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where denotes point-wise multiplication. It also works the other way around:
By applying the inverse Fourier transform , we can write:
and:
The relationships above are only valid for the form of the Fourier transform shown in the Proof section below. The transform may be normalized in other ways, in which case constant scaling factors (typically or ) will appear in the relationships above.
The proof here is shown for a particular normalization of the Fourier transform. As mentioned above, if the transform is normalized differently, then constant scaling factors will appear in the derivation.
Let belong to the Lp-space . Let be the Fourier transform of and be the Fourier transform of :
but must be "rapidly decreasing" towards and
in order to guarantee the existence of both, convolution and multiplication product.
Equivalently, if is a smooth "slowly growing"
ordinary function, it guarantees the existence of both, multiplication and convolution product.
[2][3][4].
In particular, every compactly supported tempered distribution,
such as the Dirac Delta, is "rapidly decreasing".
Equivalently, bandlimited functions, such as the function that is constantly
are smooth "slowly growing" ordinary functions.
If, for example, is the Dirac comb both equations yield the Poisson Summation Formula and if, furthermore, is the Dirac delta then is constantly one and these equations yield the Dirac comb identity.
Functions of discrete variable sequences
The analogous convolution theorem for discrete sequences and is:
Under certain conditions, a sub-sequence of is equivalent to linear (aperiodic) convolution of and , which is usually the desired result. (see Example) And when the transforms are efficiently implemented with the Fast Fourier transform algorithm, this calculation is much more efficient than linear convolution.
Convolution theorem for Fourier series coefficients
Two convolution theorems exist for the Fourier series coefficients of a periodic function:
The first convolution theorem states that if and are in , the Fourier series coefficients of the 2π-periodic convolution of and are given by:
The second convolution theorem states that the Fourier series coefficients of the product of and are given by the discrete convolution of the and sequences:
^
McGillem, Clare D.; Cooper, George R. (1984). Continuous and Discrete Signal and System Analysis (2 ed.). Holt, Rinehart and Winston. p. 118 (3-102). ISBN0-03-061703-0.
^Horváth, John (1966). Topological Vector Spaces and Distributions. Reading, MA: Addison-Wesley Publishing Company.
^Barros-Neto, José (1973). An Introduction to the Theory of Distributions. New York, NY: Dekker.
^Petersen, Bent E. (1983). Introduction to the Fourier Transform and Pseudo-Differential Operators. Boston, MA: Pitman Publishing.
Katznelson, Yitzhak (1976), An introduction to Harmonic Analysis, Dover, ISBN0-486-63331-4
Li, Bing; Babu, G. Jogesh (2019), "Convolution Theorem and Asymptotic Efficiency", A Graduate Course on Statistical Inference, New York: Springer, pp. 295–327, ISBN978-1-4939-9759-6