^n\right)$ such that their spectrum (the support of their Fourier transform) is contained in a given compact set in ${\mathbb{R}}^n$.

KEYWORDS:

• Real Paley–Wiener theorem
• $L_p$-spaces
• Fourier transform

AMS SUBJECT CLASSIFICATION:

• 46F12

Acknowledgments

A part of this work was done when V.N.Huy was working at the Vietnam Institute for Advanced Study in Mathematics (VIASM). The author would like to thank the VIASM for providing a fruitful research environment and working condition.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research of V.N. Huy is funded by the Vietnam National University, Hanoi (VNU) [grant number QG.16.08].