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var('t') h1(t)=(11/3)*t^2; h2(t)=-(5/6)*2^t+3; f(t)=piecewise([[(-pi,0),h1(t)],[(0,pi),h2(t)]]) uta=plot(f(t),t,-pi,pi) uta
t
w1(t)=sin(t); w2(t)=sin(2*t); w3(t)=sin(3*t); nor=integral((w1(t))^2,t,-pi,pi) nor
pi
#si se sacan las otras normas se observara que son iguales# k1=integral(w1(t)*h1(t),t,-pi,0)+integral(w1(t)*h2(t),t,0,pi); k2=integral(w2(t)*h1(t),t,-pi,0)+integral(w2(t)*h2(t),t,0,pi); k3=integral(w3(t)*h1(t),t,-pi,0)+integral(w3(t)*h2(t),t,0,pi); theta1(t)=(k1/nor)*w1(t); theta2(t)=(k2/nor)*w2(t); theta3(t)=(k3/nor)*w3(t); pt(t)=theta1(t)+theta2(t)+theta3(t) uu=plot(pt(t),(t,-pi,pi),color='black') uv=plot(f(t),(t,-pi,pi),color='red') uw=uu+uv uw.show()
w5(t)=1 w6(t)=cos(t) w7(t)=cos(3*t) w8(t)=cos(5*t) nor0=integral(w5(t)^2,t,-pi,pi) nor0
2*pi
nor1=integral(w6(t)^2,t,-pi,pi) nor1
pi
#la norma de w6,w7 y w8 es la misma la diferente es la primera norma
#aproximacion conjunto#1 #si se sacan las otras normas se observara que son iguales# k1=integral(w1(t)*h1(t),t,-pi,0)+integral(w1(t)*h2(t),t,0,pi); k2=integral(w2(t)*h1(t),t,-pi,0)+integral(w2(t)*h2(t),t,0,pi); k3=integral(w3(t)*h1(t),t,-pi,0)+integral(w3(t)*h2(t),t,0,pi); theta1(t)=(k1/nor)*w1(t); theta2(t)=(k2/nor)*w2(t); theta3(t)=(k3/nor)*w3(t); pt(t)=theta1(t)+theta2(t)+theta3(t) u=plot(pt(t),(t,-pi,pi),color='black') #aproximacion conjunto #2 #hacemos las proyecciones del vector sobre las funciones dadas #hacemos el producto punto de cada funcion q0(t)=integral(w5(t)*h1(t),t,-pi,0)+integral(w5(t)*h2(t),t,0,pi) q1(t)=integral(w6(t)*h1(t),t,-pi,0)+integral(w6(t)*h2(t),t,0,pi) q2(t)=integral(w7(t)*h1(t),t,-pi,0)+integral(w7(t)*h2(t),t,0,pi) q3(t)=integral(w8(t)*h1(t),t,-pi,0)+integral(w8(t)*h2(t),t,0,pi) #con el producto punto se hacen las respectivas proyecciones e0(t)=q0(t)*w5(t)/nor0 e1(t)=q1(t)*w6(t)/nor1 e2(t)=q2(t)*w7(t)/nor1 e3(t)=q3(t)*w8(t)/nor1 #sumamos las proyecciones para aproximar et(t)=e0(t)+e1(t)+e2(t)+e3(t) uu=plot(et(t),(t,-pi,pi),color='green') uv=plot(f(t),(t,-pi,pi),color='blue') uw=uu+uv uw.show()
r0(t)=1 r1(t)=sin(t) r2(t)=sin(2*t) r3(t)=sin(3*t) r4(t)=cos(t) r5(t)=cos(3*t) r6(t)=cos(5*t) #al hacer la norma todas son iguales acxepto r0 nor0=integral(r0(t)^2,t,-pi,pi) nor0
2*pi
nor1=integral(r1(t)^2,t,-pi,pi) nor1
pi
#hacemos las proyecciones del vector sobre las funciones dadas #hacemos el producto punto de cada funcion z0(t)=integral(r0(t)*h1(t),t,-pi,0)+integral(r0(t)*h2(t),t,0,pi) z1(t)=integral(r1(t)*h1(t),t,-pi,0)+integral(r1(t)*h2(t),t,0,pi) z2(t)=integral(r2(t)*h1(t),t,-pi,0)+integral(r2(t)*h2(t),t,0,pi) z3(t)=integral(r3(t)*h1(t),t,-pi,0)+integral(r3(t)*h2(t),t,0,pi) z4(t)=integral(r4(t)*h1(t),t,-pi,0)+integral(r4(t)*h2(t),t,0,pi) z5(t)=integral(r5(t)*h1(t),t,-pi,0)+integral(r5(t)*h2(t),t,0,pi) z6(t)=integral(r6(t)*h1(t),t,-pi,0)+integral(r6(t)*h2(t),t,0,pi) #cada proyección b0(t)=z0(t)*r0(t)/nor0 b1(t)=z1(t)*r1(t)/nor1 b2(t)=z2(t)*r2(t)/nor1 b3(t)=z3(t)*r3(t)/nor1 b4(t)=z4(t)*r4(t)/nor1 b5(t)=z5(t)*r5(t)/nor1 b6(t)=z6(t)*r6(t)/nor1 #la suma de las proyecciones para aproximar bt(t)=b0(t)+b1(t)+b2(t)+b3(t)+b4(t)+b5(t)+b6(t) uu=plot(bt(t),(t,-pi,pi),color='red') uv=plot(f(t),(t,-pi,pi),color='blue') uw=uu+uv uw.show()