File:Test8trubes300.jpg

Test of the Discrete Bessel transform of the truncated Bessel function

$f(x)=J_0(x)\,$UnitStep$(\lambda\!-\!x)$

where $~ \lambda=\,$BesselJZero$[0,1]\approx 2.404825557695773$

Description
Small red circle represent the input function, represented at the grid of $M\!=\!8$ points; the coordinates of these circles are $(x_n,f(x_n))$ for $n=1..8$, where $x_n$ are abscissas of the grid for the discrete Bessel.

These circles follow the smooth profile $y\!=\!J_0(x)$, shown with thin black curve

The input array is determined with $f_n\!=\! f(x_n)\, w_n$ for $n=1.8$

The transform matrix $T$ of the Discrete Bessel determines the output $\displaystyle g_n=\sum_{m=1}^M T_{n,m}\, f_m$

This output is represented by the big blue circles wit coordinates $(x_n,g_n/w_n)$ for $n=1.8$

It follows the expected smooth curve

$\displaystyle y=g(x)=$ $\displaystyle \int_0^\infty f(t) \, J_0(t \, x) \, t \, \mathrm d t=$ $\displaystyle \int_0^\lambda J_0(t) \, J_0(t \, x) \, t \, \mathrm d t=$ $\displaystyle \lambda \, J_1(\lambda) \, \frac{J_0(\lambda\, x)}{1\!-\!x^2} $

Curve $y=g(x)$ approaches the ordinate axis $x\!=\!0$ with zero derivative at point

$y\!=\!\lambda \, J_1(\lambda) \approx 1.2484591696955065$

The pole at $x\!=\!1$ due to denominator in the expression for $g(x)$ is compensated by the zero at $x\!=\!1$ in the numerator; so, the first zero of $g(x)$ is

$x=\,$BesselJZero$[0,2]/\lambda = \,$BesselJZero$[0,2]/$BesselJZero$[0,1]\,$ $\approx 2.2954172674276943$

C++ generator of curves
DB jnp(int n,DB x){ return .5*( jn(n-1,x)-jn(n+1,x) ) ; } // Derivative of n th Bessel DB jnz(int v, int k){ DB x,t; t=M_PI*(k+.5*v-.25); x= t - (v*v-.25)*.5/t; x-= jn(v,x)/jnp(v,x); // Newton adjustment of the root x-= jn(v,x)/jnp(v,x); x-= jn(v,x)/jnp(v,x); return x; } // the k th zero of v th Bessel
 * 1) include
 * 2) include
 * 3) include
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)
 * 1) include"ado.cin"

int main{ int m,n,v,k; DB s, x,y; FILE *o; int M=8; DB X[M+1],W[M+1],T[M+1][M+1],TT[M+1][M+1], F[M],G[M]; DB S=jnz(0,M+1); DB jd=jnz(0,1); DB qs=sqrt(1./S); DB q=sqrt(2./S); for(n=1;n<M+1;n++){ x=jnz(0,n); X[n]=x*qs; y=W[n]=q/fabs(j1(x)); printf("%3d %20.16lf %20.16lf\n",n,X[n],W[n]); } for(m=1;m<=M;m++){// printf("%2d",m); for(n=1;n<=M;n++){ T[m][n]=W[m]*j0(X[m]*X[n])*W[n]; }} //for(m=1;m<=M;m++){printf("\n"); //for(n=1;n<=M;n++){printf("%14.10lf",T[m][n]); }} //printf("\n"); //for(m=1;m<=M;m++){printf("\n"); //for(n=1;n<=M;n++){ s=0.; //                       for(k=1;k<=M;k++) s+=T[m][k]*T[k][n] ; //                       TT[m][n]=s;printf("%14.10lf",TT[m][n]); //               }} //printf("\n\n"); //for(m=1;m<=M;m++){x=X[m]; y=exp(-x*x/2.); F[m]=y*W[m]; /* printf("%14.10lf",F[m]); */ } for(m=1;m<=M;m++){x=X[m]; if(x<jd) y=j0(x); else y=0.; F[m]=y*W[m]; /*printf("%14.10lf",F[m]);*/ } /*printf("\n");*/ for(m=1;m<=M;m++){ s=0.; for(n=1;n<=M;n++) s+=T[m][n]*F[n]; G[m]=s;} //for(m=1;m<=M;m++) printf("%14.10lf",G[m]); //printf("\n"); o=fopen("test8trube.eps", "w"); ado(o,520,220); fprintf(o,"10 60 translate 100 100 scale\n"); DO(n,6){M(n,1)L(n,0)} DO(n,3){M(0,n/2.)L(5,n/2.)} fprintf(o,"2 setlinecap 2 setlinejoin .006 W S\n");
 * 1) define M(x,y) fprintf(o,"%8.4lf %8.4lf M\n",0.+x, 0.+y);
 * 2) define L(x,y) fprintf(o,"%8.4lf %8.4lf L\n",0.+x, 0.+y);
 * 3) define O(x,y) fprintf(o,"%8.4lf %8.4lf O\n",0.+x, 0.+y);
 * 4) define o(x,y) fprintf(o,"%8.4lf %8.4lf o\n",0.+x, 0.+y);

