On Function Spaces Characterized by the Wigner Transform

Abstract

Let ω i be weight functions on R, (i=1,2,3,4). In this work, we define CW p,q,r,s,τ ω 1 ,ω 2 ,ω 3 ,ω 4 (R) to be vector space of (f, g) ∈ L p ω 1 × L q ω 2 (R) such that the τ −Wigner transforms Wτ (f, .) and Wτ (., g) belong to L r ω 3 R 2 and L s ω 4 R 2 respectively for 1 ≤ p, q, r, s < ∞, τ ∈ (0, 1). We endow this space with a sum norm and prove that CW p,q,r,s,τ ω 1 ,ω 2 ,ω 3 ,ω 4 (R) is a Banach space. We also show that CW p,q,r,s,τ ω 1 ,ω 2 ,ω 3 ,ω 4 (R) becomes an essential Banach module over L 1 ω 1 × L 1 ω 2 (R). We then consider approximate identities.2 f t s for all x, ω ∈ R, 0 = s ∈ R, respectively. The parameters in wavelet theory are "time"x and "scale" s. Dilation operator D s preserves the shape of f , but it changes the scale, [7].Given any fixed 0 = g ∈ L 2 (R)(called the window function), the short-time Fourier transform (STFT) of a function f ∈ L 2 (R) with respect to g is defined by

Inclusion Theorems in the Function Spaces With Wigner Transform

“…is satisfied [9]. Also the binary translation mapping (f, g) −→ T z (f, g) is continuous from CW p,q,r,s,τ ω1,ω2,ω3,ω4 (R) into CW p,q,r,s,τ ω1,ω2,ω3,ω4 (R) for every fixed z ∈ R and the mapping z → T z (f, g) is continuous from R into CW p,q,r,s,τ ω1,ω2,ω3,ω4 (R) [9]. Let f , g, h, k be Borel measurable functions on R. The binary convolution is defined by…”

“…In the literature, many function spaces have been defined using various timefrequency operators [3], [8], [9], [10], [10], [11], [13]. One of them is the space CW p,q,r,s,τ ω1,ω2,ω3,ω4 (R) defined by Kulak and Ömerbeyoglu [9] using the τ −Wigner transform. Now let's give the definition of this space and talk about some of its properties.…”

“…| dy < ∞ must be required for the binary convolution to be defined [9]. Let's the weights ω 3 and ω 4 be constants.…”

“…for all (f, g) ∈ CW p,q,r,s,τ ω1,ω2,ω3,ω4 (R) [9]. As can be understood, some important properties and approximate units of the space CW p,q,r,s,τ ω1,ω2,ω3,ω4 (R) have been investigated in [9].…”

“…for all (f, g) ∈ CW p,q,r,s,τ ω1,ω2,ω3,ω4 (R) [9]. As can be understood, some important properties and approximate units of the space CW p,q,r,s,τ ω1,ω2,ω3,ω4 (R) have been investigated in [9]. In this study, inclusion relations and compact embedding theorems of this space will be discussed.…”