Question

In Exercises 5-14, the space is \(C\left[ {0,2\pi } \right]\) with inner product (6).

13. Explain why a Fourier coefficient of the sum of two functions is the sum of the corresponding Fourier coefficients of the two functions.

Short answer

The linearity property of the inner product is the reason behinda Fourier coefficient of the sum of two functions being the sum of the corresponding Fourier coefficients of the two functions.

Step-by-step solution

Inner Product. 

The Inner Productfor any two arbitrary functions is given by:

\(\left\langle {f,g} \right\rangle = \int_0^{2\pi } {f\left( t \right)g\left( t \right)dt} \)

Find the Fourier Coefficients.

As per the question, let\(f{\rm{ and }}g{\rm{ in }}C\left[ {0,2\pi } \right]\), then for\(m\)being a nonnegative integer, we have:

\(\left\langle {\left( {f + g} \right),\cos mt} \right\rangle {\rm{ and }}\left\langle {\left( {f + g} \right),\sin mt} \right\rangle \)

Usinglinearity of the inner product, we have:

\[\begin{array}{c}\left\langle {\left( {f + g} \right),\cos mt} \right\rangle = \left\langle {f,\cos mt} \right\rangle + \left\langle {g,\cos mt} \right\rangle \\\left\langle {\left( {f + g} \right),\sin mt} \right\rangle = \left\langle {f,\sin mt} \right\rangle + \left\langle {g,\sin mt} \right\rangle \end{array}\]

When these expressions get divided by trigonometric values, they will yield similar coefficients for both functions.

Hence, the linearity property of the inner product is the reason behind a Fourier coefficient of the sum of two functions being the sum of the corresponding Fourier coefficients of the two functions.