(p,q)-frames in shift-invariant subspaces of mixed Lebesgue spaces L^p,q(R× R^d)

2020/01/20 by Yingchun Jiang, Jiao Li, Jiang, Yingchun +1
Mathematics · #Mathematical Analysis and Transform Methods #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems

paper · pdf · doi:10.48550/arxiv.2001.08519

Abstract

In this paper, we mainly discuss the (p,q)-frame in shift-invariant subspace V_p,q(Φ)=\∑_i=1^r∑_j₁∈ Z∑_j₂∈ Z^dd_i(j₁,j₂)ϕ_i(⋅-j₁,⋅-j₂):(d_i(j₁,j₂))_(j₁,j₂)∈ Z\timesZ^d∈ ℓ^p,q(Z\timesZ^d)\ of mixed Lebesgue space L^p,q(R× R^d). Some equivalent conditions for \ϕ_i(⋅-j₁,⋅-j₂):(j₁,j₂)\inZ\timesZ^d,1≤ i≤ r\ to constitute a (p,q)-frame of V_p,q(Φ) are given. Moreover, the result shows that V_p,q(Φ) is closed under these equivalent conditions of (p,q)-frame for the family \ϕ_i(⋅-j₁,⋅-j₂):(j₁,j₂)\inZ\timesZ^d,1≤ i≤ r\, although the general result is not correct.

Citations

• The wavelet transform, time-frequency localization and signal analysis

Related

• Construction of frames for shift-invariant spaces
• Frames for weighted shift-invariant spaces
• Reconstruction of functions in principal shift-invariant subspaces of mixed Lebesgue spaces
• Weyl-Heisenberg frames for subspaces of L²(R)
• Frames by orbits of two operators that commute