Average sampling and reconstruction in shift-invariant spaces and variable bandwidth spaces

ABSTRACT

In this paper, we prove that any f belongs to a shift-invariant space $V\left(\varphi \right)$ can be reconstructed uniquely and stably from its local averages on some discrete sets, where the generator ϕ is a continuously differentiable function satisfying certain decay conditions. As a special case, we obtain the average sampling expansion for shift-invariant spaces generated by B-splines and Meyer scaling function, and also the results can be extended to variable bandwidth spaces. Moreover, an explicit pair of dual frame from local averages is given by using quasi-interpolation as well as piecewise linear approximation. In addition, it has been shown that any f belongs to $V\left(\varphi \right)$ can be reconstructed uniquely and stably from its local averages by using iterative reconstruction algorithm. As a consequence, we obtain two concrete pairs of dual frames from local averages and iterative reconstruction algorithms for functions belonging to shift-invariant spaces generated by B-splines and Meyer scaling function, and also for functions belonging to variable bandwidth spaces.

COMMUNICATED BY:

• Zuhair Nashed

KEYWORDS:

• Shift-invariant spaces
• variable bandwidth spaces
• B-spline function
• Meyer scaling function
• uniform and nonuniform sampling
• Bernstein inequality
• frames
• Riesz basis and Wirtinger inequality

MATHEMATICS SUBJECT CLASSIFICATION 2010::

• 94A20
• 42C15

Acknowledgements

We thank the referees for meticulously reading the manuscript and giving us several valuable suggestions and comments which helped us to improve the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.