In this paper, combining the non-decaying properties with the mixed-norm properties, the revelent sampling problems are studied under the target space of \(L_{\vec{p},\frac{1}{\omega }}(\mathbb{R}^{d})\). Firstly, we will give a stability theorem for the shift-invariant subspace \(V_{\vec{p},\frac{1}{\omega }}(\varphi )\). Secondly, an ideal sampling in \(W_{\vec{p},\frac{1}{\omega }}^{s}(\mathbb{R}^{d})\) is proved, and thirdly, a convergence theorem (or algorithm) is shown for \(V_{\vec{p},\frac{1}{\omega }}(\varphi )\). It should be pointed out that the auxiliary function \(\varphi \) enjoys the membership in a Wiener amalgam space.

本文结合非衰减性质和混合范数性质，研究了目标空间\(L_{\vec{p},\frac{1}{\omega }}(\ mathbb{R}^{d})\)。首先，我们给出平移不变子空间\(V_{\vec{p},\frac{1}{\omega }}(\varphi )\) 的稳定性定理。其次，证明了 \(W_{\vec{p},\frac{1}{\omega }}^{s}(\mathbb{R}^{d})\)中的理想采样，第三，显示了\(V_{\vec{p},\frac{1}{\omega }}(\varphi )\) 的收敛定理（或算法） 。需要指出的是，辅助函数\(\varphi \)享有维纳汞齐空间中的隶属关系。