Acta Applicandae Mathematicae

, Volume 145, Issue 1, pp 175–192 | Cite as

Sampling and Reconstruction in Shift Invariant Spaces of \(B\)-Spline Functions

  • A. Antony Selvan
  • R. Radha


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 \(g\).


B-Splines Bernstein’s inequality Frames Hermite interpolation Krein–Favard constants Nonuniform sampling Riesz basis Wirtinger–Sobolev inequality 

Mathematics Subject Classification

65D07 94A20 



We wish to thank the referees for meticulously reading the manuscript and giving us several valuable suggestions to improve our earlier version of the manuscript.


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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology MadrasChennaiIndia

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