Biortogonale

Biortogonale refers to a pair of sequences or functions that have a specific orthogonality property. In mathematics, orthogonality generally means that the inner product of two distinct elements is zero. Biortogonale sequences, however, extend this concept to two different sequences. Let's consider two sequences, {u_i} and {v_j}, indexed by i and j. These sequences are biortogonale if their inner product, typically defined as an integral or a sum, is zero when the indices are different (i ≠ j) and non-zero when the indices are the same (i = j).

More formally, for discrete sequences {u_i}_{i=1}^n and {v_j}_{j=1}^n, they are biortogonale if <u_i, v_j> = c_i δ_{ij}, where <.,.> denotes an inner product and δ_{ij} is the Kronecker delta, which is 1 if i=j and 0 if i≠j. The constants c_i are generally non-zero. For continuous functions, the condition would involve integrals: ∫ u_i(x) v_j(x) dx = c_i δ_{ij}.

The concept of biortogonale sequences is important in various fields, including approximation theory, numerical analysis, and signal processing. They are particularly useful in the construction of basis functions and for tasks like signal decomposition and reconstruction. For instance, in the context of orthogonal polynomials, a sequence of polynomials and a corresponding sequence of dual polynomials can form a biortogonale set. This property allows for efficient expansion of functions in terms of these polynomial bases. The dual sequence plays a crucial role in determining the coefficients of the expansion.