$p$-Adic multiresolution analysis and wavelet frames

Understand

We study $p$-adic multiresolution analyses (MRAs).

• A complete characterisation of test functions generating MRAs (scaling functions) is given.
• We prove that only 1-periodic test functions may be taken as orthogonal scaling functions.
• We also suggest a method for the construction of wavelet functions and prove that any wavelet function generates a $p$-adic wavelet frame.