[PDF] periodic function contains no constant term.

What if a function is not periodic?

Any function that is not periodic is called aperiodic . A function f is said to be periodic if, for some nonzero constant P, it is the case that for all values of x in the domain. A nonzero constant P for which this is the case is called a period of the function.

Can a convergent Fourier series be used for periodic functions?

Some periodic functions can be described by Fourier series. For instance, for L2 functions, Carleson's theorem states that they have a pointwise ( Lebesgue) almost everywhere convergent Fourier series. Fourier series can only be used for periodic functions, or for functions on a bounded (compact) interval.

Are arbitrary periodic functions a sum of trigonometric functions with matching periods?

The trigonometric functions sine and cosine are common periodic functions, with period (see the figure on the right). The subject of Fourier series investigates the idea that an 'arbitrary' periodic function is a sum of trigonometric functions with matching periods.

How many times can a periodic function take on a value?

Periodic functions can take on values many times. More specifically, if a function is periodic with period , then for all in the domain of and all positive integers , If is a function with period , then , where is a non-zero real number such that is within the domain of , is periodic with period .