Given a set of functions ={?1,…,??}⊂?2(ℝ), we construct a principal shift-invariant space V nearest to in the sense that V minimizes the expression ∑?=1?‖??−????‖2, among all the principal shift-invariant spaces with an orthonormal generator which are also translation invariant, or among all the principal shift-invariant spaces with an orthonormal generator which are also 1?ℤ -invariant for some fixed ?∈ℕ .
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