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master > master: notebook - wk10 - cleanup

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RD 2022-12-15 07:35:09 +01:00
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notebooks/week-10.ipynb
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@ -101,26 +101,28 @@
"Für $i\\in\\{0,1,\\ldots,n-1\\}$ ist die $i$-te von $\\mathcal{F}_{n}\\,u_{n,k}$\n",
"\n",
"$$\n",
" \\begin{array}{rcl}\n",
" (\\mathcal{F}_{n}\\,u_{n,k})_{i}\n",
" = \\sum_{j=0}^{n-1}\n",
" (\\mathcal{F}_{n})_{ij}\\,(u_{n,k})_{j}\n",
" = \\sum_{j=0}^{n-1}\n",
" &= &\\displaystyle\\sum_{j=0}^{n-1}\n",
" (\\mathcal{F}_{n})_{ij}\\,(u_{n,k})_{j}\\\\\n",
" &= &\\displaystyle\\sum_{j=0}^{n-1}\n",
" \\tfrac{1}{\\sqrt{n}}\n",
" \\exp(-\\imath\\tfrac{2\\pi\\,ij}{n})\n",
" \\cdot\n",
" \\tfrac{1}{\\sqrt{n}}\n",
" \\exp(\\imath\\tfrac{2\\pi\\,kj}{n})\n",
" = \\frac{1}{n}\\sum_{j=0}^{n-1}\n",
" \\exp(\\imath\\tfrac{2\\pi\\,kj}{n})\\\\\n",
" &= &\\frac{1}{n}\\displaystyle\\sum_{j=0}^{n-1}\n",
" \\underbrace{\n",
" \\exp(\\imath\\tfrac{2\\pi\\,(k-i)j}{n})\n",
" }_{= r^{j}}\n",
" }_{= r^{j}}\\\\\n",
" \\end{array}\n",
"$$\n",
"\n",
"wobei $r \\colonequals \\exp(\\imath\\tfrac{2\\pi\\,(k-i)}{n})$.\n",
"wobei $r := \\exp(\\imath\\tfrac{2\\pi\\,(k-i)}{n})$.\n",
"Wir haben, dass\n",
" $r = 1$\n",
" $\\Leftrightarrow$\n",
" $\\tfrac{k-i}{n} \\in \\Z$\n",
" $\\tfrac{k-i}{n} \\in \\mathbb{Z}$\n",
" $\\Leftrightarrow$\n",
" $i = k$,\n",
"da $0\\leq i,k \\leq n-1$.\n",
@ -130,23 +132,34 @@
"Aus der o.s. Berechnung erhält man also\n",
"\n",
"$$\n",
" \\begin{array}{rcl}\n",
" (\\mathcal{F}_{n}\\,u_{n,k})_{i}\n",
" = \\tfrac{1}{n}\\displaystyle\\sum_{j=0}^{n-1}r^{j}\n",
" = \\tfrac{1}{n}\n",
" \\begin{cases}\n",
" &= &\\tfrac{1}{n}\\displaystyle\\sum_{j=0}^{n-1}r^{j}\\\\\n",
" &= &\\tfrac{1}{n}\n",
" \\left\\{\n",
" \\begin{array}{lcl}\n",
" \\displaystyle\\sum_{j=0}^{n-1}1 &: &i = k\\\\\n",
" \\frac{1 - r^{n}}{1 - r} &: &\\text{sonst}\\\\\n",
" \\end{cases}\n",
" = \\tfrac{1}{n}\n",
" \\begin{cases}\n",
" \\dfrac{1 - r^{n}}{1 - r} &: &\\text{sonst}\\\\\n",
" \\end{array}\n",
" \\right.\\\\\n",
" &= &\\tfrac{1}{n}\n",
" \\left\\{\n",
" \\begin{array}{lcl}\n",
" n &: &i = k\\\\\n",
" \\frac{1 - 1}{1 - r} &: &\\text{sonst}\\\\\n",
" \\end{cases}\n",
" = \\delta_{ik}.\n",
" \\dfrac{1 - 1}{1 - r} &: &\\text{sonst}\\\\\n",
" \\end{array}\n",
" \\right.\\\\\n",
" &= &\\left\\{\n",
" \\begin{array}{lcl}\n",
" 1 &: &i = k\\\\\n",
" 0 &: &\\text{sonst}\\\\\n",
" \\end{array}\n",
" \\right.\\\\\n",
" &= &\\delta_{ik}.\\\\\n",
" \\end{array}\n",
"$$\n",
"\n",
"Darum $\n",
" \\mathcal{F}_{n}\\,u_{n,k}\n",
"Darum $\\mathcal{F}_{n}\\,u_{n,k}\n",
" = \\left(\\begin{matrix}\n",
" 0\\\\\n",
" 0\\\\\n",
@ -163,8 +176,7 @@
" \\\\\n",
" \\\\\n",
" \\end{matrix}\n",
" = \\mathbf{e}_{k}\n",
"$.\n"
" = \\mathbf{e}_{k}$.\n"
]
},
{