Approximation using Simple Functions

> ks <- 1:5
> cols <- rainbow(length(ks))
> plot.new()
> j <- 1
> for (j in seq_along(ks)) {
+     xlim <- c(0, 5)
+     ylim <- c(0, 20)
+     k <- ks[j]
+     x <- seq(0, 5, 0.001)
+     z <- data.frame(x = x, y = 0)
+     n.intervals <- k * 2^k
+     intervals <- seq(0, k, 1/(2^k))
+     ys <- 1:n.intervals
+     vals <- (ys - 1)/2^k
+     i <- 1
+     for (i in 1:n.intervals) {
+         test <- c(intervals[i], intervals[(i + 1)])
+         test <- sqrt(test)
+         condition <- z$x >= test[1] & z$x < test[2]
+         z[condition, 2] <- vals[i]
+     }
+     condition <- z$x >= sqrt(tail(intervals, 1))
+     z[condition, 2] <- k
+     par(new = T)
+     plot(z$x, z$y, type = "l", xlim = c(0, 2), ylim = c(0, 2),
+         xlab = "", ylab = "", lwd = 2, col = cols[j])
+ }
> par(new = T)
> x <- seq(0, 5, 0.1)
> plot(x, x^2, type = "l", xlim = c(0, 2), ylim = c(0, 2), col = "black",
+     main = expression(paste(y == x^2, "approximation")), lty = "dashed",
+     lwd = 3, xlab = "", ylab = "")
> label <- c(expression(paste(phi, 1)), expression(paste(phi, 2)),
+     expression(paste(phi, 3)), expression(paste(phi, 4)), expression(paste(phi,
+         5)), expression(y == x^2))
> legend("topleft", legend = label, fill = c((cols), "black"),
+     cex = 1)