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In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the product of their Fourier transforms. More generally, convolution in one domain (e.g., time domain ) equals point-wise multiplication in the other domain (e.g., frequency domain ).
The Laplace transform is used frequently in engineering and physics; the output of a linear time-invariant system can be calculated by convolving its unit impulse response with the input signal. Performing this calculation in Laplace space turns the convolution into a multiplication; the latter being easier to solve because of its algebraic form.
This theorem is proved by applying the inverse Laplace transform on the convolution theorem in form of the cross-correlation. Let f ( t ) {\displaystyle f(t)} be a function with bilateral Laplace transform F ( s ) {\displaystyle F(s)} in the strip of convergence α < ℜ s < β {\displaystyle \alpha <\Re s<\beta } .
Proof (using convolution theorem): () ... Versions of this theorem also hold for the Laplace transform, two-sided Laplace transform, Z-transform and Mellin transform.
The Laplace–Stieltjes transform in the case of a scalar-valued function is thus seen to be a special case of the Laplace transform of a Stieltjes measure. To wit, = (). In particular, it shares many properties with the usual Laplace transform. For instance, the convolution theorem holds:
In mathematics, the Riemann–Lebesgue lemma, named after Bernhard Riemann and Henri Lebesgue, states that the Fourier transform or Laplace transform of an L 1 function vanishes at infinity. It is of importance in harmonic analysis and asymptotic analysis .
The Riesz potential can be defined more generally in a weak sense as the convolution I α f = f ∗ K α {\displaystyle I_{\alpha }f=f*K_{\alpha }} where K α is the locally integrable function:
The multidimensional Laplace transform is useful for the solution of boundary value problems. Boundary value problems in two or more variables characterized by partial differential equations can be solved by a direct use of the Laplace transform. [3] The Laplace transform for an M-dimensional case is defined [3] as