Approximate Oblique Dual Frames for Krein Spaces

ABSTRACT Due to its significance in mathematics and engineering, the operator theory of Krein spaces and Krein space approaches have been being attracted attention of many mathematicians. Recently, the concept of frame has been introduced in Krein spaces. This paper addresses the approximate oblique dual frames for Krein spaces. We present some parametric expressions and constructions of approximate oblique dual frames of a given frame sequence and prove that there exists a bijection between the approximate oblique dual frames of a given frame sequence and its portrait under a bounded invertible operator on and between the approximate oblique dual frames of original and perturbed frame sequences. Also, we estimate the deviation of the canonical approximate oblique dual frames of original and perturbed frame sequences and give an explicit characterization for the best approximation of the approximate oblique dual frame of original frame sequence using that of perturbed frame sequence. Finally, applying our results to shift‐invariant systems, we derive some new results in shift‐invariant spaces.