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Homework 11
- Section 6.3 Fourier Series: Scalar Product Spaces
- Read Section 6.3. Do Problem 6.37.
- A Geogebra activity on Fourier approximation
- Homework: 6.39, 6.40, 6.41 (use cosines), 6.43, 6.45, 6.46, 6.50, 6.51, 6.53
- Hand In: 6.41*, 6.43*, 6.50
- 6.41*. Complete Problem 6.41 on Page 351, but with the function \(g(x)\) defined to be \[g(x) = \left\{\matrix{3\vert x\vert&\mbox{for}\ -\frac{1}{3}\leq x\leq \frac{1}{3}\cr 1&\mbox{otherwise}}\right.\] for \(-1 \leq x\leq 1\). (This is NOT the wave shown in Figure 6.12a.) (Hints: (a) Make use of the even nature of the functions you are integrating to halve the amount of calculation (but don't forget to incorporate the values of the integrals you don't need to work out) and (b) There may be a need to consider various cases of \(n\) beyond \(n\) odd or \(n\) even; thus you may find yourself considering when \(n\) has the forms \(6m, 6m+3\) (two cases when \(n\) is a multiple of \(3\)), \(6m\pm 1, 6m\pm 2\) (four cases when \(n\) is not a multiple of \(3\)).)
- 6.43*. Complete Problem 6.43 on Page 352, but where the underlying space is \(\mathcal{F}((-1,2])\), the space of piecewise continuous functions on the interval \((-1,2]\). Thus the inner product is \[\langle f(x),g(x)\rangle = \int_{-1}^2 f(x)g(x)\ dx.\] Use the same initial set \(\mathcal{W}\).
- 6.50. Complete Problem 6.50, as stated in the text. Assume that the vectors \(V_i\) are all nonzero.
- Read Section 6.4. Do the True/False questions (pp. 363-364). Do Problems 6.37.
- Homework: 6.60(c), 6.61, 6.62, 6.64, 6.67, 6.73(a)(c), 6.75, 6.76
- Hand In: 6.61(b)*, 6.73(c)*
- 6.61(b)*. Complete Problem 6.61(b) on Page 365, but with the matrix \[A=\frac{1}{2}\left[\matrix{1&-1&\sqrt{2}&a\cr -1&-1&0&b\cr -1&-1&0&c\cr -1&1&\sqrt{2}&d\cr}\right].\]
- 6.73(c)*. Complete Problem 6.73(c) on Page 365, but with the hyperplane \(x-3y+2z+w=0\).