Integral transforms, operational calculus

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• ConceptIntegral transform

  An integral transform turns a function $f$ into a new function $Tf$ by integrating $f$ against a fixed function of two variables, the kernel. In symbols, $(Tf)(u) = \int f(t) K(t, u)\, dt$. The Laplace transform, the Fourier transform and the Mellin transform are the main examples. A good transform turns a hard operation, such as differentiation or convolution, into an easy one, such as multiplication. It can be undone by an inverse transform.

• ConceptKernel of an integral transform

  The kernel of an integral transform is the function $K(t, u)$ of two variables that is integrated against the input function. It determines the transform completely: different transforms differ only in their kernels and in their domains of integration. Properties of the kernel, such as symmetry or decay, decide which functions can be transformed and whether the transform can be inverted.

• ConceptTransform pair

  A transform pair is a function $f$ together with its image $F = Tf$ under an integral transform $T$. It is often written $f \leftrightarrow F$ or $f(t) \leftrightarrow F(s)$. Tables of transform pairs are the main working tool of the Laplace transform and the Fourier transform. They let one go back from $F$ to $f$ without computing an inverse transform.

• ConceptInverse transform

  The inverse transform of an integral transform $T$ recovers a function $f$ from its image $Tf$. It exists when $T$ is one-to-one on the class of functions in use. Often the inverse is itself an integral transform with its own kernel. Examples are the Fourier inversion formula and the Bromwich integral for the inverse Laplace transform. A typical use of a transform is: transform the problem, solve it on the transform side, then apply the inverse transform.

• ConceptIntegral operator

  An integral operator is a linear map between spaces of functions that acts by integration against a kernel: $(Tf)(x) = \int K(x, y) f(y)\, dy$. Integral operators are the infinite-dimensional analogue of matrices. They appear as integral transforms, as solution operators of differential equations, and in Volterra integral equations and Fredholm integral equations. With a square-integrable kernel they are compact operators, which gives them a spectral theory close to that of matrices.

• ConceptSymmetric and self-reciprocal kernels

  A kernel $K(x, y)$ of an integral transform is symmetric when it does not change if $x$ and $y$ are swapped. It is self-reciprocal (a Fourier kernel) when the transform is its own inverse: applying it twice gives back the original function. The Fourier cosine and sine kernels and the Hankel transform kernel $\sqrt{xy}\, J_\nu(xy)$ are both symmetric and self-reciprocal. For such kernels the transform and its inverse transform are given by the same formula, which makes the theory and the tables symmetric.