Biorthogonal multiresolution analysis on a triangle and applications

• Wavelets have unfolded their full computation efficiently in numerical and applied analysis.
• The properties of wavelet bases provide a rigorous analysis for dynamical systems.
• We construct in this work biorthogonal wavelet bases on a triangle.
• These bases are adapted to the study of the Sobolev spaces.
• The bases allow many concrete numerical examples as numerical simulation for elliptic problems or image processing.

We present in this paper new constructions of biorthogonal multiresolution analysis on the triangle ΔΔ. We use direct method based on the tensor product to construct dual scaling spaces on ΔΔ. Next, we construct the associated wavelet spaces and we prove that the associated wavelets have compact support and preserve the original regularity. Finally, we describe some regular results which are very useful to establish the norm equivalences. As applications, we prove that the wavelet bases constructed in this paper are adapted for the study of the Sobolev spaces H0s(Δ) and Hs(Δ)Hs(Δ) (s∈Ns∈N) and are easy to implement.