Series Approximation of Functions: Taylor/McLaurin, Fourier & More

[gulsen]

I wonder if there's any other series that can be used to approximate a function, other that Taylor/McLaurin and Fourier. For instance, can we expand a function in terms of Gaussians (Aexp(-bx^2))? Maybe something else?

 

[daveb]

From what I remember of functional analysis (it was a while ago), as long as the individual functions in the series form a basis for the function space, then yes, you can expand it in terms of that series. For example, the sine/cosine functions form a basis and are used in Fourier expansions.

 

[tacman]

Yes, there are indeed other series that can be used to approximate a function. One example is the Chebyshev series, which uses a combination of Chebyshev polynomials to approximate a function. Another example is the Legendre series, which uses Legendre polynomials for approximation. Both of these series have their own unique properties and can be useful in different scenarios.

There are also other types of series expansions that can be used, such as the Hermite series, which uses Hermite polynomials, and the Bessel series, which uses Bessel functions. These series expansions are often used in physics and engineering applications.

Regarding your question about expanding a function in terms of Gaussians, yes, it is possible to do so. This type of series expansion is called a Gaussian series or a Gaussian basis expansion. It is commonly used in the field of machine learning and signal processing, where it is used to approximate functions with a sum of Gaussian functions.

In summary, while Taylor/McLaurin and Fourier series are commonly used for function approximation, there are many other series expansions that can also be used, each with their own advantages and applications.