(p,q)-frames in shift-invariant subspaces of mixed Lebesgue spaces L^p,q(R×R^d)
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In this paper, we mainly discuss the (p,q)-frame in shift-invariant subspace V_p,q(Φ)={∑_i=1^r∑_j_1∈Z∑_j_2∈Z^dd_i(j_1,j_2)ϕ_i(·-j_1,·-j_2):(d_i(j_1,j_2))_(j_1,j_2)∈Z×Z^d∈ℓ^p,q(Z×Z^d)} of mixed Lebesgue space L^p,q(R×R^d). Some equivalent conditions for {ϕ_i(·-j_1,·-j_2):(j_1,j_2)∈Z×Z^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(·-j_1,·-j_2):(j_1,j_2)∈Z×Z^d,1≤ i≤ r}, although the general result is not correct.
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