Quantitative bounds for unconditional pairs of frames
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We formulate a quantitative finite-dimensional conjecture about frame multipliers and prove that it is equivalent to Conjecture 1 in [SB2]. We then present solutions to the conjecture for certain classes of frame multipliers. In particular, we prove that there is a universal constant κ>0 so that for all C,β>0 and N∈ℕ the following is true. Let (x_j)_j=1^N and (f_j)_j=1^N be sequences in a finite dimensional Hilbert space which satisfy x_j=f_j for all 1≤ j≤ N and ∑_j=1^N ε_j⟨ x,f_j⟩ x_j≤ Cx, for all x∈ℓ_2^M and |ε_j|=1. If the frame operator for (f_j)_j=1^N has eigenvalues λ_1≥...≥λ_M and λ_1≤β M^-1∑_j=1^Mλ_j then (f_j)_j=1^N has Bessel bound κβ^2 C. The same holds for (x_j)_j=1^N.
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