= Integral transform
{wiki}
An <integral transform> constructs a new <function> by integrating an input against a kernel:
$$
(Tf)(s)=\int_X K(s,x)f(x)\,d\mu(x).
$$
The kernel $K$ specifies how each input location contributes to an output indexed by $s$. Whenever the <integrals> exist, <linearity> gives $T(af+bg)=aTf+bTg$. The <function spaces>, convergence conditions and normalization are part of the definition; a formal kernel alone need not define an invertible or even everywhere-defined <linear operator>.<nist-digital-library-of-mathematical-functions/1 Algebraic and Analytic Methods>[§1.14, Fourier, Laplace and Mellin integral-transform definitions]{ref}
The <Laplace transform> uses $K(s,t)=e^{-st}$ on nonnegative time. For $f(t)=e^{-at}$ it gives $1/(s+a)$ when $\operatorname{Re}(s+a)>0$. The <Fourier transform> decomposes a <function> into oscillatory <frequencies>, while the <Mellin transform> is adapted to changes of scale. A smoothing <Gaussian> kernel instead gives the <Weierstrass transform>.
The useful operation is often a change of representation: the <Laplace transform of a derivative> turns a constant-coefficient <differential equation> into an algebraic equation with initial data. The inverse step then requires a theorem, such as the <Bromwich inversion formula>, with its own regularity and convergence assumptions.
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