Orthogonal Functions and Transforms
The coefficients am may be obtained as
i.e., am is the projection of x(t) on to ? m(t).
The set {?m(t)} is said to be complete or closed if there exists no square-integrable function x(t) for which
If this is true, x(t) should be a member of the set.
When the set {?m(t)} is complete, it is said to be an orthogonal basis, and may be used for accurate representation of signals, e.g., the Fourier series
Note: x(t) and the ?m(t)’s must be square-integrable.