prb4.tex (5668B)
1 \include{preamble.tex} 2 3 \begin{document} 4 \maketitle 5 \tableofcontents 6 7 \section{Sheet 4} 8 9 \subsection{Fourier Series} 10 The Fourier series of a $p$ periodic function $f$, integrable on 11 $[-\frac{p}{2}, \frac{p}{2}]$ is 12 \begin{align} 13 f(x) = \frac{a_0}{2} + \sum_{n=1}^\infty \left(a_n \cos(\frac{2\pi n x}{p}) 14 b_n sin(\frac{2\pi n x}{p})\right). 15 \end{align} 16 The coefficients $a_n$ and $b_n$ are called the Fourier coefficients of $f$ 17 and are given by 18 \begin{align} 19 a_n &= \frac{2}{p} \int_{-\frac{p}{2}}^{\frac{p}{2}} f(x) \sin(\frac{2\pi 20 n x}{p}) dx, \;\;\;\;\; n\geq 0 \\ 21 b_n &= \frac{2}{p} \int_{-\frac{p}{2}}^{\frac{p}{2}} f(x) \cos(\frac{2\pi 22 n x}{p}) dx, \;\;\;\;\; n\geq 1 23 \end{align} 24 Let us compute the Fourier series of $f(t) = t$ for $t \in [-\frac{1}{2}, 25 \frac{1}{2}]$. The Fourier coefficients are 26 \begin{align} 27 a_n &= 2\int_{-\frac{1}{2}}^{\frac{1}{2}} t \cos(2\pi n t)\ dt = 0 28 \;\;\;\;\; \text{(odd: g(-t) = -g(t))},\\ 29 \nonumber\\ 30 b_n &= 2\int_{-\frac{1}{2}}^{\frac{1}{2}} t \sin(2\pi n t)\ dt = \\ 31 &= 2 \left(-\frac{1}{2\pi n} \cos(2\pi n 32 t)\bigg|_{-\frac{1}{2}}^{\frac{1}{2}} 33 +\int_{\frac{1}{2}}^{\frac{1}{2}} \frac{1}{2 \pi n}\cos(2\pi n t)\ dt 34 \right) =\\ 35 &= -\frac{1}{\pi n}\left( -\cos(\pi n) + \frac{1}{\pi n }\sin(\pi 36 n)\right) = 37 \frac{\sin(\pi n) - \pi n \cos(\pi n)}{(\pi n)^2}. 38 \end{align} 39 Thereby the Fourier series of $f(t) = t$ is 40 \begin{align} 41 f(t) = \sum_{n=1}^\infty \left(\frac{\sin(\pi n) - \pi n \cos(\pi n)}{(\pi 42 n)^2}\right) \sin(2\pi n t) = t 43 \end{align} 44 \subsection{Truncation Error} 45 The truncation error of the trigonometric polynomial $(Sf_N)$ of degree $N$ is 46 \begin{align} 47 \sum_{|k| > N} |\hat{f}(k)|^2 = \lVert f - S_N\rVert_2^2 = 48 \int_{-\frac{1}{2}}^{\frac{1}{2}} |E_N(t)|^2 dt. 49 \end{align} 50 Computations for $N = 3$ and $N = 9$ were done in python with a integration error of 51 around $10^{-15}$, resulting in the overall truncation errors of 52 \begin{align} 53 \sum_{|k| > 3} |\hat{f}(k)|^2 = 0.0053,\\ 54 \sum_{|k| > 9} |\hat{f}(k)|^2 = 0.0143. 55 \end{align} 56 To achieve $\lVert E_N\rVert^2_2 < 0.1 \lVert f \rVert^2_2$, the number of 57 coefficients needed are about $61$. This was done using a while loop and 58 evaluating $\lVert E_N\rVert^2_2$ for $N$ until the above condition is met. 59 60 \subsection{Orthonormal Bases} 61 Here we will go through the most important properties of orthonormal bases. 62 So let $\{b_n\}_{n\in \mathbb{N}}$ be an ONB of a vector space $\mathcal{H}$, 63 then for every $x\in \mathcal{H}$ we may write 64 \begin{align} 65 x = \sum_{b_n} \langle b_n, x\rangle b_n, 66 \end{align} 67 and 68 \begin{align} 69 \lVert x \rVert^2 = \sum_{b_n} |\langle b_n, x\rangle|^2. 