Let V M l , UM l+1,j , and BM l+1,j be defined as in Section 18.4. Show that                                  BM l+1 = BM l+1,0 ∪···∪ BM l+1,2l−1 is the basis of the orthogonal complement of V M l in...

Let V M l , UM l+1,j , and BM l+1,j be defined as in Section 18.4. Show that

                                 BM l+1 = BM l+1,0 ∪···∪ BM l+1,2l−1

is the basis of the orthogonal complement of V M l in V M l+1. Hint: Show that:

(1) the functions in BM l+1 are contained in V M l+1;

(2) the functions in BM l+1 are orthogonal to each function in V M l ;

(3) the functions in BM l+1 are orthonormal; and

(4) if f ∈ V M l+1 and f is orthogonal to each function in V M l , then f · IAl j can be represented as a linear combination of the functions from BM l+1,j (and hence f can be represented as a linear combination of the functions from BM l+1).