(a) Explanation Given: The linearly independent set of real valued functions that are continuous on an interval [ a , b ] is { f 0 ( x ) , f 1 ( x ) , f 2 ( x ) , … } . The inner product ( f n , ϕ n ) = ∫ a b f n ( x ) ϕ n ( x ) d x define the function in the set B ′ = { ϕ 0 ( x ) , ϕ 1 ( x ) , ϕ 2 ( x ) ⋯ } . Calculation: According to Gram-Schmidt orthogonalization process, the set B ′ = { ϕ 0 ( x ) , ϕ 1 ( x ) , ϕ 2 ( x ) ⋯ } is a orthogonal set constructed from the orthogonal set { f 0 ( x ) , f 1 ( x ) , f 2 ( x ) , … } , if ϕ 0 ( x ) , ϕ 1 ( x ) … ϕ n ( x ) satisfy the condition as follows: ϕ 0 ( x ) = f 0 ( x ) ϕ 1 ( x ) = f 1 ( x ) − ( f 1 , ϕ 0 ) ‖ ϕ 0 ‖ 2 ϕ 0 ( x ) ϕ 2 ( x ) = f 2 ( x ) − ( f 2 , ϕ 0 ) (b) To determine To show: The functions ϕ 0 ( x ) , ϕ 1 ( x ) and ϕ 2 ( x ) are mutually orthogonal.

Differential Equations with Boundary-Value Problems (MindTap Course List)

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 11.1, Problem 26E

(a)

To determine

The orthogonal function ϕ3(x) in the set B′.

(b)

To determine

To show: The functions ϕ0(x),ϕ1(x) and ϕ2(x) are mutually orthogonal.

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Chapter 11.1, Problem 25E

Chapter 11.1, Problem 27E

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Differential Equations with Boundary-Value Problems (MindTap Course List)

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The orthogonal function ϕ 3 ( x ) in the set B ′ .

Solution Summary: The author describes the orthogonal function varphi, which is a linearly independent set of real valued functions that are continuous on an interval.