515 – Plotting Féjer’s kernels

Féjer’s kernels are the functions that play the role of “approximations to the Dirac delta” in the computations we will use to obtain Weierstrass approximation theorem. The nth approximation is given by

\displaystyle K_n(s)= \frac1{n+1}\left(\frac{\sin\frac{(n+1)s}2}{\sin\frac s2}\right)^2 for s\ne0, K_n(0)=n+1.

I preemptively apologize for the roughness of my code. Using the following program in Sage gives the diagram shown below, displaying the kernels K_0,K_1,K_2,K_4,K_8,K_{15},K_N for N=100. Additional details can be made more prominent by varying the size of the parameter {\tt eps}, or by using additional features of the {\tt plot} function. In this example, {\tt eps}=\pi. To extend the graph to all values of s, simply reflect about the line s=\pi, and extend periodically with period 2\pi.

{\tt var('s')}

{\tt N=100}

{\tt eps=pi}

{\tt def\ K(n,s):}

\hspace{1in} {\tt return\ (sin((n+1)*s/2)/sin(s/2))^2/(n+1)}

{\tt P0=plot(K(0,s),\ -eps,\ eps,\ color='red')}

{\tt P1=plot(K(1,s),\ -eps,\ eps,\ color='brown')}

{\tt P2=plot(K(2,s),\ -eps,\ eps,\ color='blue')}

{\tt P4=plot(K(4,s),\ -eps,\ eps,\ color='black')}

{\tt P8=plot(K(8,s),\ -eps,\ eps,\ color='purple')}

{\tt P15=plot(K(15,s),\ -eps,\ eps,\ color='orange')}

{\tt pN=plot(K(N,s),\ -eps,\ eps,\ color='cyan')}

{\tt P=P0+P1+P2+P4+P8+P15+pN}

{\tt P}

Here is the corresponding outcome (click to enlarge):


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