Probability - University of Michigan
4.3 Partial sum of Fourier series and the Dirichlet kernel.
In this section we derive a formula for the partial sums of a Fourier series involving what is called the Dirichlet kernel and we show some properties of the Dirichlet kernel. We begin with a trigonometric identity.
Proposition 1.
(1) + cos x + cos 2x + … + cos nx =
Proof. Let L = + cos x + cos 2x + … + cos nx. We multiply L by 2 sin giving
2 (sin ) L = sin + 2 sin cos x + 2 sin cos 2x + … + 2 sin cos nx
Now we use the identity sin A cos B = giving
2 (sin ) L = sin + sin - sin + sin - sin + … + sinx - sin x
= sinx
and (1) follows. //
Definition 1. The quantity + cos x + cos 2x + … + cos nx = appearing in (1) is denoted by Dn(x) and called the Dirichlet kernel.
Proposition 2. Let f(x) be piecewise continuous for - ( < x < (. Let
(2) Sn(x) = a0 + (ak cos kx + bk sin kx)
be the nth partial sum of the Fourier series of f(x). Then
(3) Sn(x) = f(s) Dn(x – s) ds
Proof. One has
Sn(x) = f(s) ds + ( cos kx f(x) cos ks ds + sin kx f(x) sin kx dx)
= f(s) ds + ( f(x) cos kx cos ks ds + f(x) sin kx sin ks ds)
= f(s) [ + cos k(x – s)] ds = f(s) Dn(x – s) ds
which proves (3). //
Here are some properties of Dn(x).
Proposition 3. Dn(x) is continuous, even, periodic of period 2( and f(x) Dn(x) dx = ( and = .
Proof. All of these properties follow from the fact that Dn(x) is equal to + cos x + cos 2x + … + cos nx. Note that cos kx dx = 0. //
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