Hilbert空间L~2(R)上的小波保持子

研究了小波的运算性质与保持小波的算子的性质。首先,研究了函数空间L2(R)中的全体小波构成的集合W(L2(R))的代数性质,证明了GW(L2(R)):=W(L2(R))∪{0}(0与小波之集)在数乘、加法及卷积运算下是封闭的,进而形成一个交换赋范代数;其次,讨论了Hilbert空间L2(R)上将小波映射为小波的有界线性算子(称为小波保持子)的性质,证明了这些算子的全体WP(L2(R))构成一个含幺乘法半群;最后,研究了L2(R)上将小波映射为小波或0函数的有界线性算子(称为广义小波保持子),证明了这些算子的全体GWP(L2(R))构成了Banach算子代数B(L2(R))的一个含幺赋范子代数。同时,还给出了L2(R)上有界算线性算子成为小波保持子的一个充分条件。

Wavelet preservers on the Hilbert space L~2(R)

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.