Shift-invariant sequences and non-homogeneous dual wavelet frames on local fields

Abstract Recent research has increasingly explored the construction of wavelets and frames in function spaces extending beyond standard domains like $\mathbb{R}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> </mml:math> and $\mathbb{C}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math> . This paper addresses these developments within the context of local fields. We provide comprehensive characterizations of three shift-invariant structures: Bessel sequences, frame sequences, and Parseval frame sequences. Furthermore, we establish equivalence among non-homogeneous dual wavelet frames across different levels and demonstrate that every non-homogeneous dual wavelet frame can derive its corresponding homogeneous counterpart.