# Relation Between Fourier and Taylor Series

## Abstract

Infinite series can converge in various ways to give the resultant function. Particularly, here, we consider the Fourier series and compare it with its Taylor equivalent both of which are convergent infinite series in their own rights. However, these are valid under separate limiting conditions. We consider what happens if we try to derive one series from the other or see if such a derivation is possible at all and its implications. An expansion of the Dirichlet kernel, while using a form of the Dirac delta function has been shown to yield the Taylor series in its form. However, it introduced certain restrictions on both the local and global nature of such a function.

### Keywords

Fourier series Taylor series Dirichlet kernel Dirac delta function Heaviside step functionThe foundation of Fourier series^{1} is based on the principle of orthogonality of functions. Now, the coefficients for the Fourier Series can easily be derived by using the orthogonality condition. If we try to apply a similar analogy to Taylor Series coefficients, then the function which obtains all the Taylor coefficients turns out to be not quite a function, but a distribution called the Dirac delta function. This will explicitly be used later to derive one from the other later.

*by parts*. Performing the first 2 iterations gives us:

^{2}

Thus on expanding, it looks very similar to the Taylor series, but not quite of the same form. However, on closer inspection, we get the same form between \( - \pi \) to \(\pi \).

It must be noted that the above formulation of equality holds only for the specific transformation relation. Generalizing, this shows that the global properties of Fourier series can be matched by the local properties of expanding a function using a certain expansion coefficient. This above formulation could be replaced for a general Fourier series with the period replacing the limits \(-\pi \) to \(\pi \).

Thus, a neat relation has been shown to exist between a function’s Fourier and Taylor series in a certain domain.

## Footnotes

## Copyright information

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