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Haar system on a product of zero-dimensional compact groups

Sergei Lukomskii (2011)

Open Mathematics

In this work, we study the problem of constructing Haar bases on a product of arbitrary compact zero-dimensional Abelian groups. A general scheme for the construction of Haar functions is given for arbitrary dimension. For dimension d=2, we describe all Haar functions.

Haar wavelets on the Lebesgue spaces of local fields of positive characteristic

Biswaranjan Behera (2014)

Colloquium Mathematicae

We construct the Haar wavelets on a local field K of positive characteristic and show that the Haar wavelet system forms an unconditional basis for L p ( K ) , 1 < p < ∞. We also prove that this system, normalized in L p ( K ) , is a democratic basis of L p ( K ) . This also proves that the Haar system is a greedy basis of L p ( K ) for 1 < p < ∞.

Halfway to a solution of X 2 - D Y 2 = - 3

R. A. Mollin, A. J. Van der Poorten, H. C. Williams (1994)

Journal de théorie des nombres de Bordeaux

It is well known that the continued fraction expansion of D readily displays the midpoint of the principal cycle of ideals, that is, the point halfway to a solution of x 2 - D y 2 = ± 1 . Here we notice that, analogously, the point halfway to a solution of x 2 - D y 2 = - 3 can be recognised. We explain what is going on.

Hankel determinants of the Thue-Morse sequence

Jean-Paul Allouche, Jacques Peyrière, Zhi-Xiong Wen, Zhi-Ying Wen (1998)

Annales de l'institut Fourier

Let ϵ = ( ϵ n ) n ≥ 0 be the Thue-Morse sequence, i.e., the sequence defined by the recurrence equations: ϵ 0 = 1 , ϵ 2 n = ϵ n , ϵ 2 n + 1 = 1 - ϵ n . We consider { | ℰ n p | } n ≥ 1 , p ≥ 0 , the double sequence of Hankel determinants (modulo 2) associated with the Thue-Morse sequence. Together with three other sequences, it obeys a set of sixteen recurrence equations. It is shown to be automatic. Applications are given, namely to combinatorial properties of the Thue-Morse sequence and to the existence of certain Padé approximants of the power series ∑ n ≥ 0 ( - 1 ) ϵ n x n .

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