A spectral Paley-Wiener theorem for the Heisenberg group and a support theorem for the twisted spherical means on \mathbb{C}^n

Eswar, Narayanan; Thangavelu, Sundaram

doi:10.5802/aif.2189

Cited by 11 publications

(9 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Some comments on the relations between results in this paper and results in the literature are in order. The study of bandlimited functions and Paley-Wiener-type results recently has been an active branch of research, for noncommutative groups [15,25,18,16,2], and for symmetric spaces, see e.g. [20,14,22].…”

Section: Discussionmentioning

confidence: 99%

“…See e.g. [25,18,2] for such results for certain classes of nilpotent groups including the Heisenberg group, and [15,16] for more general cases. By contrast, we consider an "inverse" version: For a function f with compactly supported Fourier transform (the latter being defined by spectral theory of the sub-Laplacian), we study complex-analytic properties of f .…”

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

Paley‐Wiener estimates for the Heisenberg group

Führ

2010

Mathematische Nachrichten

View full text Add to dashboard Cite

The Paley-Wiener space P W (G) on a stratified Lie group G is defined via the spectral decomposition of the associated sub-Laplacian. In this paper, we show that functions in P W (H), where H denotes the Heisenberg group, extend to an entire function on the complexification H C , satisfying a growth estimate of exponential order two. We also show that a converse, characterizing elements of P W (H) only in terms of pointwise growth behaviour of the entire extension, is not available.

show abstract

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Paley‐Wiener estimates for the Heisenberg group

Führ

2010

Mathematische Nachrichten

View full text Add to dashboard Cite

show abstract

“…There is no answer so far for the Gelfand pair (U(n) ⋉ H n , U(n)), where H n is the 2n + 1-dimensional Heisenberg group. Even though some attempts have been made to address this Paley-Wiener theorem for the Heisenberg group, [9,24,25,23]. The Fourier analysis for symmetric spaces of noncompact type is well understood by the work of Helgason and Gangolli, [10,14].…”

Section: Examples Show That Most Likely There Is No Universal Answer mentioning

confidence: 99%

Paley-Wiener theorems with respect to the spectral parameter

Dann¹

2011

New Developments in Lie Theory and Its Applications

View full text Add to dashboard Cite

One of the important questions related to any integral transform on a manifold M or on a homogeneous space G/K is the description of the image of a given space of functions. If M = G/K, where (G, K) is a Gelfand pair, then the harmonic analysis is closely related to the representations of G and the direct integral decomposition of L 2 (M ) into irreducible representations. We give a short overview of the Fourier transform on such spaces and then ask if one can describe the image of the space of smooth compactly supported functions in terms of the spectral parameter, i.e., the parameterization of the set of irreducible representations in the support of the Plancherel measure for L 2 (M ). We then discuss the Euclidean motion group, semisimple symmetric spaces, and some limits of those spaces. f (λ) = F R n (f )(λ) = f (x)e −2πix·λ dx .

show abstract

“…There is a wide literature on Paley-Wiener theorems on the Heisenberg group. The earliest result is due to Ando [2], followed by Thangavelu [21,22,23], Arnal and Ludwig [3], Narayanan and Thangavelu [19]. Results are mostly related to the group (operator-valued)…”

Section: Introductionmentioning

confidence: 99%

Paley–Wiener theorems for the $${\text {U}(n)}$$ U ( n ) -spherical transform on the Heisenberg group

Astengo

Blasio²,

Ricci

2014

Annali di Matematica

View full text Add to dashboard Cite

We prove several Paley–Wiener-type theorems related to the spherical transform on the Gelfand pair H_n xU(n), U(n) , where H_n is the 2n + 1-dimensional Heisenberg group. Adopting the standard realization of the Gelfand spectrum as the Heisenberg fan in R^2, we prove that spherical transforms of U(n)-invariant functions and distributions with compact support in H_n admit unique entire extensions to C^2 , and we find real-variable characterizations of such transforms. Next, we characterize the inverse spherical transforms of compactly supported functions and distributions on the fan, giving analogous characterizations

show abstract

A spectral Paley-Wiener theorem for the Heisenberg group and a support theorem for the twisted spherical means on \mathbb{C}^n

Cited by 11 publications

References 8 publications

Paley‐Wiener estimates for the Heisenberg group

Paley‐Wiener estimates for the Heisenberg group

Paley-Wiener theorems with respect to the spectral parameter

Paley–Wiener theorems for the $${\text {U}(n)}$$ U ( n ) -spherical transform on the Heisenberg group

Contact Info

Product

Resources

About