We discuss some perturbation results concerning certain pairs of sequences of vectors in a Hilbert space H and producing new sequences, which share, with the original ones, reconstruction formulas on a dense subspace of H or on the whole space. We also propose some preliminary results on the same issue, but in a distributional settings.

1.

A sequence Fξ of elements of H is complete if its linear span is dense in H.

2.

A complex sequence {cn} is finite is cn ≠ 0 for only a finite number of elements.

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In any case, (2.6) are always true in weak sense by (2.3), meaning that f,g=n=0φ̃n,fφn,g=n=0ψn,fψ̃n,g, f,gH.

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This means that Qf,g=n=0f,φnψn,g, f,gH.

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We are using F(x) and G(x) to indicate the tempered distributions since we have in mind only those regular distributions for which this is possible. This is possible since we are not claiming that all the elements of S(R) can be multiplied.

31.

Note that in this formula, we are using ‖·‖1, ‖·‖2, and ‖·‖ since they all appear. In the rest of this paper, we have mostly simply used ‖·‖ to indicate the norm in L2(R), since there is no possible misunderstanding.

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