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On group decompositions of bounded cosine sequences

Wojciech Chojnacki (2007)

Studia Mathematica

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