TY - JOUR
T1 - Shift Invariant Spaces and Shift Preserving Operators on Locally Compact Abelian Groups
TT -
JF - IJMSI
JO - IJMSI
VL - 6
IS - 2
UR - http://ijmsi.ir/article-1-234-en.html
Y1 - 2011
SP - 21
EP - 32
KW - locally compact abelian group
KW - shift invariant space
KW - frame
KW - range function
KW - shift preserving operator
KW - range operator.
N2 - We investigate shift invariant subspaces of $L^2(G)$, where $G$ is a locally compact abelian group. We show that every shift invariant space can be decomposed as an orthogonal sum of spaces each of which is generated by a single function whose shifts form a Parseval frame. For a second countable locally compact abelian group $G$ we prove a useful Hilbert space isomorphism, introduce range functions and give a characterization of shift invariant subspaces of $L^2(G)$ in terms of range functions. Finally, we investigate shift preserving operators on locally compact abelian groups. We show that there is a one-to-one correspondence between shift preserving operators and range operators on $L^2(G)$ where $G$ is a locally compact abelian group.
M3 10.7508/ijmsi.2011.02.003
ER -