数学 - Python - JavaScript - 解析学 - 連続写像の空間 - ストーン・ワイエルシュトラスの定理(直接証明、漸化式、帰納法、極限、一様収束)

解析入門〈3〉

学習環境

• Surface 3 (4G LTE)、Surface 3 タイプ カバー、Surface ペン(端末)
• Windows 10 Pro (OS)
• Nebo(Windows アプリ)
• iPad Pro + Apple Pencil
• MyScript Nebo(iPad アプリ)
• 参考書籍
  • Pythonからはじめる数学入門
  • JavaScriptリファレンス 第6版
  • インタラクティブ・データビジュアライゼーション

解析入門〈3〉(松坂 和夫(著)、岩波書店)の第13章(連続写像の空間)、13.2(ストーン・ワイエルシュトラスの定理)、問題3.を取り組んでみる。

3. 
   
   $\begin{array}{}\left|x\right|-\left(P_n\left(x\right)+\frac{x^2-{\left(P_n\left(x\right)\right)}^2}{2}\right)\\ =\left|x\right|-P_n\left(x\right)+\frac{\left(\left|x\right|+P_n\left(x\right)\right)\left(\left|x\right|-P_n\left(x\right)\right)}{2}\\ =\left(\left|x\right|-P_n\left(x\right)\right)\left(1+\frac{\left|x\right|+P_n\left(x\right)}{2}\right)\end{array}$

   よって、

   $\left|x\right|-P_{n+1}\left(x\right)=\left(\left|x\right|-P_n\left(x\right)\right)\left(1+\frac{\left|x\right|+P_n\left(x\right)}{2}\right)$

   また、

   $\begin{array}{}{P}_{n+1}\left(x\right)-P_n\left(x\right)\\ =\frac{x^2-{\left(P_n\left(x\right)\right)}^2}{2}\\ \ge \; <!--\; greater-than\; or\; equal\; to\; -->\frac{x^2}{2}\\ \ge \; <!--\; greater-than\; or\; equal\; to\; -->0\end{array}$
   $\begin{array}{}\left|x\right|\le \; <!--\; less-than\; or\; equal\; to\; -->1\\ P_1\left(x\right)=0+\frac{x^2-0^2}{2}=\frac{x^2}{2}\le \; <!--\; less-than\; or\; equal\; to\; -->{\left|x\right|}^2\le \; <!--\; less-than\; or\; equal\; to\; -->\left|x\right|\\ 0\le \; <!--\; less-than\; or\; equal\; to\; -->P_0\left(x\right)\le \; <!--\; less-than\; or\; equal\; to\; -->P_1\left(x\right)\le \; <!--\; less-than\; or\; equal\; to\; -->\left|x\right|\end{array}$
   $\begin{array}{}\left|x\right|\le \; <!--\; less-than\; or\; equal\; to\; -->1\\ P_{n+1}\left(x\right)\\ =P_n\left(x\right)+\frac{x^2-{\left(P_n\left(x\right)\right)}^2}{2}\\ =\frac{x^2+2P_n\left(x\right)-{\left(P_n\left(x\right)\right)}^2}{2}\\ \le \; <!--\; less-than\; or\; equal\; to\; -->\frac{x^2+2P_n\left(x\right)+{\left(P_n\left(x\right)\right)}^2}{2}\\ =\frac{{\left(x+P_n\left(x\right)\right)}^2}{2}\\ \le \; <!--\; less-than\; or\; equal\; to\; -->\frac{{\left(\left|x\right|+\left|x\right|\right)}^2}{2}\\ ={\left|x\right|}^2\\ \le \; <!--\; less-than\; or\; equal\; to\; -->\left|x\right|\end{array}$

   よって、帰納法により、

   $P_n\left(x\right)\le \; <!--\; less-than\; or\; equal\; to\; -->\left|x\right|$

   ゆえに、

   $0\le \; <!--\; less-than\; or\; equal\; to\; -->P_n\left(x\right)\le \; <!--\; less-than\; or\; equal\; to\; -->P_{n+1}\left(x\right)\le \; <!--\; less-than\; or\; equal\; to\; -->\left|x\right|$

   また、

   $\begin{array}{}\left|x\right|-P_{n+1}\left(x\right)\\ =\left(\left|x\right|-P_n\left(x\right)\right)\left(1-\frac{\left|x\right|+P_n\left(x\right)}{2}\right)\\ \le \; <!--\; less-than\; or\; equal\; to\; -->\left|x\right|{\left(1-\frac{\left|x\right|}{2}\right)}^n\left(1-\frac{\left|x\right|}{2}\right)\\ \le \; <!--\; less-than\; or\; equal\; to\; -->\left|x\right|{\left(1-\frac{\left|x\right|}{2}\right)}^n\end{array}$

   よって、帰納法より

   $\left|x\right|-P_n\left(x\right)\le \; <!--\; less-than\; or\; equal\; to\; -->\left|x\right|{\left(1-\frac{\left|x\right|}{2}\right)}^n$

