restart;phi1:=(x)->x+1;

NiM+SSVwaGkxRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCY5JCIiIkYuRi5GJUYlRiU=

phi2:=(x)->x^3+3*x;

NiM+SSVwaGkyRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYqJDkkIiIkIiIiRi5GL0YlRiVGJQ==

phi3:=(x)->x^5+5*x;

NiM+SSVwaGkzRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYqJDkkIiImIiIiRi5GL0YlRiVGJQ==

g:=c[1]*phi1(x)+c[2]*phi2(x)+c[3]*phi3(x);

NiM+SSJnRzYiLCgmSSJjR0YlNiMiIiJGKiomJkYoNiMiIiNGKiwmKiRJInhHRiUiIiRGKkYxRjJGKkYqKiYmRig2I0YyRiosJiokRjEiIiZGKkYxRjhGKkYq

solve(identity(g,x),{c[1],c[2],c[3]});

NiM8JS8mSSJjRzYiNiMiIiMiIiEvJkYmNiMiIiJGKi8mRiY2IyIiJEYq

restart;phi1:=(x)->1;

NiM+SSVwaGkxRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlIiIiRiVGJUYl

phi2:=(x)->x+1;

NiM+SSVwaGkyRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCY5JCIiIkYuRi5GJUYlRiU=

phi3:=(x)->2*x+3;

NiM+SSVwaGkzRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCY5JCIiIyIiJCIiIkYlRiVGJQ==

g:=c[1]*phi1(x)+c[2]*phi2(x)+c[3]*phi3(x);

NiM+SSJnRzYiLCgmSSJjR0YlNiMiIiJGKiomJkYoNiMiIiNGKiwmSSJ4R0YlRipGKkYqRipGKiomJkYoNiMiIiRGKiwmRjBGLkY0RipGKkYq

solve(identity(g,x),{c[1],c[2],c[3]});

NiM8JS8mSSJjRzYiNiMiIiMsJCZGJjYjIiIkISIjLyZGJjYjIiIiLCRGKyEiIi9GK0Yr

restart;L := proc( n , x ) if n = 0 then return 1; elif n = 1 then return x; else return expand(1/n*((2*(n-1)+1)*x*L(n-1,x)-(n-1)*L(n-2,x))); end if; end proc:lTestOrt := proc( w , li , a , b ) local res, j, k, g; res := true; for j from 1 to nops(li) do for k from 1 to nops(li) do g := int(w(x)*op(j,li)*op(k,li),x=a..b); if j = k then if g <= 0 then res := false; end if; end if; if j <> k then if g <> 0 then res := false; end if; end if; end do; end do; return res; end proc:w:=(x)->1;

NiM+SSJ3RzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlIiIiRiVGJUYl

liste := [seq(L(i,x),i=0..5)];

NiM+SSZsaXN0ZUc2IjcoIiIiSSJ4R0YlLCYqJEYoIiIjIyIiJEYrIyEiIkYrRicsJiokRihGLSMiIiZGK0YoIyEiJEYrLCgqJEYoIiIlIyIjTiIiKUYqIyEjOkY4I0YtRjtGJywoKiRGKEYzIyIjakY7RjEjISNORjhGKCMiIzpGOw==

lTestOrt(w,liste,-1,1);

NiNJJXRydWVHSSpwcm90ZWN0ZWRHRiQ=

restart;T := proc ( n , x ) if n = 0 then return 1; elif n = 1 then return x; else return 2*x*T(n-1,x)-T(n-2,x); end if; end proc:lTestOrt := proc( w , li , a , b ) local j, k, g; for j from 1 to nops(li) do for k from 1 to nops(li) do g := evalf(int(w(x)*op(j,li)*op(k,li),x=a..b)); if j = k then if g <= 0 then printf("g: %a\tj: %a\tk: %a\n", g, j, k); return false; end if; end if; if j <> k then if g <> 0 then printf("g: %a\tj: %a\tk: %a\n", g, j, k); return false; end if; end if; end do; end do; return true; end proc:w:=(x)->1/sqrt(1-x^2);

