Wavelet bases on a triangle

In the present paper we find new constructions of orthonormal multiresolution analyses on the triangle Δ. In the first one, we describe a direct method to define an orthonormal multiresolution analysis which is adapted for the study of the Sobolev spaces $H_0^s(\mathrm{\Delta })$ ( $s\in ‐{N}$ ). In the second one, we add boundary conditions for constructing an orthonormal multiresolution analysis which is adapted for the study of the Sobolev spaces H ^s (Δ) ( $s\in ‐{N}$ ). The associated wavelets preserve the original regularity and are easy to implement.