Generalized Paley–wiener Theorems

Chen, Qiuhui; Li, Lijie; Ren, Guangbin

doi:10.1142/s0219691312500208

Cited by 14 publications

(4 citation statements)

References 9 publications

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“…The real Paley‐Wiener theorem has been extended to higher dimensions as well as to other integral transforms instead of the derivative operator; see previous works 3–8 . In particular, the real Paley‐Wiener theorem has been extended to the Bülow and Sommer version 9 of the quaternion Fourier transform

F_{ℍ} (f) (w) = \int_{ℝ^{2}} f (x) e^{- i w_{1} x_{1}} e^{- j w_{2} x_{2}} d x;

see Fu and Li 10 .…”

Section: Introductionmentioning

confidence: 99%

Real Paley‐Wiener theorem for the octonion Fourier transform

Ren

2021

Math Methods in App Sciences

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F_{ℍ} (f) (w) = \int_{ℝ^{2}} f (x) e^{- i w_{1} x_{1}} e^{- j w_{2} x_{2}} d x;

see Fu and Li 10 .…”

Section: Introductionmentioning

confidence: 99%

Real Paley‐Wiener theorem for the octonion Fourier transform

Ren

2021

Math Methods in App Sciences

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“…He then demonstrates the applications of his method to the FT and proves a series of Paley–Wiener‐type theorems for functions with bounded, unbounded, convex, and nonconvex domains. Because Tuan's proofs are mainly based on the Parseval's identity for the FT and a property that many other integral transforms share, so a wide number of papers have been successfully devoted to the extension of the theory on Hankel transform, Dunkl transform, singular Sturm–Liouville integral transform, and many other important integral transforms (see other studies and the references therein). A comprehensive overview of the literature of real Paley–Wiener theorem was included in Andersen et al…”

Section: Introductionmentioning

confidence: 99%

Real Paley–Wiener theorems for the (k,a)‐generalized Fourier transform

Leng

Fei

2020

Math Methods in App Sciences

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In this paper, we obtain several versions of the real Paley-Wiener theorems for the (k, a)-generalized Fourier transform recently investigated at length by Ben Saïd, Kobayashi, and Ørsted. This generalized Fourier transform can be regarded as a two-parameter generalization of Howe's description of classical Fourier transform, where k is a multiplicity function for the Dunkl operators on R d ⧵ {0} and a > 0 arises from the interpolation of the two Lie algebra (2, R) actions on the Weil representation of Mp(d, R) and the minimal unitary representation of the O(d + 1, 2).

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“…He then demonstrates the applications of his method to the Fourier transform and proves a series of Paley‐Wiener type theorems for functions with bounded, unbounded, convex, and nonconvex domains. Since Tuan's proofs are mainly based on the Parseval's identity for the Fourier transform and a property that many other integral transforms share, so a wide number of papers have been successfully devoted to the extension of the theory on many other integral transforms(see the previous studies and the references therein).…”

Section: Introductionmentioning

confidence: 99%