Fourier Series - Properties

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 :

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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.