I would like to be able to expand certain functions as linear combinations of other specified functions when I know that the original function can be represented in such a way e.g. we know that $\sin^{2}(x)$ can be represented as a finite linear combination of $1$ and $\cos(2x)$, so I want to be able to tell Mathematica to find the coefficients in the linear combination (which here would be $1/2$ and $- 1/2$, respectively).

Is there a command or a simple piece of code which will do this:

  1. without being specific to Fourier series, so can be applied to combinations of other (pre-specified) functions more generally, and
  2. doesn't rely on numerical evaluation of the functions anywhere?

Thanks in advance for any help.

  • 1
    $\begingroup$ The whole problem simplifies, if the specified set of functions--basis--is orthonormal. In this case the expansion coefficients can be computed by projection. In application to functional basis, the projection would be integration. $\endgroup$
    – yarchik
    Oct 17, 2021 at 21:16

1 Answer 1


Normally I'd use SolveAlways, but unfortunately that will not always work. For instance:

SolveAlways[u*Cos[x]^2 + v*Sin[x]^2 == 1, x]
(* {{}, {u -> v + Sec[x]^2 - v Sec[x]^2}, {v -> 1}} *)

_solvealways fails

Another way is to do a minimization of the total square difference (or could be Abs difference) between your linear combo and the target function. This actually works here, entirely symbolically:

  Integrate[({u, v} . {1, Cos[2 x]} - Sin[x]^2)^2, {x, 0, 2 Pi}], {u, v}]

(* {1/2, -(1/2)} *)

Alternatively, there is Orthogonalize. First you need to choose an interval you want it to be linear on. It could be $[0,2\pi]$ or $(-\infty,\infty)$ for example. You then define an inner product which must converge on your interval. Use Orthogonalize to produce a new orthogonal basis which is properly normalized.

a = 0;
b = 2 Pi;
innerproduct[u_, v_] := Integrate[u v, {x, a, b}]
unnormedbasis = {1, Cos[2 x]};
basis = Orthogonalize[unnormedbasis, innerproduct];
consts = basis/unnormedbasis;
projection = innerproduct[Sin[x]^2, #] & /@ basis
Sin[x]^2 == projection . basis
FullSimplify[(projection*consts) . unnormedbasis == Sin[x]^2]
(* True *)

(* {1/2, -(1/2)} *)

In cases where you don't have an exact linear combination, you can use this technique to work out approximations. For instance, this finds an approximation to $\sqrt{x}$ in terms of a linear combination of some basis polynomials:

a = 0;
b = 1;
innerproduct[u_, v_] := Integrate[u v, {x, a, b}]
unnormedbasis = {1, x, x^2, x^3};
basis = Orthogonalize[unnormedbasis, innerproduct];
consts = basis/unnormedbasis;
projection = innerproduct[Sqrt[x], #] & /@ basis

result = FullSimplify[projection . basis]
(* 8/63 (1 + x (15 + x (-15 + 7 x))) *)

Plot[{Sqrt[x], result}, {x, a, b}]

orthogonalization approximation of sqrt


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.