Sampling and Reconstruction in Shift Invariant Spaces of $B$-Spline Functions

Abstract

A Kadec-type theorem is proved for functions belonging to the shift invariant space $V(Q_{r})$, where $Q_{r}$ denotes the $B$-spline function of even order $r$. It is also shown that if a non-zero function $f\in V(Q_{m})$ has infinitely many simple or double zeros on the real axis which are separated, then there exists at least one pair of consecutive zeros whose distance apart is greater than or equal to a certain number which depends on Krein–Favard constants. Further, iterative reconstruction algorithms are provided for functions in $V(Q_{m})$. Finally, a sampling density theorem for irregular Gabor-type frames $\{M_{y_{j}}T_{x_{i,j}}g^{(l)}:l=0, 1, \dots , {k-1}, i, j \in \mathbb{Z} \}$ is proved for $g\in V(Q_{m})\cap L^{1}( \mathbb{R})$ under certain sufficient conditions on $g$.