3. 三角多項式近似：ぱらぱらめくる『A Short Course on Approximation Theory』

• Weierstrass's Second Theorem
  • 任意の連続な周期関数は三角関数の多項式によって任意の精度で近似できる

asc <- seq(0, 330, by = 30)
dec <- c(408, 89, -66, 10, 338, 807, 1238, 1511, 1583, 1462, 1183, 804)
dec<-sample(dec)
plot(2*pi*asc/360, dec, pch = "+", col = "red", xlim = c(0, 2*pi), ylim = c(-500, 2000),
     xlab = "Ascension [radians]", ylab = "Declination [minutes]",
     main = "Gauss' Pallas Data")
grid()
points(2*pi*asc/360, dec, pch = "o", col = "red")
ts <- seq(0, 2*pi, len = 100)
xs <- trigApprox(ts ,dec, 1)
lines(ts, xs, col = "black")
xs <- trigApprox(ts ,dec, 6)
lines(ts, xs, col = "blue")
legend(3, 0, c("Trig. Regression of degree 1", "Trig. Regression of degree 6",
                  "Astronomical position"), col = c("black", "blue", "red"),
                lty = c(1,1,0), pch = c("", "", "+"))