//M(0,1) DO(n,500){x=.02*(n+1); y=exp(-x*x/2.); L(x,y)} M(0,1) DO(n,127){x=.04*(n+1); y=j0(x); L(x,y)} fprintf(o,"1 setlinecap 1 setlinejoin S\n");

DO(n,127){x=.04*(n+.2); y=j0(jd*x)/(1.-x*x)*jd*j1(jd); if(n==0)M(x,y) else L(x,y) printf("%6.4lf %6.3lf\n",x,y);} fprintf(o,"1 setlinecap 1 setlinejoin S\n");

fprintf(o,"1 0 0 RGB .016 W\n"); for(m=1;m<=M;m++) o(X[m],F[m]/W[m]) fprintf(o,"0 0 1 RGB .01 W\n"); for(m=1;m<=M;m++) O(X[m],G[m]/W[m])

fprintf(o,"showpage\n%c%cTrailer\n",'%','%'); fclose(o); system("epstopdf test8trube.eps"); system("open    test8trube.pdf"); return 0; }

Latex generator of labels
\documentclass[12pt]{article} \usepackage{geometry} \usepackage{graphicx} \usepackage{rotating} \paperwidth 514pt \paperheight 188pt \topmargin -110pt \oddsidemargin -92pt \newcommand \ing {\includegraphics} \newcommand \sx {\scalebox} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \begin{document} \begin{picture}(410,212) %\put(2,4){\ing{bessel8testSte}} \put(2,4){\ing{test8trube}} \put(2,190){\sx{1.4}{\rot{-2}$\displaystyle y\!=\!\lambda J_1(\lambda) \frac{J_0(\lambda\, x)}{1\!-\!x^2}$ \ero}} %\put(1,189){\sx{1.4}{$\frac 3 2$}} \put(2,159){\sx{1.4}{$1$}} \put(14,150){\sx{1.4}{\rot{-6}$\displaystyle y\!=\! J_0(x)$ \ero}} \put(2,110){\sx{1.42}{$\frac 1 2$}} \put(2,59){\sx{1.4}{0}} \put(10,50){\sx{1.4}{0}} \put(108,50){\sx{1.4}{1}} \put(208,50){\sx{1.4}{2}} \put(308,50){\sx{1.4}{3}} \put(409,50){\sx{1.4}{4}} \put(504,50){\sx{1.5}{$x$}} \put(366,30){\sx{1.4}{\rot{0}$\displaystyle y\!=\! J_0(x)$ \ero}} %\put(123,114){\sx{1.4}{\rot{-52}$y\!=\! 2 J_1(2x)/x$\ero }} %\put(184,141){\sx{1.2}{step }} %\put(210,134){\sx{1.2}{\rot{-90}step\ero }} \end{picture} \end{document}