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Non-Lebesgue multiresolution analyses

Lawrence Baggett (2010)

Colloquium Mathematicae

Classical notions of wavelets and multiresolution analyses deal with the Hilbert space L²(ℝ) and the standard translation and dilation operators. Key in the study of these subjects is the low-pass filter, which is a periodic function h ∈ L²([0,1)) that satisfies the classical quadrature mirror filter equation |h(x)|²+|h(x+1/2)|² = 2. This equation is satisfied almost everywhere with respect to Lebesgue measure on the torus. Generalized multiresolution analyses and wavelets exist in abstract Hilbert...

Non-MSF Wavelets for the Hardy Space H²(ℝ)

Biswaranjan Behera (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

All wavelets constructed so far for the Hardy space H²(ℝ) are MSF wavelets. We construct a family of H²-wavelets which are not MSF. An equivalence relation on H²-wavelets is introduced and it is shown that the corresponding equivalence classes are non-empty. Finally, we construct a family of H²-wavelets with Fourier transform not vanishing in any neighbourhood of the origin.

On a gap series of Mark Kac

Katusi Fukuyama (1999)

Colloquium Mathematicae

Mark Kac gave an example of a function f on the unit interval such that f cannot be written as f(t)=g(2t)-g(t) with an integrable function g, but the limiting variance of n - 1 / 2 ∑ k = 0 n - 1 f ( 2 k t ) vanishes. It is proved that there is no measurable g such that f(t)=g(2t)-g(t). It is also proved that there is a non-measurable g which satisfies this equality.

On | A , δ | k -summability of orthogonal series

Xhevat Z. Krasniqi (2012)

Mathematica Bohemica

In the paper, we prove two theorems on | A , δ | k summability, 1 ≤ k ≤ 2 , of orthogonal series. Several known and new results are also deduced as corollaries of the main results.

On (C,1) summability for Vilenkin-like systems

G. Gát (2001)

Studia Mathematica

We give a common generalization of the Walsh system, Vilenkin system, the character system of the group of 2-adic (m-adic) integers, the product system of normalized coordinate functions for continuous irreducible unitary representations of the coordinate groups of noncommutative Vilenkin groups, the UDMD product systems (defined by F. Schipp) and some other systems. We prove that for integrable functions σₙf → f (n → ∞) a.e., where σₙf is the nth (C,1) mean of f. (For the character system of the...

On the a b c -problem in Weyl-Heisenberg frames

Xinggang He, Haixiong Li (2014)

Czechoslovak Mathematical Journal

Let a , b , c > 0 . We investigate the characterization problem which asks for a classification of all the triples ( a , b , c ) such that the Weyl-Heisenberg system { e 2 π i m b x ⁢ χ [ n a , n a + c ) : m , n ∈ ℤ } is a frame for L 2 ( ℝ ) . It turns out that the answer to the problem is quite complicated, see Gu and Han (2008) and Janssen (2003). Using a dilation technique, one can reduce the problem to the case where b = 1 and only let a and c vary. In this paper, we extend the Zak transform technique and use the Fourier analysis technique to study the problem for the case of...

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