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Riesz Bases in Subspaces of L2 (R+ )

Riesz Bases in Subspaces of L2 (R+ ),10.1007/s003650010019,Constructive Approximation,Tim N. T. Goodman,Charles A. Micchelli,Zuowei Shen

Riesz Bases in Subspaces of L2 (R+ )   (Citations: 2)
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.    In a recent investigation [8] concerning the asymptotic behavior of Gram—Schmidt orthonormalization procedure applied to the nonnegative integer shifts of a given function, the problem of determining whether or not such functions form a Riesz system in arose. In this paper, we provide a sufficient condition to determine whether the nonnegative translates form a Riesz system on . This result is applied to identify a large class of functions for which very general translates enjoy the Riesz basis property in .
Journal: Constructive Approximation - CONSTR APPROX , vol. 17, no. 1, pp. 39-46, 2001
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    • ...Similarly, whereas there are methods for generating Riesz bases in subspaces of L2(R) and L2(R+) [13, 18], we are not aware of general methods for generating Riesz bases in subspaces of L2(Rd) for d ≥ 2, except for grids of sampling points with, apart...

    Cornelis V. M. van der Meeet al. A Method for Generating Infinite Positive Self-adjoint Test Matrices a...

    • ...Using the corresponding sampling results on S = R and a recent result of Goodman et al. [16] on deriving Riesz bases of functions of the half-line...
    • ...However, it will turn out that the sequence of unilateral translates {φ( ·− tj )}∞ j =0 is a Riesz basis of a suitable RKHS on R + which can be described explicitly in terms of Hardy spaces, if the sequence of bilateral translates {φ( ·− tj )}∞ j =−∞ is a Riesz basis of H P 2 . A major tool in deriving these results will be the main result of [16] which allows one to derive certain sampling expansions on RKHS of functions on R ...
    • ...is bounded and strictly positive selfadjoint on � 2(Z+) or, equivalently, that the functions {φ( ·− tj )}j ∈Z+ on the positive half-line form a Riesz basis of a suitable closed subspace of the RKHS H P 2 defined in section 3.1. Its proof is based on [16], theorem 2.4...
    • ...Proof. In view of theorem 2.4 of Goodman et al. [16] and theorem 3.2 above, it suffices to prove the following: (i) The functions {φ(·−tj )}j ∈Z+ form a Riesz basis of some closed subspace of L2(R)...

    Cornelis V. M. Van Der Meeet al. Sampling Expansions and Interpolation in Unitarily Translation Invaria...

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