The spectral function of shift-invariant spaces on general lattices

Shift Invariant Spaces and Shift Preserving Operators on Locally Compact Abelian Groups

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 funct...

Convolution and Homogeneous Spaces

Introduction to Shift-Invariant Spaces I: Linear Independence

Shift-invariant spaces play an increasingly important role in various areas of mathematical analysis and its applications. They appear either implicitly or explicitly in studies of wavelets, splines, radial basis function approximation, regular sampling, Gabor systems, uniform subdivision schemes, and perhaps in some other areas. One must keep in mind, however, that the shift-invariant system e...

On generalized topological molecular lattices

In this paper, we introduce the concept of the generalized topological molecular lattices as a generalization of Wang's topological molecular lattices,  topological spaces, fuzzy topological spaces, L-fuzzy topological spaces and soft topological spaces. Topological molecular lattices were defined by closed elements, but in this new structure we present the concept of the open elements and defi...

Approximation Orders of and Approximation Maps from Local Principal Shift-invariant Spaces Approximation Orders of and Approximation Maps from Local Principal Shift-invariant Spaces

Approximation orders of shift-invariant subspaces of L p (IR d), 2 p 1, generated by the shifts of one compactly supported function are considered. In that course, explicit approximation maps are constructed. The approach avoids quasi-interpolation and applies to stationary and non-stationary reenements. The general results are specialized to box spline spaces, to obtain new results on their ap...