12 1 Orthogonal Functions Introduction a function is

12. 1 Orthogonal Functions Introduction a function is considered to be a generalization of a vector. We will see the concepts of inner product, norm, orthogonal (perpendicular), …. . VECTORS 1) Inner Product FUNCTIONS 1) Inner Product Defined in [a, b] 2) Norm (length) 2) Norm Example: Compute

12. 1 Orthogonal Functions VECTORS 3) Orthogonal (perpendicular) FUNCTIONS 3) Orthogonal Example: We say they are Orthogonal if Unlike vector analysis, where the word orthogonal is a synonym for perpendicular, in this present context the term orthogonal have no geometric significance. 4) Orthogonal set We say it is orthogonal set if Example: Show that they are orthogonal

12. 1 Orthogonal Functions VECTORS FUNCTIONS 4) Orthogonal set We say it is orthogonal set if 5) Orthonormal set We say it is orthonormal set if Example: Find an orthonormal set

12. 1 Orthogonal Functions VECTORS FUNCTIONS 6) Linear Combination orthonormal set Example: Write the vector u as a linear combination orthonormal set Example: Write the function f as a linear combination Remark:

12. 1 Orthogonal Functions We are primarily interested in infinite sets of orthogonal functions. 4) Orthogonal set We say it is orthogonal set if Example: orthogonal set

12. 1 Orthogonal Functions 4) Orthogonal set Example: We say it is orthogonal set if orthogonal set Example: 4) Complete orthogonal set Write as linear combination The only continuous function orthogonal to each member of the set is the zero function. Remark: Is this possible for any function f ?

12. 1 Orthogonal Functions 4) Orthogonal set Example: We say it is orthogonal set if orthogonal set Example: orthogonal set

12. 1 Orthogonal Functions 4) Orthogonal with respect w is said to be orthogonal with respect to a weight function w(x) on an interval [a, b] if Example: verify by direct integration that the functions are orthogonal with respect to the indicated weight function on the given interval.

12. 1 Orthogonal Functions The Gram-Schmidt process for constructing an orthogonal set from a linearly independent set of real-valued functions linearly independent set orthogonal set Example: Construct orthogonal set