12.1 Orthogonal Functions a function is considered to be a generalization of a vector. We will see the concepts of inner product, norm, orthogonal (perpendicular),

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Presentation transcript:

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

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

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

12.1 Orthogonal Functions VECTORS FUNCTIONS Example: Write the vector u as a linear combination 6) Linear Combination orthonormal set 6) Linear Combination 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 We say it is orthogonal set if Example: orthogonal set The only continuous function orthogonal to each member of the set is the zero function. Example: Write as linear combination 4) Complete orthogonal set Remark: Is this possible for any function f ?

12.1 Orthogonal Functions 4) Orthogonal set We say it is orthogonal set if Example: 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 verify by direct integration that the functions are orthogonal with respect to the indicated weight function on the given interval. Example: verify by direct integration that the functions are orthogonal with respect to the indicated weight function on the given interval. Example:

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