We show that there exists a bounded subset of $\mathbb{R}$ such no system exponentials can be Riesz basis for the corresponding Hilbert space. An additional result gives lower bound constant any putative two dimensional disk.

More generally, in the first claim we can replace the Lp(Q) norm with any norm || · || on F with the Riesz property, which is that |f | ≤ |g| (pointwise) implies ||f || ≤ ||g||. For the second claim, we can use any norm that has the Riesz property and for which the norm of the constant function f ≡ 1 is 1, which is clearly satisfied by Lp(Q) when Q is a probability measure. With these more gene...

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.

This is the first in a series of two papers in which we construct a convolution product for associated homogeneous distributions (AHDs) with support in R. In this paper, we show that if f and g are AHDs with degrees of homogeneity a−1 and b−1, the convolution fa∗gb exists as an AHD, provided the resulting degree of homogeneity a + b − 1 is not a natural number. Under this restriction, it is fou...

and Applied Analysis 3 Wj and W̃j of Vj and Ṽj , respectively, satisfying Vj 1 Vj ̇Wj , Wj ⊥ Ṽj and Ṽj 1 Ṽj ̇W̃j , W̃j ⊥ Vj . The corresponding biorthogonal wavelets are given by ψj ∑ n∈Z −1 ã j 1−nφj 1 2 · −n , ψ̃j ∑ n∈Z −1 a j 1−nφ̃j 1 2 · −n . 1.4 A generalization of the biorthogonal wavelets of Cohen-Daubechies-Feuveau 1 was introduced that was based on exponential B-splines 12 . By generalizing...

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...