70 \end{align} 71 For any $x, y \in \mathcal{H}$ we can write the scalar product as 72 \begin{align} 73 \langle x, y\rangle = \sum_{b_n} \langle b_n, x\rangle \langle b_n, 74 y\rangle, 75 \end{align} 76 Furthermore there exists a linear projection $\Phi\ : \mathcal{H} 77 \rightarrow l^2(\{b_n\}_n)$ such that 78 \begin{align} 79 \langle \Phi(x), \Phi(y)\rangle = \langle x, y \rangle\;\;\; \forall x, y 80 \in \mathcal{H}. 81 \end{align} 82 83 An example of an orthonormal basis, which spans $L^2([-\frac{p}{2}, 84 \frac{p}{2}])$ is $\mathcal{T}_p = \{e_n := \frac{e^{2\pi i 85 \frac{n}{p}x}}{\sqrt{p}}\}_{n\in\mathbb{Z}}$. The $e_n$'s are orthonormal in 86 $L^2$ which can be easily seen by using the scalar product of $L^2$, so for 87 $n, m \in \mathbb{Z}$ 88 \begin{align} 89 \langle e_n, e_m\rangle_{L^2([-\frac{p}{2}, \frac{p}{2})} &= 90 \frac{1}{p}\int_{[-\frac{p}{2}, \frac{p}{2}]}e_n \cdot e_m^* \ dx=\\ 91 &=\frac{1}{p}\int_{[-\frac{p}{2}, \frac{p}{2}]} e^{2\pi i \frac{(n-m)}{p} x} \ dx=\\ 92 &=\frac{\sin(\pi (n-m))}{\pi(n-m)} = 93 \begin{cases} 94 0 \;\;\;\; n\neq m\\ 95 1 \;\;\;\; n=m 96 \end{cases} 97 \end{align} 98 \subsection{Dirichlet Kernel} 99 The function 100 \begin{align} 101 D_t(x) := \sum_{\lVert k \rVert_\infty \leq t} e_k(x), \;\;\;\;\; x\in 102 \mathbb{R}^d 103 \end{align} 104 is called the Dirichlet Kernel. For $0 < t \in \mathbb{N}$ we have 105 \begin{align} 106 (S_tf)(x) = \int_{I^d} f(y) D_t(x-y) dy, 107 \end{align} 108 where $S_t$ represents the orthogonal projection onto the trigonometric 109 polynomials $\Pi_t$ of degree $t$, by 110 \begin{align} 111 &S_t:\ L^1(\mathbb{T}^d) \rightarrow \Pi_t \\ 112 &f \mapsto \sum_{\lVert k \rVert \leq t} \langle f, 113 e_k\rangle_{L^2(\mathbb{T}^d)} e_k \;\;\;\;\; k \in \mathbb{Z}^d 114 \end{align} 115 And furthermore the Dirichlet Kernel satisfies 116 \begin{align} 117 D_t(x) = \prod_{i=1}^d \frac{e_{t+1}(x_i) - e_{-t}(x_i)}{e_1(x_i) - 1} 118 \end{align} 119 To show the convolution property, we start off by applying the orthogonal 120 projection into the trigonometric polynomials $S_t$ onto a function $f \in 121 L(\mathbb{T}^d)$ 122 \begin{align} 123 (S_tf) &= \sum_{\lvert k\rVert_\infty \leq t} \int_{I^d} f(y) e^{-2\pi i 124 \langle k, y\rangle}\ dy\ e^{2\pi i\langle k, x\rangle} =\\ 125 &= \int_{I^d}f(y) \sum_{\lvert k\rVert_\infty \leq t} e^{2\pi i \langle 126 k, (x- y)\rangle}\ dy =\\ 127 &= (f * D_t) (x) = \int_{I^d} f(y) D_t(x - y)\ dy. 128 \end{align} 129 To show the reformulation of the Dirichlet kernel, we need to simply 130 calculate it directly 131 \begin{align} 132 \sum_{\lVert k \rVert_\infty \leq t} e^{2\pi i \langle k , x\rangle} &= 133 \prod_{j=1}^d \sum_{k_j = -t}^t e^{2\pi i k_j x_j} =\\ 134 &= \prod_{j=1}^d e^{-2\pi i t x_j} \sum_{k_j = 0}^{2t} e^{2\pi i k_j 135 x_j}=;\;\;\;\; \text{(trigonometric series)}\\ 136 &= \prod_{j=1}^d e^{-2\pi i t x_j} \frac{e^{2\pi i (2t + 1)x_j} - 137 1}{e^{2\pi i x_j} - 1} =\\ 138 &= \prod_{j = 1} \frac{e_{t+1}(x_j) - e_{-t}(x_j)}{e_1(x_j) - 1}. 139 \end{align} 140 %\printbibliography 141 \end{document}