Eugenio Hernández

Smoothing Minimally Supported Frequency Wavelets: Part II

The main purpose of this paper is to give a procedure to “mollify” the low-pass filters of a large number ofMinimally Supported Frequency (MSF) wavelets so that the smoother functions obtained in this way are also low-pass filters for an MRA. Hence, we are able to approximate (in the L2-norm) MSF wavelets by wavelets with any desired degree of smoothness on the Fourier transform side. Although the MSF wavelets we consider are bandlimited, this may not be true for their smooth approximations. This phenomena is related to the invariant cycles under the transformation x ↦2x (mod2π). We also give a characterization of all low-pass filters for MSF wavelets. Throughout the paper new and interesting examples of wavelets are described.

Estimates for the Kakeya Maximal Operator on Radial Functions in Rn

For a real number N > 1, the Kakeya maximal operator K N is defined on locally integrable functions fof R n as

Greedy bases in variable Lebesgue spaces

{K_N}f\left( x \right) = \mathop {\sup }\limits_{x \in R \in {B_N}} \frac{1}{{\left| R \right|}}\int {_R} \left| {f\left( y \right)} \right|dy

Further results on the connectivity of Parseval frame wavelets

where B N denotes the class of all rectangles in R n of eccentricity N, that is, congruent with any dilate of the rectangle [0,1]n-1x [0, N], and where x007C;Ax007C; represents the Lebesgue measure of the set A.

The Zak Transform(s)

It is well known that the compactly supported wavelets cannot belong to the class C∞(R) ∩ L(R). This is also true for wavelets with exponential decay. We show that one can construct wavelets in the class C∞(R)∩L2(R) that are “almost” of exponential decay and, moreover, they are band-limited. We do this by showing that we can adapt the construction of the Lemarie-Meyer wavelets [LM] that is found in [BSW] so that we obtain band-limited, C∞-wavelets on R that have subexponential decay, that is, for every 0 0 such that |ψ(x)| ≤ Ce e−|x| 1−e , x ∈ R. Moreover, all of its derivatives have also subexponential decay. The proof is constructive and uses the Gevrey classes of functions. ∗Research supported by grant 2 P301 052 07 from KNB, Poland. †Research supported by grant PB94-149 from DGICYT (Ministerio de Educacion y Ciencia, Spain) and by a grant from Southwestern Bell Telephone Company. Math Subject Classification. Primary 42C15

Embeddings and Lebesgue-Type Inequalities for the Greedy Algorithm in Banach Spaces

We compute the right and left democracy functions of admissible wavelet bases in variable Lebesgue spaces defined on

Group Riesz and frame sequences: the Bracket and the Gramian

\mathbb R^n