Orthonormal Sequences in L 2(R d ) and Time Frequency Localization

Abstract

We prove that there does not exist an orthonormal basis {b n } for L 2 (R) such that the sequences {μ(b n )}, {μ( b n )}, and { (b n ) ( b n )} are bounded. A higher dimensional version of this result that involves generalized dispersions is also obtained. The main tool is a time-frequency localization inequality for orthonormal sequences in L 2 (R d ). On the other hand, for d > 1 we construct a basis {b n } for L 2 (R d ) such that the sequences {μ(b n )}, {μ( b n )}, and { (b n ) ( b n )} are bounded.

Phase space localization of orthonormal sequences in Lα2(R+)

“…The equality in (1.7) is attained for the sequence of Hermite functions and the higher dimensional version of this result that involving generalized dispersions ∥ |x| s f n ∥ 2 L 2 (R d ) and ∥ |ξ | s F( f n )∥ 2 L 2 (R d ) , s > 0, for orthonormal sequences in L 2 (R d ) was obtained by Malinnikova [16].…”

“…Our proof here is inspired from similar results established in [16]. To do so we will use the time-limiting and the frequency-limiting operators on L 2 α (R + ) defined by…”

“…[14,16]). The first of these results is an extension of Shapiro's result for orthonormal sequences in L 2 α (R + ).…”

“…Nevertheless based on an idea of Malinnikova [16] we adapt the proof of [16,Theorem 2] to establish the localization inequality (3.43) which implies the following strong uncertainty principle for orthonormal bases:…”

Localization of Orthonormal Sequences in the Spherical Mean Setting

“…993], and the same proof may be reformulated in our case, however for sake of simplicity we shall just deduce it. Let λ be a positive real number then according to De Branges [4, pp 449] there is a nonzero function g ∈ L 2 (dμ n ) which vanishes almost everywhere on [0, λ] along with its Hankel transform H n−1 2 (g), and by the same way according to Malinnikova [14] there exists h ∈ L 2 (R n ) such that h vanishes almost everywhere on C λ ∈ {x ∈ R n | |x| λ} along with its Fourier transform h, then by considering f (r, x 1 , . .…”