NiM+SSJ3RzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiQtSSVzcXJ0R0YlNiMsJiIiIkYxKiQ5JCIiIyEiIkY1RiVGJUYl

liste := [seq(T(i,x),i=0..5)];

NiM+SSZsaXN0ZUc2IjcoIiIiSSJ4R0YlLCYqJEYoIiIjRishIiJGJywmKiZGKEYnRilGJ0YrRihGLCwoKiZGKEYnRi1GJ0YrRiohIiNGJ0YnLCgqJkYoRidGL0YnRitGLkYxRihGJw==

lTestOrt(w,liste,-1,1);

NiNJJXRydWVHSSpwcm90ZWN0ZWRHRiQ=

restart;tr1:=(a,b)->1/2*((b-a)*x+a+b);

NiM+SSR0cjFHNiJmKjYkSSJhR0YlSSJiR0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUsKComLCY5JSIiIjkkISIiRjFJInhHRiVGMSNGMSIiI0YyRjVGMEY1RiVGJUYl

tr2:=(a,b)->(2*x-a-b)/(b-a);

NiM+SSR0cjJHNiJmKjYkSSJhR0YlSSJiR0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUqJiwoSSJ4R0YlIiIjOSQhIiI5JUYyIiIiLCZGM0Y0RjFGMkYyRiVGJUYl

L := proc( n ) option remember; if n = 0 then return 1; elif n = 1 then return x; else return expand(1/n*((2*n-1)*x*L(n-1)-(n-1)*L(n-2))); end if; end proc:ak := proc( k , f , a , b ) option remember; return ((2*k+1)/2)*int(f(tr1(a,b))*L(k),x=-1..1); end proc:P := proc( n , f , a , b ) local res, aks, ls; option remember; aks := [seq(ak(k,f,a,b),k=1..n+1)]; ls := [seq(L(k),k=1..n+1)]; res := [seq(op(k,aks)*op(k,ls),k=1..n+1)]; res := simplify(sum(op(k,res),k=1..n+1)); res := sort(expand(res)); print(plot(res,x=a..b)); print(plot(abs(res-f(x)),x=a..b)); return res;end proc:

g1:=P(3,x->sin(x*Pi),-1,1);

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

NiM+SSNnMUc2IiwqKiZJI1BpR0kqcHJvdGVjdGVkR0YpISIkSSJ4R0YlIiIkIyEkRCYiIiMqJkYoISIiRitGLCMiI05GLyomRihGMUYrIiIiIyEjOkYvKiZGKEYqRitGNSMiJDokRi8=

g2:=P(6,x->exp(x),-1,3);

