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