Undersampled Windowed Exponentials and Their Applications
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Undersampled Windowed Exponentials and Their Applications Chun-Kit Lai1 · Sui Tang2
Received: 2 March 2018 / Accepted: 1 November 2018 © Springer Nature B.V. 2018
Abstract We characterize the completeness and frame/basis property of a union of undersampled windowed exponentials of the form F (g) := e2π inx : n ≥ 0 ∪ g(x)e2π inx : n < 0 for L2 [−1/2, 1/2] by the spectra of the Toeplitz operators with the symbol g. Using this characterization, we classify all real-valued functions g such that F (g) is complete or forms a frame/basis. Conversely, we use the classical non-harmonic Fourier series theory to determine all ξ such that the Toeplitz operators with the symbol e2π iξ x is injective or invertible. These results demonstrate an elegant interaction between frame theory of windowed exponentials and Toeplitz operators. Finally, we use our results to answer some open questions in dynamical sampling, and derivative samplings on Paley-Wiener spaces of bandlimited functions. Keywords Completeness · Frames · Spectra · Toeplitz operators · Windowed exponentials Mathematics Subject Classification 94O20 · 42C15 · 42C30
1 Introduction Background Let Ω be a measurable set of finite Lebesgue measure in Rd . Suppose that g ∈ L2 (Ω) \ {0} and Λ is a countable subset of Rd . We define the collection of windowed
B S. Tang
[email protected] C.-K. Lai [email protected]
1
Department of Mathematics, San Francisco State University, 1600 Holloway Avenue, San Francisco, CA 94132, USA
2
Department of Mathematics, Johns Hopkins University, 3400 North Charles Street, Baltimore, MD 21218, USA
C.-K. Lai, S. Tang
exponentials by
E (g, Λ) := g(x)e2π iλ,x : λ ∈ Λ .
A windowed exponentials Nnatural question is to determine when a union of finitely many 2 j =1 E (gj , Λj ) is complete or forms a frame/Riesz basis for L (Ω). Let us recall the definitions as below: Definition 1.1 The collection of windowed exponentials L2 (Ω) if there exists 0 < A ≤ B < ∞ such that A f 2 ≤
N
j =1 E (gj , Λj )
forms a frame for
2 N f (x)gj (x)e−2π iλ,x dx ≤ B f 2 . j =1 λ∈Λj
It is called a Riesz basis if the collection forms a frame with the property that every f ∈ L2 (Ω) is expanded uniquely as αj,λ gj (x)e2π iλ,x . It is called minimal if no sin 2 gle windowed exponential of N j =1 E (gj , Λj ) lies in the L span of the other windowed N exponentials in j =1 E (gj , Λj ). The collection of windowed exponentials is called complete if the equations f (x)gj (x)e−2π iλ,x dx = 0, ∀λ ∈ Λj , j = 1, . . . , N, Ω
then imply that f = 0 a.e. on Ω. We say that {g1 , . . . , gN } is admissible if there exists some Λj such that N j =1 E (gj , Λj ) forms a frame of windowed exponentials for L2 (Ω). The question in determining the completeness and frame/basis properties of windowed exponentials is closely related to wavelet theory and Gabor analysis since problems about the frame of translate in the multi-resolution analysis and regular Gabor system can be reduced to those of windowed exponentials via respectivel
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