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LSUlUExPVEc2Ji0lJ0NVUlZFU0c2JDdXNyQkISIiIiIhJCIzUSw7YjFhdkhcISM8NyQkITNQTExMTFE2RyIqISM9JCIzXzlZKFwxN0YqW0YvNyQkITNobW1tVC5ccCQpRjMkIjNdeiZlMzZbRSZbRi83JCQhM0xMTEwkKSlRal4oRjMkIjNpQVAtdSY9bHolRi83JCQhM1RMTEwkPUt2bCdGMyQiM0N2ZXp3NmREWkYvNyQkITNobm1tVHMhRyFlRjMkIjNNR2c8Yl5iUFlGLzckJCEzaUxMTDN5TzVdRjMkIjMmKVFXcTs+UFBYRi83JCQhM2orKyt2RSUpKj0lRjMkIjMzRG5WOHo5NldGLzckJCEzKVJMTCQzV0RUTEYzJCIzOHksXTNJKD1EJUYvNyQkITMnNCsrXWQoUSZcI0YzJCIzaDJXIVImcEVlU0YvNyQkITM6bW1tbSY0YGkiRjMkIjNnNSZvb0srYiJRRi83JCQhM0dLTExMUVcqZSkhIz4kIjNXbnMvcVQmemIkRi83JCQiM0hJIyoqKioqKipILFEhI0AkIjNSYFh4LSZbJjRLRi83JCQiM1EoKioqKioqKlwqM3EpRmBvJCIzOyQqW1UqPVRheSNGLzckJCIzISoqKioqKioqcD1ccSJGMyQiM3l2b21ITl8jSCNGLzckJCIzXm1tbSJmQklZI0YzJCIzKWZXZj5tJykqZjxGLzckJCIzeUtMTExPW2tMRjMkIjNtJjMhMyR6KHAsNUYvNyQkIjMuS0xMTCZRIkdURjMkIjNZK2BbRlojb0wjRjM3JCQiMysqKioqKlxzXWssJkYzJCIzYURlZnIzJ2ZLKUYzNyQkIjNXSkxMTHZ2LWVGMyQiM3k/XlJYMSMqZj5GLzckJCIzJzMrK11zZ2FtJ0YzJCIzOiVcciIzKkhGViRGLzckJCIzRisrK3YiZXBbKEYzJCIzRD8iKVwvPCwyXkYvNyQkIjMkUUxMTGUvVE0pRjMkIjMlXGgjPiIqZjohPihGLzckJCIzOUxMTGVEQkoiKkYzJCIzIz5AJClIS1AxWSpGLzckJCIzSW1tbTtrRCEpKipGMyQiMzg1UUIqb2tqQiIhIzs3JCQiM01tbTtmYEAnMyJGLyQiMy8lNHNBM2twZiJGZXM3JCQiM3oqKioqXG5aKUg7IkYvJCIzKVxOJFJyP25vPkZlczckJCIzWW1tbUp5KmVDIkYvJCIzOUJAdWA8QFRDRmVzNyQkIjMnKSoqKioqKlJeYko4Ri8kIjMuSmIpNGh5LS0kRmVzNyQkIjNmKioqKipcNWFgVCJGLyQiMzNHb1UoZSR6InAkRmVzNyQkIjNvKioqKlw3UlYnXCJGLyQiMy8iM0RLTU9zWCVGZXM3JCQiM2wqKioqKlxAZmtlIkYvJCIzRChbNDxbcEtZJkZlczckJCIzMExMTGA0Tm47Ri8kIjMlcCl6KSk0WmZJbEZlczckJCIzIyoqKioqKipcLHNgPEYvJCIzc1wtaTlCQnB5RmVzNyQkIjNbbW07ek0pPiQ9Ri8kIjNpVmVndmZEKEcqRmVzNyQkIjMkKioqKioqKnBmYTw+Ri8kIjM0akgvJVw4JjQ2ISM6NyQkIjMjSExMZWdgISkqPkYvJCIzYl1neHEtJHlJIkZddzckJCIzdyoqKipcI0cyQTMjRi8kIjMxaDNyXVlyWzpGXXc3JCQiMztMTEwkKUdba0BGLyQiM0wiKWU+Ujo9QT1GXXc3JCQiMyMpKioqKlw3eWhdQUYvJCIzXD43TmZNW2FARl13NyQkIjN4bW1tJylmZExCRi8kIjNwKHBnTExqX18jRl13NyQkIjNibW1tLEZUPUNGLyQiMyJbVFteL3FJJ0hGXXc3JCQiM0ZMTCRlI3BhLURGLyQiMyoqXEQxc1pwak1GXXc3JCQiMyopKioqKioqUnYmKXpERi8kIjMpKXk9UVgpZSYqKVJGXXc3JCQiM0hMTExHVVlvRUYvJCIzIUhFby0wZCh6WUZddzckJCIzIioqKioqXG4nKjMzRkYvJCIzbDZjKT4wVDwtJkZddzckJCIzX21tbTFeclpGRi8kIjNYeGlYJWYhKmZRJkZddzckJCIzNExMZSozayoqeSNGLyQiMypINStFdShIK2VGXXc3JCQiMzQrK11zSUBLR0YvJCIzLGBgbT85JUhDJ0ZddzckJCIzNCsrK1MybHNHRi8kIjMhPW0+KTNKcCVwJ0ZddzckJCIzNCsrXTIlKTM4SEYvJCIzNFRhKDRldWE8KEZddzckJCIzLysrdi5VYWNIRi8kIjNkLko7PyVSanMoRl13NyQkIiIkRiwkIjMxKm88I29zbjkkKUZddy0lJkNPTE9SRzYmJSRSR0JHJCIjNUYrJEYsRitGaVxsLSUrQVhFU0xBQkVMU0c2JFEieDYiUSFGXl1sLSUlRk9OVEc2JCUqSEVMVkVUSUNBR0ZoXGwtJSVWSUVXRzYkOyQhIzVGKyQiI0lGKzskITEpPUI5JSkqNFI7ISM5JCIxJVwoM3hTIzRbKSEjOA==

