2. 代数多項式近似：ぱらぱらめくる『A Short Course on Approximation Theory』

• Stone–Weierstrass theorem(Weierstrass approximation theorem)
  • 閉区間上のどんな連続関数も多項式関数によって任意の精度で一様に近似できる

library(pracma)
my.function<-function(x){
	sin(x)+cos(x^2)
}

n<-1:20
precision<-rep(0,length(n))
for(i in 1:length(n)){
	out<-polyApprox(my.function, -pi/2, pi/2, n[i])
	precision[i]<-out$estim.prec
}
par(mfcol=c(1,2))
plot(n,precision)
f <- polyApprox(my.function, -pi/2, pi/2, 100)$f
x <- seq(-pi/2, pi/2, length.out = 100)
y <- my.function(x) - f(x)
plot(x, y, type = "l", col = "blue")
grid()
par(mfcol=c(1,1))