Construction of dual bases

Let Bn:={b0,b1,…,bn}(n=0,1,…,N;N∈N) be the sets of linearly independent functions. We give a simple method of construction, the dual functions Dn:={d0(n),d1(n),…,dn(n)}(0≤n≤N) satisfying the following conditions: spanDn=spanBn and =δij(0≤i,j≤n≤N), where δii=1δii=1, δij=0δij=0 for i≠ji≠j, and <⋅,⋅><⋅,⋅> is a given inner product. The proposed algorithm allows us to construct all the sets of the dual functions D0,D1,…,DND0,D1,…,DN in the time O(N3)O(N3), where NN is a natural number. Four illustrative examples presenting the possible applications of obtained results are given.