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Problem 1: Constructing the Gram Matrix for Fourier Basis Functions
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Given Equations
Fourier Series with basis functions:

Scalar (inner) product:

Wronskian:

Find: Gram Matrix and Properties
A. Construct the Gram Matrix for the given Fourier basis functions using the scalar product. B. Determine the property of this Gram Matrix. C. Show that $$

\left \{ \cos \theta\ \, \sin \theta\ \ \right \} $$ are linearly independent using the Wronskian.

Solution
A. From Wolfram Mathworld the Gram matrix is defined as a set $$ \mathbf{V} $$ of $$ \mathbf{m} $$ vectors (points in $$ \mathbb R ^n $$), the Gram Matrix $$ \mathbf{G} $$ is the matrix of all possible inner products of $$ \mathbf{V} $$, i.e. where $$ \mathbf{A}^T $$ denotes the transpose. The Gram matrix determines the vectors $$ \mathbf{v}_i $$ up to isometry. For this problem the value of $$ \mathbf{v}_i = g(x)= \cos n \omega\ \theta\ \ $$ and $$ \mathbf{v}_j = h(x)= \sin n \omega\ \theta\ \ $$ Then,

Problem 2: Pose Question to be solved
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Problem 3: Pose Question to be solved
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Problem 4: Pose Question to be solved
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Problem 6: Pose Question to be solved
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Problem 7: Pose Question to be solved
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