On matrix-valued wave packet frames in $$L^2({\mathbb {R}}^d, {\mathbb {C}}^{s\times r})$$ L 2 ( R d , C s
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On matrix-valued wave packet frames in L2 (Rd , Cs×r ) Jyoti1 · Lalit Kumar Vashisht1 Received: 23 February 2019 / Accepted: 19 October 2020 © Springer Nature Switzerland AG 2020
Abstract In this paper, we study matrix-valued wave packet frames for the matrix-valued function space L 2 (Rd , Cs×r ). An interplay between matrix-valued wave packet frames and its associated atomic wave packet frames is discussed. This is inspired by examples which show that frame properties cannot be carried from matrix-valued wave packet scaling functions to its associated atomic wave packet scaling functions and vice versa. Construction of matrix-valued wave packet frames for L 2 (Rd , Cs×r ) from corresponding atomic wave packet frames for L 2 (Rd ) (and conversely) are given. Some special classes of matrix-valued scaling functions are given. A characterization of tight matrix-valued wave packet frames in terms of orthogonality of Bessel sequences has been obtained. Further, we provide a characterization of superframes which can generate matrix-valued frames. Finally, a Paley-Wiener type perturbation result with respect to matrix-valued wave packet scaling functions is given. Keywords Frame · Bessel sequence · Gabor frame · Wave packet system · Perturbation Mathematics Subject Classification 42C15 · 42C30 · 42C40 · 43A32
1 Introduction A wave packet is a function defined on a domain D ⊂ Rd which is “well localized in phase space”. The Gaussian function is one of the standard examples of a wave
The research of Jyoti is supported by the Council of Scientific & Industrial Research (CSIR), India, Grant No. 09/045(1374)/2015-EMR-I.
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Lalit Kumar Vashisht [email protected] Jyoti [email protected]
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Department of Mathematics, University of Delhi, Delhi 110007, India 0123456789().: V,-vol
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packet function. By applying dilation, modulation and translation operators to the Gaussian function in the study of some classes of singular integral operators, Cordoba and Fefferman [11] introduced the concept of a wave packet system. To be precise, in order to reduce a pseudo-differential operator to a multiplier, and analysis of the Egorov operator which is given by −n ei(•,ξ ) f (ξ )dξ, E( f (•)) = (2π ) 2 Rd
Cordoba and Fefferman [11] introduced the notion of “wave packet transform”. Let H+ be the Siegel upper half-space, that is, the set of all symmetric d-by-d matrices X such that ImX > 0 and let 1 φ(y, η, X ) = X expi η.(x − y) + (x − y).X (x − y) , (y, η, X ) ∈ Rd × Rd × H+ , 2 1
where X = (det ImX) 4 . Then, for all f ∈ L 2 (Rd ), the wave packet transform W is given by W : (y, η, X ) → f , φ(y, η, X ). Later, Labate, Weiss, and Wilson [36] adopted the same expression to study frame properties of wave packet systems, in particular, for the construction of Parseval frames of wave packets in the Lebesgue space L 2 (Rd ). In [30], Hernández, Labate, Weiss, and Wilson gave a characterization of Parseval wave packet frames for L 2 (Rd ). Many researchers have investiga
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