p03.ipynb

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  {
   "cell_type": "markdown",
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   "source": [
    "# p03: Band-limited interpolation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will interpolate following functions using band limited interpolation on an infinite grid.\n",
    "\n",
    "**Delta function**\n",
    "$$\n",
    "v(x) = \\begin{cases}\n",
    "1, & x =0 \\\\\n",
    "0, & \\textrm{otherwise}\n",
    "\\end{cases}\n",
    "$$\n",
    "\n",
    "**Square wave**\n",
    "$$\n",
    "v(x) = \\begin{cases}\n",
    "1, & |x| \\le 3 \\\\\n",
    "0, & \\textrm{otherwise}\n",
    "\\end{cases}\n",
    "$$\n",
    "\n",
    "**Hat function**\n",
    "$$\n",
    "v(x) = \\max(0, 1-|x|/3)\n",
    "$$\n",
    "\n",
    "Since all functions are zero away from origin, we can restrict to some finite interval, say $[-10,10]$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format='svg'\n",
    "from numpy import arange,maximum,abs,zeros,sin,pi\n",
    "from matplotlib.pyplot import subplot,figure,plot,grid,axis"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
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      "text/plain": [
       "<Figure size 1000x1000 with 3 Axes>"
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   "source": [
    "h = 1.0;\n",
    "xmax = 10.0;\n",
    "x = arange(-xmax,xmax+h,h)\n",
    "xx = arange(-xmax-h/20, xmax+h/20, h/10) # for plotting\n",
    "figure(figsize=(10,10))\n",
    "for pl in range(3):\n",
    "    subplot(4,1,pl+1)\n",
    "    if pl==0:\n",
    "        v = (x==0)                      # delta function\n",
    "    elif pl==1:\n",
    "        v = (abs(x) <= 3.0)             # square wave\n",
    "    else:\n",
    "        v = maximum(0.0,1.0-abs(x)/3.0) # hat function\n",
    "    plot(x,v,'o'); grid(True)\n",
    "    \n",
    "    # Evaluate interpolant on xx\n",
    "    p = zeros(len(xx))\n",
    "    for i in range(len(x)):\n",
    "        p = p + v[i]*sin(pi*(xx-x[i])/h)/(pi*(xx-x[i])/h)\n",
    "    plot(xx,p)\n",
    "    axis([-xmax,xmax,-0.5,1.5]);"
   ]
  }
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