Integral Transform - Importance of Orthogonality

Importance of Orthogonality

The individual basis functions have to be orthogonal. That is, the product of two dissimilar basis functions—integrated over their domain—must be zero. An integral transform, in actuality, just changes the representation of a function from one orthogonal basis to another. Each point in the representation of the transformed function in the image of the transform corresponds to the contribution of a given orthogonal basis function to the expansion. The process of expanding a function from its "standard" representation to a sum of a number of orthonormal basis functions, suitably scaled and shifted, is termed "spectral factorization." This is similar in concept to the description of a point in space in terms of three discrete components, namely, its x, y, and z coordinates. Each axis correlates only to itself and nothing to the other orthogonal axes. Note the terminological consistency: the determination of the amount by which an individual orthonormal basis function must be scaled in the spectral factorization of a function, F, is termed the "projection" of F onto that basis function.

The normal Cartesian graph per se of a function can be thought of as an orthonormal expansion. Indeed, each point just reflects the contribution of a given orthonormal basis function to the sum. In this way, the graph of a continuous real-valued function in the plane corresponds to an infinite set of basis functions; if the number of basis functions were finite, the curve would consist of a discrete set of points rather than a continuous contour.