Biorthogonal

Biorthogonal describes a relation between two families of vectors or functions in a space equipped with a bilinear form, typically an inner product. A pair of families {φ_i} and {ψ_j} is biorthogonal if the inner product satisfies ⟨φ_i, ψ_j⟩ = δ_ij for all indices i and j. The two families are orthogonal to each other across the pairing, but elements within the same family need not be mutually orthogonal.

In finite-dimensional spaces this situation often arises with dual bases. If {φ_i} is a basis and {ψ_j} is its dual with respect to the inner product, then ⟨φ_i, ψ_j⟩ = δ_ij. The two families need not be identical; they are simply biorthogonal to each other. In infinite-dimensional spaces biorthogonal systems can be complete without each family being orthonormal, and they can support stable representations under suitable conditions.

Biorthogonal structures also occur with polynomials and wavelets. Biorthogonal polynomials consist of two sequences {p_n} and {q_n} that satisfy ∫ p_n(x) q_m(x) w(x) dx = δ_nm for a chosen weight w. This generalizes orthogonality to non-self-adjoint settings and has applications in approximation theory and numerical analysis. In signal processing, biorthogonal wavelet bases provide separate analysis and synthesis systems, allowing for symmetric wavelets with practical filter designs while maintaining biorthogonality between the analysis and synthesis components.

Overall, biorthogonality enables reconstruction and expansion in non-orthogonal settings by pairing two dual families that interact through the inner product to yield delta-function behavior.