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An iterative scheme for constructing compactly supported orthonormal (o.n.) multi-wavelets with vanishing moments of arbitrarily high order is established. Precisely, let φ=[φ1,. . .,φr]⊤ be an r-dimensional o.n. scaling function vector with polynomial preservation of order (p.p.o.) m, and ψ=[ψ1,. . .,ψr]⊤ an o.n. multi-wavelet corresponding to φ, with two-scale symbols P and Q, respectively. Then a new (r+1)-dimensional o.n. scaling function vector φ♯:=[φ⊤,φr+1]⊤ and some corresponding o.n. multi-wavelet ψ♯ are constructed in such a way that φ♯ has p.p.o.=n>m and their two-scale symbols P♯ and Q♯ are lower and upper triangular block matrices, respectively, without increasing the size of the supports. For instance, for r=1, if we consider the mth order Daubechies o.n. scaling function φ [ABSTRACT FROM AUTHOR]