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On group decompositions of bounded cosine sequences

Wojciech Chojnacki (2007)

Studia Mathematica

A two-sided sequence ( c ₙ ) n ∈ ℤ with values in a complex unital Banach algebra is a cosine sequence if it satisfies c n + m + c n - m = 2 c ₙ c ₘ for any n,m ∈ ℤ with c₀ equal to the unity of the algebra. A cosine sequence ( c ₙ ) n ∈ ℤ is bounded if s u p n ∈ ℤ | | c ₙ | | < ∞ . A (bounded) group decomposition for a cosine sequence c = ( c ₙ ) n ∈ ℤ is a representation of c as c ₙ = ( b ⁿ + b - n ) / 2 for every n ∈ ℤ, where b is an invertible element of the algebra (satisfying s u p n ∈ ℤ | | b ⁿ | | < ∞ , respectively). It is known that every bounded cosine sequence possesses a universally defined group decomposition, here referred...

On infinitely smooth almost-wavelets with compact support.

M. Berkolaiko, I. Novikov (1993)

Collectanea Mathematica

There are known wavelets with exponential decay on infinity [2,3,4] and wavelets with compact support [5]. But these functions have finite smoothness. It is known that there do not exist infinitely differentiable compactly supported wavelets.

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...

On the adaptive wavelet estimation of a multidimensional regression function under α -mixing dependence: Beyond the standard assumptions on the noise

Christophe Chesneau (2013)

Commentationes Mathematicae Universitatis Carolinae

We investigate the estimation of a multidimensional regression function f from n observations of an α -mixing process ( Y , X ) , where Y = f ( X ) + ξ , X represents the design and ξ the noise. We concentrate on wavelet methods. In most papers considering this problem, either the proposed wavelet estimator is not adaptive (i.e., it depends on the knowledge of the smoothness of f in its construction) or it is supposed that ξ is bounded or/and has a known distribution. In this paper, we go far beyond this classical framework....

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