NiM+SSNnMkc2IixCKiYtSSRleHBHNiRJKnByb3RlY3RlZEdGK0koX3N5c2xpYkdGJTYjISIiIiIiSSJ4R0YlIiIoIyIrTkhmP2IiJSM+KSomLUYpNiMiIiRGL0YwRjEjISpiSjYsIkY0KiZGKEYvRjAiIicjISokeXZIPiIlJzQlKiZGNkYvRjBGPCMiKEpYYCRGPyomRihGL0YwIiImIyErMltvJykpKUY0KiZGNkYvRjBGRCMiKi5jd2kiRjQqJkYoRi9GMCIiJSMiKnYlZjtFRj8qJkY2Ri9GMEZLIyEoJjQjeiVGPyomRihGL0YwRjgjIitsW0U+U0Y0KiZGNkYvRjBGOCMhKURSaHRGNComRihGL0YwIiIjIyEpbFlZJylGPyomRjZGL0YwRlgjIihsWmUiRj8qJkYoRi9GMEYvIyEqWCJvSldGNComRjZGL0YwRi8jIihEIj4iKUY0RigjIihIdzAlRj9GNiMhJnBaKEY/

restart;tr1:=(x,a,b)->1/2*((b-a)*x+a+b);

NiM+SSR0cjFHNiJmKjYlSSJ4R0YlSSJhR0YlSSJiR0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUsKComLCY5JiIiIjklISIiRjI5JEYyI0YyIiIjRjNGNkYxRjZGJUYlRiU=

tr2:=(x,a,b)->(2*x-a-b)/(b-a);

NiM+SSR0cjJHNiJmKjYlSSJ4R0YlSSJhR0YlSSJiR0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUqJiwoOSQiIiM5JSEiIjkmRjMiIiIsJkY0RjVGMkYzRjNGJUYlRiU=

T := proc ( n , x ) option remember; if n = 0 then return 1; elif n = 1 then return x; else return expand(2*x*T(n-1,x)-T(n-2,x)); end if; end proc:c := proc( k , f , a , b ) option remember; if k = 0 then return evalf(1/Pi*int(f(tr1(x,a,b))/sqrt(1-x^2),x=-1..1)); else return evalf(2/Pi*int((f(tr1(a,b))*T(k,x))/sqrt(1-x^2),x=-1..1)); end if; end proc:P := proc( n , f , a , b ) option remember; return expand(sum(c(n,f,a,b)*T(n,tr2(a,b)),k=0..n));end proc:

P(4,x->cos(x),-1,1);

Error, (in tr1) tr1 uses a 3rd argument, b, which is missing

P(9,x->ln(x),0.5,5);

Error, (in tr1) tr1 uses a 3rd argument, b, which is missing

-