Properties
We say that ƒ belongs to if ƒ is a 2π-periodic function on R which is k times differentiable, and its kth derivative is continuous.
- If ƒ is a 2π-periodic odd function, then for all n.
- If ƒ is a 2π-periodic even function, then for all n.
- If ƒ is integrable, and This result is known as the Riemann–Lebesgue lemma.
- A doubly infinite sequence in is the sequence of Fourier coefficients of a function in if and only if it is a convolution of two sequences in . See
- If, then the Fourier coefficients of the derivative can be expressed in terms of the Fourier coefficients of the function, via the formula .
- If, then . In particular, since tends to zero, we have that tends to zero, which means that the Fourier coefficients converge to zero faster than the kth power of n.
- Parseval's theorem. If, then .
- Plancherel's theorem. If are coefficients and then there is a unique function such that for every n.
- The first convolution theorem states that if ƒ and g are in L1, then, where ƒ ∗ g denotes the 2π-periodic convolution of ƒ and g. (The factor is not necessary for 1-periodic functions.)
- The second convolution theorem states that .
- The Poisson summation formula states that the periodic summation of a function, has a Fourier series representation whose coefficients are proportional to discrete samples of the continuous Fourier transform of :
- .
- Similarly, the periodic summation of has a Fourier series representation whose coefficients are proportional to discrete samples of, a fact which provides a pictorial understanding of aliasing and the famous sampling theorem.
- Also see variants of Fourier analysis.
Read more about this topic: Fourier Series
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (18031882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (16321704)