We give an example of an indefinite weight Sturm-Liouville problem whose eigenfunctions form a Riesz basis under Dirichlet boundary conditions but not under anti-periodic boundary conditions.

In this paper we show that every g-frame for a Hilbert space H can be represented as a linear combination of two g-orthonormal bases if and only if it is a g-Riesz basis. We also show that every g-frame can be written as a sum of two tight g-frames with g-frame bounds one or a sum of a g-orthonormal basis and a g-Riesz basis for H . We further give necessary and sufficient conditions on g-Besse...

In this paper we investigate compactly supported wavelet bases for Sobolev spaces. Starting with a pair of compactly supported refinable functions φ and φ̃ in L2(R) satisfying a very mild condition, we provide a general principle for constructing a wavelet ψ such that the wavelets ψjk := 2j/2ψ(2j · − k) (j, k ∈ Z) form a Riesz basis for L2(R). If, in addition, φ lies in the Sobolev space H(R), t...

A new wavelet-based geometric mesh compression algorithm was developed recently in the area of computer graphics by Khodakovsky, Schröder, and Sweldens in their interesting paper [23]. The new wavelets used in [23] were designed from the Loop scheme by using ideas and methods of [26, 27], where orthogonal wavelets with exponential decay and pre-wavelets with compact support were constructed. Th...

In this paper we rst review the construction of stable Riesz bases for nite element spaces with respect to Sobolev norms. Then, we construct optimal order multilevel preconditioners for the matrices in the normal form of the equations arising in the nite element discretization of non{symmetric second order elliptic equations. The optimality of the AMLI methods is proven under H 2 {regularity as...

In this paper we present topology aspects in non-localization results. A well-known such no-go result is the BalianLow theorem that states the generator of a Weyl-Heisenberg Riesz basis cannot be well time-frequency localized. More general, the statement applies to multi WH Riesz bases, or super frames as well. These results turn out to be connected to non-triviality of a complex vector bundle....

Let {X1, . . . ,X2n,T} be a real basis for the Heisenberg Lie algebra hn. In this paper, we calculate the explicit kernels of the Riesz transforms Rj = XjL 1 2 , j = 1, 2, · · · , 2n on the Heisenberg group Hn. Here L = − ∑2n j=1 X 2 j is the sub-Laplacian operator. We show that ∫∞ −∞Rjdt = Rj for j = 1, 2, · · · , 2n where Rj ’s are the regular Riesz transforms on R . We also construct the “mi...

Shift-invariance \ berization techniques" are applied for the study of the synthesis and analysis operators of a ne (wavelet) systems. In this approach, one has rst to circumvent the fact that a ne systems are not shift-invariant. The results obtained include characterizations of the Bessel property, the Riesz basis property and the frame property of such sets in terms of the behaviour of simpl...

We consider a regular indefinite Sturm-Liouville problem with two self-adjoint boundary conditions, one being affinely dependent on the eigen-parameter. We give sufficient conditions under which a basis of each root subspace for this Sturm-Liouville problem can be selected so that the union of all these bases constitutes a Riesz basis of a corresponding weighted Hilbert space.

We extend to several dimensions the result of K. Seip and Y.I. Lyubarskii that proves the existence of Riesz basis of exponentials for a finite union of intervals with equals lengths.