On a Characterization of Riesz Bases via Biorthogonal Sequences
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(2020) 26:67
LETTER TO THE EDITORS
On a Characterization of Riesz Bases via Biorthogonal Sequences Diana T. Stoeva1 Received: 21 March 2020 © The Author(s) 2020
Abstract It is well known that a sequence in a Hilbert space is a Riesz basis if and only if it is a complete Bessel sequence with biorthogonal sequence which is also a complete Bessel sequence. Here we prove that the completeness of one (any one) of the biorthogonal sequences can be removed from the characterization. Keywords Riesz basis · Biorthogonal sequence · Complete sequence · Bessel sequence · Gabor system Mathematics Subject Classification 42C15 · 46B15
1 Introduction and motivation A Riesz basis for a separable Hilbert space H is a sequence of the form (V ek )∞ k=1 with (ek )∞ k=1 being an orthonormal basis for H and V being a bounded bijective operator from H onto H. Riesz bases were introduced by Bari [3,4] and already in [4] many properties and equivalent characterizations were determined. Below we collect the standard equivalences of Riesz bases from [6,9,12], some of which appeared already in [4,7]: Theorem 1.1 For a sequence ( f k )∞ k=1 in a Hilbert space H, the following conditions are equivalent: (R1 ) ( f k )∞ k=1 forms a Riesz basis for H. (R2 ) ( f k )∞ k=1 is a complete Bessel sequence in H and it has a biorthogonal sequence (gk )∞ k=1 which is also a complete Bessel sequence in H.
Communicated by Hans G. Feichtinger.
B 1
Diana T. Stoeva [email protected] Acoustics Research Institute, Austrian Academy of Sciences, Wohllebengasse 12-14, 1040 Vienna, Austria 0123456789().: V,-vol
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Journal of Fourier Analysis and Applications
(2020) 26:67
(R3 ) ( f k )∞ is complete in H and it has a complete biorthogonal sequence (gk )∞ k=1 k=1 ∞ 2 2 so that ∞ k=1 | f , f k | < ∞ and k=1 | f , gk | < ∞ for every f ∈ H. (R4 ) ( f k )∞ k=1 is complete in H and there exist positive constants A and B so that A
|ck |2 ≤
ck f k 2 ≤ B
|ck |2
(1.1)
2 for every finite scalar sequence (ck ) (and hence for every (ck )∞ k=1 ∈ ). ∞ ∞ (R5 ) ( f k )k=1 is complete in H and its Gram matrix ( f k , f j ) j,k=1 determines a bounded bijective operator on 2 . basis for H. (R6 ) ( f k )∞ k=1 is a bounded unconditional ∞ is a basis for H such that (R7 ) ( f k )∞ k=1 ck f k converges in H if and only if ∞k=1 2 |c | < ∞. k k=1
The main purpose of this paper is to show that one may remove the condition for ∞ completeness of one (any one) of the sequences ( f k )∞ k=1 and (gk )k=1 in (R2 ), and thus in (R3 ) as well. In general, it is a more difficult task to check the lower Riesz basis condition in (1.1) compare to the upper one. From this point of view, (R2 ) is a useful characterization of Riesz bases, avoiding the verification of the lower condition. On the other hand, completeness might not be simple to check either. From this point of view, removing the verification of completeness of one of the sequences from (R2 ) is of significant importance when checking the Riesz basis property. It can also be used in furt
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