   ゆえに

   $0\le \; <!--\; less-than\; or\; equal\; to\; -->\left|x\right|-P_n\left(x\right)\le \; <!--\; less-than\; or\; equal\; to\; -->\frac{1}{2^n}<\frac{2}{n+1}$

   以上のことから

   $\underset{n\to \; <!--\; rightwards\; arrow\; -->\mathrm{\infty \; <!--\; infinity\; -->}}{\lim }{P}_n\left(x\right)=\left|x\right|$

   となる。

   （証明終）

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, plot

x = symbols('x')


def pn(n):
    if n == 0:
        return 0
    else:
        return pn(n - 1) + (x ** 2 - pn(n - 1) ** 2) / 2

pns = [pn(n0) for n0 in range(10)]
p = plot(abs(x), *pns, (x, -1, 1), show=False)
p[0].line_color = 'red'
p.save('sample3.svg')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample3.py
$

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="0.5">
<label for="dx">dx = </label>
<input id="dx" type="number" min="0" step="0.0001" value="0.001">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-1">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="1">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="0">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="2">
<br>

<label for="n0">n0 = </label>
<input id="n0" type="number" min="1" step="1" value="5">

<button id="draw0">draw</button>
<button id="clear0">clear</button>

<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/d3/4.2.6/d3.min.js" integrity="sha256-5idA201uSwHAROtCops7codXJ0vja+6wbBrZdQ6ETQc=" crossorigin="anonymous"></script>

<script src="sample3.js"></script>    

JavaScript

let div0 = document.querySelector('#graph0'),
    pre0 = document.querySelector('#output0'),
    width = 600,
    height = 600,
    padding = 50,
    btn0 = document.querySelector('#draw0'),
    btn1 = document.querySelector('#clear0'),
    input_r = document.querySelector('#r0'),
    input_dx = document.querySelector('#dx'),
    input_x1 = document.querySelector('#x1'),
    input_x2 = document.querySelector('#x2'),
    input_y1 = document.querySelector('#y1'),
    input_y2 = document.querySelector('#y2'),
    input_n0 = document.querySelector('#n0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_n0],
    p = (x) => pre0.textContent += x + '\n',
    range = (start, end, step=1) => {
        let res = [];
        for (let i = start; i < end; i += step) {
            res.push(i);
        }
        return res;
    };

let f = (n) => (x) => {
    if (n === 0) {
        return 0;
    }
    return f(n - 1)(x) + (x ** 2 - f(n - 1)(x) ** 2) / 2;
};

let draw = () => {
    pre0.textContent = '';


    let r = parseFloat(input_r.value),
        dx = parseFloat(input_dx.value),
        x1 = parseFloat(input_x1.value),
        x2 = parseFloat(input_x2.value),
        y1 = parseFloat(input_y1.value),
        y2 = parseFloat(input_y2.value),
        n0 = parseInt(input_n0.value, 10);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [],
        lines = [],
        fns = [[(x) => Math.abs(x), 'red']]
        .concat(range(0, n0).map((m) => [f(m), 'green']));

    fns
        .forEach((o) => {
            let [fn, color] = o;
            
            for (let x = x1; x <= x2; x += dx) {
                let y = fn(x);
                
                if (Math.abs(y) < Infinity) {
                    points.push([x, y, color]);
                }
            }
        });
    
    let xscale = d3.scaleLinear()
        .domain([x1, x2])
        .range([padding, width - padding]);

    let yscale = d3.scaleLinear()
        .domain([y1, y2])
        .range([height - padding, padding]);

    let xaxis = d3.axisBottom().scale(xscale);
    let yaxis = d3.axisLeft().scale(yscale);
    div0.innerHTML = '';
    let svg = d3.select('#graph0')
        .append('svg')
        .attr('width', width)
        .attr('height', height);

    svg.selectAll('circle')
        .data(points)
        .enter()
        .append('circle')
        .attr('cx', (d) => xscale(d[0]))
        .attr('cy', (d) => yscale(d[1]))
        .attr('r', r)
        .attr('fill', (d) => d[2] || 'green');

    svg.selectAll('line')
        .data([[x1, 0, x2, 0], [0, y1, 0, y2]].concat(lines))
        .enter()
        .append('line')
        .attr('x1', (d) => xscale(d[0]))
        .attr('y1', (d) => yscale(d[1]))
        .attr('x2', (d) => xscale(d[2]))
        .attr('y2', (d) => yscale(d[3]))
        .attr('stroke', (d) => d[4] || 'black');
    
    svg.append('g')
        .attr('transform', `translate(0, ${height - padding})`)
        .call(xaxis);

    svg.append('g')
        .attr('transform', `translate(${padding}, 0)`)
        .call(yaxis);
    p(fns.join('\n'));
};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();

r = dx =
x1 = x2 =
y1 = y2 =
n0 =