Dual multiwavelet frames with symmetry from two-direction re nable functions

‎Motivated by [B‎. ‎Han and Q‎. ‎Mo‎, ‎{\em Adv‎. ‎Comp‎. ‎Math.} {\bf 18}‎ ‎(2003) 211-245] and [B‎. ‎Han and Z‎. ‎Shen‎, ‎{\em Constr‎. ‎Approx.}‎ ‎{\bf 29} (2009) 369-406]‎, ‎we propose dual two-direction frames in‎ ‎dual Sobolev spaces $(H^{s}(\mathbb{R}),H^{-s}(\mathbb{R})),$‎ ‎with $s > 0$‎. ‎Based on the dual two-direction frames from a pair‎ ‎of two-direction refinable functions‎, ‎dual multiwavelet frames‎ ‎with symmetry $\{\Psi^{\ell}(x):=(\psi^{\ell}_{1}(x),$‎ ‎$\psi^{\ell}_{2}(x))^{T}\}^{d}_{\ell=1}$ and‎ ‎$\{\widetilde{\Psi}^{\ell}(x):$‎ ‎$=(\widetilde{\psi}^{\ell}_{1}(x),$‎ ‎$\widetilde{\psi}^{\ell}_{2}(x))^{T}\}^{d}_{\ell=1}$ can be‎ ‎constructed very easily‎. ‎The vanishing moment of the constructed‎ ‎multiwavelet frames is discussed‎. ‎An example is given to‎ ‎illustrate our results‎.