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Added the last sentence to emphasize the use of `RuleDelayed` .
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LouisB
LouisB
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To find an orthonormal set of functions, we can do the following:

p[n_] := (1 - x^2) x^(n - 1)

s = Simplify /@ Orthogonalize[Table[p[n], {n, 1, 4}],
    Integrate[#1 #2, {x, -1, 1}] &];

check = Table[Integrate[f g, {x, -1, 1}], {f, s}, {g, s}] // 
  MatrixForm

(*  1  0  0  0
    0  1  0  0
    0  0  1  0
    0  0  0  1  *)

There is a lot of good information and examples of AngleBracket in the context of inner products in previous questions on this forum, so you will want to do that search. Meanwhile, one simple way to use AngleBracket is to enter it with the key press sequence "Esc < Esc f , g Esc > Esc" and evaluate it with a rule, as in this example:

ip = AngleBracket[f_, g_] :> Integrate[f  g, {x, -1, 1}];

〈 p[1], p[3] 〉 /. ip
〈 p[3], p[3] 〉 /. ip
〈 s[[1]], s[[3]] 〉 /. ip
(*  16/105
    16/315
    0       *)

In the above we have defined an inner product rule ip, entered our inner products in the angle bracket notation, and then applied ip to evaluate the angle bracket. Note that ip is actually a RuleDelayed defined with :>, not an immediate Rule.

To find an orthonormal set of functions, we can do the following:

p[n_] := (1 - x^2) x^(n - 1)

s = Simplify /@ Orthogonalize[Table[p[n], {n, 1, 4}],
    Integrate[#1 #2, {x, -1, 1}] &];

check = Table[Integrate[f g, {x, -1, 1}], {f, s}, {g, s}] // 
  MatrixForm

(*  1  0  0  0
    0  1  0  0
    0  0  1  0
    0  0  0  1  *)

There is a lot of good information and examples of AngleBracket in the context of inner products in previous questions on this forum, so you will want to do that search. Meanwhile, one simple way to use AngleBracket is to enter it with the key press sequence "Esc < Esc f , g Esc > Esc" and evaluate it with a rule, as in this example:

ip = AngleBracket[f_, g_] :> Integrate[f  g, {x, -1, 1}];

〈 p[1], p[3] 〉 /. ip
〈 p[3], p[3] 〉 /. ip
〈 s[[1]], s[[3]] 〉 /. ip
(*  16/105
    16/315
    0       *)

In the above we have defined an inner product rule ip, entered our inner products in the angle bracket notation, and then applied ip to evaluate the angle bracket.

To find an orthonormal set of functions, we can do the following:

p[n_] := (1 - x^2) x^(n - 1)

s = Simplify /@ Orthogonalize[Table[p[n], {n, 1, 4}],
    Integrate[#1 #2, {x, -1, 1}] &];

check = Table[Integrate[f g, {x, -1, 1}], {f, s}, {g, s}] // 
  MatrixForm

(*  1  0  0  0
    0  1  0  0
    0  0  1  0
    0  0  0  1  *)

There is a lot of good information and examples of AngleBracket in the context of inner products in previous questions on this forum, so you will want to do that search. Meanwhile, one simple way to use AngleBracket is to enter it with the key press sequence "Esc < Esc f , g Esc > Esc" and evaluate it with a rule, as in this example:

ip = AngleBracket[f_, g_] :> Integrate[f  g, {x, -1, 1}];

〈 p[1], p[3] 〉 /. ip
〈 p[3], p[3] 〉 /. ip
〈 s[[1]], s[[3]] 〉 /. ip
(*  16/105
    16/315
    0       *)

In the above we have defined an inner product rule ip, entered our inner products in the angle bracket notation, and then applied ip to evaluate the angle bracket. Note that ip is actually a RuleDelayed defined with :>, not an immediate Rule.

Source Link
LouisB
LouisB
  • 12.8k
  • 1
  • 22
  • 34

To find an orthonormal set of functions, we can do the following:

p[n_] := (1 - x^2) x^(n - 1)

s = Simplify /@ Orthogonalize[Table[p[n], {n, 1, 4}],
    Integrate[#1 #2, {x, -1, 1}] &];

check = Table[Integrate[f g, {x, -1, 1}], {f, s}, {g, s}] // 
  MatrixForm

(*  1  0  0  0
    0  1  0  0
    0  0  1  0
    0  0  0  1  *)

There is a lot of good information and examples of AngleBracket in the context of inner products in previous questions on this forum, so you will want to do that search. Meanwhile, one simple way to use AngleBracket is to enter it with the key press sequence "Esc < Esc f , g Esc > Esc" and evaluate it with a rule, as in this example:

ip = AngleBracket[f_, g_] :> Integrate[f  g, {x, -1, 1}];

〈 p[1], p[3] 〉 /. ip
〈 p[3], p[3] 〉 /. ip
〈 s[[1]], s[[3]] 〉 /. ip
(*  16/105
    16/315
    0       *)

In the above we have defined an inner product rule ip, entered our inner products in the angle bracket notation, and then applied ip to evaluate the angle bracket.