The function spaces with wigner transform

In this thesis work, some properties of Wigner and τ-Wigner transforms are investigated firstly. Let ω_i (i= 1, 2, 3, 4) be weight functions. In the findings section, the vector space 〖CW 〗_(ω_1,〖 ω〗_2,〖 ω〗_(3,) ω_4)^(p, q, r, s, τ ) (R)= {(f,g)∈L_(ω_1)^p (R)×L_(ω_2)^q (R) │ (W_τ (f,∙),〖 W〗_τ (∙ ,g))∈L_(ω_3)^r (R^2)×L_(ω_4)^s (R^2)} is defined for 1≤p, q, r, s<∞ and τ∈(0,1). This space is endowed with a norm ‖(f,g)‖_(〖CW 〗_(ω_1,〖 ω〗_2,〖 ω〗_(3,) ω_4)^(p, q, r, s, τ )=) ‖(f,g)‖_(L_(ω_1)^p×L_(ω_2)^q )+‖((W_τ (f,.),〖 W〗_τ (∙ ,g)))‖_(L_(ω_3)^r×L_(ω_4)^s ) and some of its basic properties (Banach space, continuity of translation operator etc.) are investigated. Also the spaces 〖 CW 〗_(ω_1,〖 ω〗_(3 ))^(p, r , τ ) (R) and 〖 CW 〗_(ω_2,〖 ω〗_(4 ))^(q, s , τ ) (R) to be the space of the functions f∈L_(ω_1)^p (R) such that τ-Wigner transform of in L_(ω_3)^r (R^2) and the space of the functions g∈L_(ω_2)^q (R) such that τ-Wigner transform of in L_(ω_4)^s (R^2 ), respectively are endowed with norms and some basic properties of these spaces are studied.under some conditions. The spaces 〖CW 〗_(ω_1,〖 ω〗_2,〖 ω〗_(3,) ω_4)^(p, q, r, s, τ ) (R), 〖 CW 〗_(ω_1,〖 ω〗_(3 ))^(p, r , τ ) (R) and 〖 CW 〗_(ω_2,〖 ω〗_(4 ))^(q, s , τ ) (R) are shown to be essential the Banach module, and its approximate identities are investigated. Also inclusion properties and compact embeddings are discussed between these spaces. Later the relative completions of the spaces , 〖 CW 〗_(ω_1,〖 ω〗_(3 ))^(p, r , τ ) (R) ,〖 CW 〗_(ω_2,〖 ω〗_(4 ))^(q, s , τ ) (R) are defined and their multipliers spaces are found. In the last part of thesis work, dual spaces of the spaces 〖CW 〗_(ω_1,〖 ω〗_2,〖 ω〗_(3,) ω_4)^(p, q, r, s, τ ) (R), 〖 CW 〗_(ω_1,〖 ω〗_(3 ))^(p, r , τ ) (R) and 〖 CW 〗_(ω_2,〖 ω〗_(4 ))^(q, s , τ ) (R) are obtained. Keywords: τ-Wigner transform, essential the Banach module, approximate identitiy, compact embedding, multipliers space.