Vector-Valued Affine Bi-Frames on Local Fields

Abstract

By a local field, we mean a field K and an associated topology such that both groups, group under addition and group under multiplication, have the properties of locally compactness and they are Abelian groups. So the local fields are non-Archimedean fields with the additional properties of non-discreteness and disconnectedness. Essentially, we can categorize local fields into the fields of zero characteristic and positive characteristic. Under the zero characteristics, we can include the p-adic field $$\mathbb Q_p$$ , and the Cantor dyadic group and the Vilenkin p-groups can be placed under the positive characteristic local fields. A lot of work has been done on the wavelet construction on local field of positive characteristic in the last few decades. Although the local fields of positive characteristics and local fields of zero characteristics have similarity in their structures, they differ in the wavelet construction, multiresolution analysis, and many more. Recently, the concept of vector-valued subspace on local field of positive characteristic was defined by Shah and Bhat in Vector-valued nonuniform multiresolution analysis on local fields published in the International Journal of Wavelets, Multiresolution and Information Processing in 2015 by Shah and Bhat (New Zealand J Maths24:9–20, 2016). We continued the studies and define vector-valued affine bi-frames on local fields of positive characteristic. To extend the scope of study, the characterization of affine bi-frames is provided on local fields of positive characteristic.