A note on exponential Riesz bases

Abstract

AbstractWe prove that if $$I_\ell = [a_\ell ,b_\ell )$$ I ℓ = [ a ℓ , b ℓ ) , $$\ell =1,\ldots ,L$$ ℓ = 1 , … , L , are disjoint intervals in [0, 1) with the property that the numbers $$1, a_1, \ldots , a_L, b_1, \ldots , b_L$$ 1 , a 1 , … , a L , b 1 , … , b L are linearly independent over $${\mathbb {Q}}$$ Q , then there exist pairwise disjoint sets $$\Lambda _\ell \subset {\mathbb {Z}}$$ Λ ℓ ⊂ Z , $$\ell =1, \ldots , L$$ ℓ = 1 , … , L , such that for every $$J \subset \{ 1, \ldots , L \}$$ J ⊂ { 1 , … , L } , the system $$\{e^{2\pi i \lambda x} : \lambda \in \cup _{\ell \in J} \, \Lambda _\ell \}$$ { e 2 π i λ x : λ ∈ ∪ ℓ ∈ J Λ ℓ } is a Riesz basis for $$L^2 ( \cup _{\ell \in J} \, I_\ell )$$ L 2 ( ∪ ℓ ∈ J I ℓ ) . Also, we show that for any disjoint intervals $$I_\ell $$ I ℓ , $$\ell =1, \ldots , L$$ ℓ = 1 , … , L , contained in [1, N) with $$N \in {\mathbb {N}}$$ N ∈ N , the orthonormal basis $$\{e^{2\pi i n x} : n \in {\mathbb {Z}}\}$$ { e 2 π i n x : n ∈ Z } of $$L^2[0,1)$$ L 2 [ 0 , 1 ) can be complemented by a Riesz basis $$\{e^{2\pi i \lambda x}: \lambda \in \Lambda \}$$ { e 2 π i λ x : λ ∈ Λ } for $$L^2(\cup _{\ell =1}^L \, I_{\ell })$$ L 2 ( ∪ ℓ = 1 L I ℓ ) with some set $$\Lambda \subset (\frac{1}{N} {\mathbb {Z}}) \backslash {\mathbb {Z}}$$ Λ ⊂ ( 1 N Z ) \ Z , in the sense that their union $$\{e^{2\pi i \lambda x} : \lambda \in {\mathbb {Z}}\cup \Lambda \}$$ { e 2 π i λ x : λ ∈ Z ∪ Λ } is a Riesz basis for $$L^2 ( [0,1) \cup I_1 \cup \cdots \cup I_L )$$ L 2 ( [ 0 , 1 ) ∪ I 1 ∪ ⋯ ∪ I L ) .