Frames of multi-windowed exponentials on subsets ofRd

Abstract

Given discrete subsets Λ j ⊂ R d , j = 1, . . . , q, consider the set of windowed exponentials q j=1 {g j (x)e 2π i λ,x : λ ∈ Λ j } on L 2 (Ω). We show that a necessary and sufficient condition for the windows g j to form a frame of windowed exponentials for L 2 (Ω) with some Λ j is that 0 < m max j∈ J |g j | M almost everywhere on Ω for some subset J of {1, . . . , q}. If Ω is unbounded, we show that there is no frame of windowed exponentials if the Lebesgue measure of Ω is infinite. If Ω is unbounded but of finite measure, we give a simple sufficient condition for the existence of Fourier frames on L 2 (Ω). At the same time, we also construct examples of unbounded sets with finite measure that have no tight exponential frame.

Fractional Paley-Wiener and Bernstein spaces

“…Therefore, the sampling problem for P W s a , 0 ă s ă 1 2 , is equivalent to study windowed frames for L 2 a . The following result, due to C.-K. Lai [Lai11] (see also [GL14]) implies that we cannot have real sampling sequences for P W s a . Hence, we cannot obtain an analogue of (18) with point evaluations of the function instead of point evaluations of its fractional Laplacian.…”

Fractional Paley–Wiener and Bernstein spaces

“…◻ Therefore, the sampling problem for PW s a , 0 < s < 1 2 , is equivalent to study windowed frames for L 2 a . The following result, due to [16] (see also [14]) implies that we cannot have real sampling sequences for PW s a . Hence, we cannot obtain an analogue of (18) with point evaluations of the function instead of point evaluations of its fractional Laplacian.…”

“…Theorem [14,16] The family g( )e i n n ∈Λ is a frame for L 2 a for some sequence of points Λ = { n } n∈ℤ ⊆ ℝ if and only if there exist positive constants m, M such that m ≤ g( ) ≤ M .…”

Undersampled Windowed Exponentials and Their Applications

“…More generally, if 0 < m ≤ g ≤ M < ∞ on Ω, then E(g, Λ) forms a frame for L 2 (Ω). In [20] (See also [29]), the class of all admissible windows {g 1 , ..., g N } were completely classified. The necessary and sufficient condition for {g 1 , ..., g N } to be admissible for L 2 (Ω) is that there exists c > 0 with the property that max {j: g j ∞<∞} |g j | ≥ c > 0. a.e.…”

“…If {g 1 , ..., g N } is admissible, it is known that (see [20]) one can produce a frame of multi-windowed exponentials by taking Λ j in an oversampling manner in the sense that the (Beurling) density of Λ j are strictly greater than the measure of Ω.…”