Approximating function by trigonometric polynomial

[ekkilop]

Hi!

Say that we wish to approximate a function $f(x), \, x\in [0, 2\pi]$ by a trigonometric polynomial such that

$f(x) \approx \sum_{|n|\leq N} a_n e^{inx} \qquad (1)$

The best approximation theorem says that in a function space equipped with the inner product

$(f,g) = \frac{1}{2 \pi} \int_0^{2\pi} f \bar{g} dx$

the best possible approximation is the truncated Fourier series of the function, which follows from the orthonormality of the basis functions $\{ e^{inx} \}$. But what happens if we wish to consider a smaller interval, say $x \in [0, \pi/2]$, and a corresponding inner product

$(f,g) = \frac{2}{\pi} \int_0^{\pi/2} f \bar{g} dx$

but still use the functions $\{ e^{inx} \}$ (no longer orthonormal) in our approximation $(1)$? We could of course use the Fourier coefficients for all $n$ that are multiples of 4 and set the rest to zero to get the corresponding Fourier series, but this is no longer the best possible approximation. So my question is basically, what would the best approximation be in this case?

Thank you!

 

[mfb]

$$\frac{\partial (f,g)}{\partial a_n} = 0$$ for all $a_n$ is a natural result of an optimal solution. Analyzing this equation could give some interesting results.

 

[ekkilop]

Thank you mfb for your reply!

Yes, that was my original idea as well. If $g$ is the approximation in the RHS of $(1)$, then I reasoned that the optimal result should be when $(f-g) \perp f$. However, $(f-g, f)$ is a linear function in the coefficients $a_n$ so there are no extrema (I am assuming the coefficients are independent of x). Or perhaps I misunderstood something?

 

[mfb]

Ah, small fix:
I would expect that you want to minimize (f-g,f-g). As the inner product is linear in its arguments, (f-g,f-g) = (f,f) + (g,g) - (f,g) - (g,f) = (f,f) + (g,g) - 2 Re (f,g)
(f,f) is fixed, the other two parts depend on a_n and the expression should be minimal with respect to all a_n. The (g,g) part gives a nonlinearity with a proper minimum.

 

[blue_raver22]

I would first commend you for exploring different methods for approximating a function. This shows that you are thinking critically and trying to find the most accurate representation of your data. In this case, it seems like you are considering a smaller interval and a different inner product, which can definitely affect the resulting approximation.

To answer your question, the best approximation in this case would depend on the specific function f(x) that you are trying to approximate. It may not necessarily be a trigonometric polynomial, as that is just one type of approximation method. Other methods such as spline interpolation or polynomial interpolation may provide a better fit for your function on the given interval.

However, if you are set on using a trigonometric polynomial approximation and the given inner product, then the best approach would be to minimize the mean squared error between your function and the approximation. This means finding the coefficients a_n that minimize the integral of the squared difference between f(x) and the trigonometric polynomial in the given interval. This can be done using techniques such as least squares regression or optimization methods.

In summary, the best approximation for your function will depend on the specific function and the specific interval and inner product you are using. Consider exploring different methods and techniques to find the most accurate representation for your data.