Adv. Pure Appl. Math. 2017; aop Research Article Radha Ramakrishnan* and Saswata Adhikari A sampling theorem for the twisted shift-invariant space DOI: 10.1515/apam-2016-0090 Received September 9, 2016; revised April 27, 2017; accepted May 1, 2017 Abstract: Recently, a characterization of frames in twisted shift-invariant spaces in L 2 (ℝ 2n ) has been obtained in [16]. Using this result, we prove a sampling theorem on a subspace of a twisted shift-invariant space in this paper. Keywords: Canonical dual frames, frames, sampling theorem, shift-invariant space, twisted translation MSC 2010: Primary 94A20; secondary 42C15, 42B99 1 Introduction The fundamental Shannon’s sampling theorem states that any function f belonging to the PaleyśWiener space B π ={f ∈ L 2 (ℝ) : supp  f ⊂ [−π, π]} can be reconstructed from its samples {f(k) : k ∈ ℤ} by the formula f(x)=∑ k∈ℤ f(k) sinc(x − k), where sinc y = sin πy πy and  f denotes the Fourier transform of f , given by  f (ξ)=∫ ℝ f(x)e −2πi⟨x, ξ⟩ dx, ξ ∈ℝ. Paley and Wiener extended Shannon’s sampling theorem to a non-uniform sampling set in [15]. They showed that if X ={x k ∈ℝ : k ∈ ℤ} is such that |x k − k|< 1/π 2 , then any function f belonging to the class {f ∈ L 2 (ℝ) : supp  f ⊂ [−π, π]} can be recovered from its samples {f(x k ) : k ∈ ℤ}. Dufn and Eachus [7] showed that the result is true if |x k − k|< 0.22. Later, Kadee [13] showed that the maximum bound for |x k − k| has to be less than 0.25. For a more general sampling set, the sampling condition is stated in terms of Beurling density. Sampling theorems have been studied on wavelet subspaces in [22, 23]. In particular, in [23] for any closed shift-invariant subspace V 0 of L 2 (ℝ), a necessary and sufcient condition under which there is a sam- pling expansion for every f ∈ V 0 was shown. In sampling theory, non-uniform sampling in shift-invariant spaces is given importance to for the past őfteen years. We refer to a few papers [1ś3, 8ś11, 18ś20] in this connection. Characterizations of shift-invariant spaces in L 2 (ℝ n ) in terms of range functions were studied by Bownik in [4]. The study of shift-invariant spaces and frames has been extended to locally compact abelian groups *Corresponding author: Radha Ramakrishnan: Department of Mathematics, Indian Institute of Technology Madras, Chennai 600 036, India, e-mail: radharam@iitm.ac.in Saswata Adhikari: Department of Mathematics, Indian Institute of Technology Madras, Chennai 600 036, India, e-mail: saswata.adhikari@gmail.com Brought to you by | New York University Authenticated Download Date | 6/26/17 8:11 PM