J4 ›› 2012, Vol. 47 ›› Issue (4): 57-61.

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Wavelet preservers on the Hilbert space L2(R)

YIN Jun-cheng1,2, CAO Huai-xin1   

  1. 1. College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, Shaanxi, China;
    2. College of Science, China Jiliang University, Hangzhou 310018, Zhejiang, China
  • Received:2011-07-24 Online:2012-04-20 Published:2012-06-28

Abstract:

The operational properties of mother wavelets in the Hilbert space L2(R) and operators that preserve mother wavelets are discussed. Denoted by W(L2(R)) the set of all mother wavelets in L2(R), it is proved that GW(L2(R)):=W(L2(R))∪{0}  becomes an abelian normed algebra with the usual addition, scalar multiplication, convolution and the norm ‖·‖1. A bounded linear operator A on L2(R) is said to be a wavelet preserver if it maps W(L2(R)) into W(L2(R)). It is proved that the set WP(L2(R)) of all wavelet preservers is a unital multiplicative semigroup. Furthermore, an operator in B(L2(R)) is said to be a generalized wavelet preserver (GWP) if it maps GW(L2(R)) into GW(L2(R)). It is shown that the set GWP(L2(R)) of all GWPs is a unital normed subalgebra of the Banach algebra B(L2(R)). Finally, a sufficient condition for an operator to be a wavelet preserver is obtained.

Key words:  wavelet; wavelet preserver; semigroup; normed algebra

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