Density and duality theorems for regular Gabor frames

Lemvig / magazine of practical research 270 (2016) 229–263 now have okay = N on account that okay is atomless. we will write ok = ∪i∈I Ki , the place Ki is of finite degree. Then, through a classical results of Sierpinski [37, Lemma fifty two. α], the degree μM takes a continuum of values [0, μM (Ki )] at the measurable subsets of Ki , which justifies the identify non-stop body. If F = {fk }k∈M is a easy body with bounds A and B and if CF has dense diversity, then CF |K is invertible on all of L2 (M, μM ), and we are saying that {fk }k∈M is a simple Riesz relations with bounds A and B. Equivalently, you may define simple Riesz households with bounds A and B as households of vectors F = {fk }k∈M for which DF defined on easy, integrable √ √ features is bounded lower than and above through A and B, respectively, i. e. , 2 A a 2 L2 (M ) ≤ a(k)fk dμM (k) ≤B a 2 L2 (M ) M for all easy features a on M with finite help. If M is countable and outfitted with the counting degree, a easy Riesz kinfolk is just a Riesz foundation for its closed linear span, also known as a Riesz series. Our inspiration of discrete frames may well look overly technical in comparison to the standard definition (i. e. , M countable and μM the counting measure). although, it permits us to categorise the next pathological “continuous” examples as discrete frames. instance 1. give some thought to the subsequent examples with a “continuous” index set M = R: (a) permit H = 2 (Z), allow {ek }k∈Z be its commonplace orthonormal foundation, and equip M = R with a only atomic degree, whose atoms are the periods [n, n + 1), n ∈ Z, every one with degree 1. Define {fk }k∈M ⊂ H by means of fk = e okay , ok ∈ R. (b) enable H = L2 ([0, 1]), and permit M = R. repair a ∈ R and define μM = n∈Z δn+a , the place δx denotes the Dirac degree at x ∈ R. Define {fk }k∈M ⊂ H via fk (x) = e2πikx . it isn't difficult to teach that either in case (a) and (b) the relatives {fk }k∈M is a Parseval body; it's even a Riesz kinfolk. because the degree in either instances is only atomic, the body {fk }k∈M is related to be discrete. There has lately been a few curiosity within the research of (continuous) Riesz households [2,36], also known as Riesz-type frames in [18]. instance 1 (b) is a concrete model of [2, Proposition three. 7]. the next outcome exhibits, despite the fact that, that this idea brings little new to the well-studied topic of (discrete) Riesz sequences. To be extra concise, Proposition three. three indicates that norm bounded, easy Riesz households inevitably are discrete. M. S. Jakobsen, J. Lemvig / magazine of useful research 270 (2016) 229–263 239 Proposition three. three. allow (M, Σ, μM ) be a degree area. consider that {fk }k∈M is a simple Riesz family members in H that's basically norm bounded, i. e. , C := supk∈M fk < ∞. Then {fk }k∈M is a discrete family members. in addition, it holds that the place Σ0 = {E ∈ Σ : μM (E) > zero} . inf μM (E) > zero, E∈Σ0 (3. four) evidence. allow a be an integrable uncomplicated functionality on M . From the computation a(k)fk dm(k) ≤ M |a(k)| fk dm(k) ≤ C M |a(k)| dm(k), M we see that the reduce Riesz certain implies √ A a L2 (M ) ≤C a L1 (M ) . For a = 1E with E ∈ Σ and nil < μM (E) < ∞, this in flip means that √ A μM (E) 0.5 ≤ CμM (E), and therefore μM (E) ≥ A/C 2 .