Basic Properties

Property 1. $\delta_N(x-t)$ projects periodic functions onto their least squares approximation in the subspace spanned by $\{ u_k(x) \}$ .

Property 2. (The graph, Figure 2.1)

From Figure 2.1, Eq.(2.1), and Eq.(2.2) one sees that

• $\delta _N (u)$ is an even periodic function with period $2\pi$ :
  $$\delta_N(u+n2\pi )=\delta_N(u).$$

• $\delta _N (u)$ has $2N$ equally spaced zeros which are determined by
  

  $$(N+\frac{1}{2})u = \pi ,2\pi ,3\pi ,\dots ,2N\pi \quad ,$$

  
  and thus are given by
  

  $$u = 1 \frac{\pi}{N+\frac{1}{2}}, 2\frac{\pi}{N+\frac{1}{2}},\dots ,2N\frac{\pi}{N+\frac{1}{2}}~~.$$

  
  Note that the next term in this sequence is
  $$(2N+1)\frac{\pi}{N+\frac{1}{2}}=2\pi~~,$$
  but $\delta _N (u)$ is not zero there. In fact, at all integral multiples of $2\pi$ , namely $u=0,2\pi,\dots, ~~\delta _N(u)$ has the value
  $$\delta_N(2\pi n)=\frac{1}{\pi}\left( N+\frac{1}{2}\right)\,,$$
  which is the maximum value of the function.
• $\delta _N (u)$ has $N$ maxima and $N$ minima per period.

Figure 2.1: Normalized Dirichlet kernels for N=2 (solid) and N=5 (dashed) with $u=2\pi x$ . There are N maxima and N minima per period. The number of zeroes is 2N and their spacing inside each periodic subinterval is $1/(2N+1)$ .