The Fourier series for a function truncated to order
,
namely
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This result says that a Fourier series is the best possible approximation
to a function by a trigonometric polynomial of this type.
However, the conclusion depends quite heavily on how we measure the
quality of approximation.
Below are Fourier approximations to each of 3 functions on [0,1]:
the line
, the quadratic
and the square well
. For each plot the pictures get better as
improves. However the well shaped plot shows effects of Gibb's
phenomenon: near the discontinuity in
there is an overshoot which
is very narrow and spiky. The overshoot is of a size which does not
depend on the order of approximation.
A similar discontinuity is implicit in the function since the
Fourier approximations are periodic with period 1. This means that
the approximations are equal at 0 and at 1 while
is not.
The quadratic function does have
and the Fourier
approximation is much better.
My S-plus plotting code:
lin <- function(k) { x <- seq(0, 1, length = 5000) kv <- 1:k sv <- sin(2 * pi * outer(x, kv)) y <- - sv %*% (1/(pi * kv)) + 0.5 plot(x, x, xlab = "", ylab = "", main = paste(as.character(k), "Term Fourier Approximation to y=x"), type = "l") lines(x, y, lty = 2) }shows the use of the outer function and the paste function as well as how to avoid loops using matrix arithmetic.