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  2. Convolution theorem - Wikipedia

    en.wikipedia.org/wiki/Convolution_theorem

    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 ).

  3. Laplace transform - Wikipedia

    en.wikipedia.org/wiki/Laplace_transform

    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.

  4. Two-sided Laplace transform - Wikipedia

    en.wikipedia.org/wiki/Two-sided_Laplace_transform

    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 } .

  5. Convolution - Wikipedia

    en.wikipedia.org/wiki/Convolution

    Proof (using convolution theorem): () ... Versions of this theorem also hold for the Laplace transform, two-sided Laplace transform, Z-transform and Mellin transform.

  6. Laplace–Stieltjes transform - Wikipedia

    en.wikipedia.org/wiki/Laplace–Stieltjes_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:

  7. Riemann–Lebesgue lemma - Wikipedia

    en.wikipedia.org/wiki/Riemann–Lebesgue_lemma

    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 .

  8. Riesz potential - Wikipedia

    en.wikipedia.org/wiki/Riesz_potential

    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:

  9. Multidimensional transform - Wikipedia

    en.wikipedia.org/wiki/Multidimensional_transform

    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