Tempered distribution
A tempered distribution is a continuous linear functional on the Schwartz space of rapidly decreasing smooth functions. Tempered distributions form a distinguished subspace of the space of distributions, characterized by growth conditions compatible with the Fourier transform. They provide a distributional setting for harmonic analysis, constant-coefficient partial differential equations, and the spectral description of translation-invariant operators.
The space of tempered distributions on (\mathbb{R}^n) is conventionally denoted by
[ \mathcal{S}'(\mathbb{R}^n), ]
where the prime denotes the continuous dual of (\mathcal{S}(\mathbb{R}^n)). The adjective “tempered” refers to the exclusion of generalized functions whose growth at infinity is too rapid to act continuously on every Schwartz function.
Definition
The Schwartz space (\mathcal{S}(\mathbb{R}^n)) consists of all functions
[ \varphi\in C^\infty(\mathbb{R}^n) ]
such that
[ p_{\alpha,\beta}(\varphi)
\sup_{x\in\mathbb{R}^n} \left|x^\alpha \partial^\beta\varphi(x)\right| <\infty ]
for every pair of multi-indices (\alpha) and (\beta). This condition requires each derivative of (\varphi) to decrease at infinity more rapidly than the reciprocal of any polynomial.
A linear functional
[ T:\mathcal{S}(\mathbb{R}^n)\longrightarrow\mathbb{C} ]
is a tempered distribution when it is continuous with respect to the topology generated by the seminorms (p_{\alpha,\beta}). Equivalently, there exist a constant (C>0) and a nonnegative integer (m) such that
[ |\langle T,\varphi\rangle| \leq C \sum_{\substack{|\alpha|\leq m\|\beta|\leq m}} \sup_{x\in\mathbb{R}^n} \left|x^\alpha\partial^\beta\varphi(x)\right| ]
for every (\varphi\in\mathcal{S}(\mathbb{R}^n)). The pairing (\langle T,\varphi\rangle) denotes the value of the functional (T) on the test function (\varphi).
This estimate expresses two finite constraints. Derivatives of the test function occur only up to a bounded order, while their behavior at infinity is measured using polynomial weights of bounded degree. The estimate does not imply that (T) is represented by an ordinary function, because singular functionals can satisfy the same continuity condition.
Historical development
Laurent Schwartz introduced the modern theory of distributions during the late 1940s and organized its systematic presentation in Théorie des distributions, published in 1950 and 1951. The tempered subspace arose from the requirement that the classical Fourier transform on rapidly decreasing functions extend naturally to generalized functions.
In a 1953 note, You Watanabe established that continuity on (\mathcal{S}(\mathbb{R}^n)) could be tested using the diagonal family of seminorms
[ q_m(\varphi)
\sum_{\substack{|\alpha|\leq m\|\beta|\leq m}} p_{\alpha,\beta}(\varphi), \qquad m\in\mathbb{N}. ]
This formulation is topologically equivalent to the full family (p_{\alpha,\beta}) and places the polynomial weight and differentiation order under a common index. It also yields the standard finite-seminorm estimate for an individual tempered distribution.
The tempered framework incorporated earlier methods associated with generalized functions and weak differentiation. Sergei Sobolev had developed generalized derivatives in connection with differential equations, while Schwartz’s dual-space construction supplied a unified test-function topology in which such derivatives could be treated as continuous functionals.
Regular tempered distributions
A locally integrable function (f) defines a distribution by
[ \langle T_f,\varphi\rangle
\int_{\mathbb{R}^n}f(x)\varphi(x),dx. ]
When (f) has at most polynomial growth in an appropriate integral sense, this functional is tempered. In particular, a measurable function satisfying
[ |f(x)|\leq C(1+|x|)^N ]
almost everywhere, for some constants (C) and (N), determines an element of (\mathcal{S}'(\mathbb{R}^n)). The rapid decrease of (\varphi) compensates for the polynomial growth of (f), making the defining integral absolutely convergent.
Every polynomial therefore determines a regular tempered distribution. Bounded functions also define tempered distributions, as do functions belonging to (L^p(\mathbb{R}^n)) for the usual range (1\leq p\leq\infty), with the action interpreted through the corresponding integral pairing.
Not every regular distribution is tempered. A locally integrable function with sufficiently rapid exponential growth can fail to act on all Schwartz functions, because rapid decrease in the Schwartz sense means decay beyond every polynomial rate rather than beyond every exponential rate.
Singular tempered distributions
The Dirac delta distribution at a point (a\in\mathbb{R}^n) is defined by
[ \langle \delta_a,\varphi\rangle=\varphi(a). ]
Point evaluation is continuous on the Schwartz space, so (\delta_a) is tempered. Its distributional derivatives are also tempered and satisfy
[ \langle \partial^\alpha\delta_a,\varphi\rangle
(-1)^{|\alpha|} \partial^\alpha\varphi(a). ]
More generally, every distribution with compact support is tempered. Compact support removes growth at infinity from the continuity estimate, while the local finite-order property of distributions controls the derivatives of the test function on a compact set.
Tempered distributions can also be represented structurally as finite sums of distributional derivatives of continuous functions having at most polynomial growth. This characterization connects the abstract dual-space definition with finite differentiation and controlled behavior at infinity.
Fourier transform
The Fourier transform maps (\mathcal{S}(\mathbb{R}^n)) continuously onto itself. Under the convention
[ \widehat{\varphi}(\xi)
\int_{\mathbb{R}^n} e^{-2\pi i x\cdot\xi}\varphi(x),dx, ]
the Fourier transform of a tempered distribution (T) is defined by duality:
[ \langle \widehat{T},\varphi\rangle
\langle T,\widehat{\varphi}\rangle. ]
Other normalization conventions replace (\widehat{\varphi}) on the right-hand side by the inverse transform or modify the exponential factor. Each convention produces the same theory after the corresponding constants and signs are adjusted.
The dual extension is an automorphism of (\mathcal{S}'(\mathbb{R}^n)). It converts distributional differentiation into multiplication by a polynomial:
[ \widehat{\partial^\alpha T}(\xi)
(2\pi i\xi)^\alpha\widehat{T}(\xi). ]
Conversely, multiplication of (T) by the coordinate monomial (x^\alpha) becomes differentiation in the frequency variable:
[ \widehat{x^\alpha T}
\left(-\frac{1}{2\pi i}\right)^{|\alpha|} \partial^\alpha\widehat{T}. ]
The Fourier transform of the Dirac distribution at the origin is the constant distribution (1), while the transform of the constant distribution is (\delta_0). A translated Dirac distribution transforms into the smooth oscillatory function (e^{-2\pi i a\cdot\xi}).
Differentiation and multiplication
Every tempered distribution possesses derivatives of arbitrary order. For a multi-index (\alpha), the derivative is determined by
[ \langle\partial^\alpha T,\varphi\rangle
(-1)^{|\alpha|} \langle T,\partial^\alpha\varphi\rangle. ]
Since differentiation acts continuously on the Schwartz space, the resulting functional remains tempered.
Multiplication by a polynomial also preserves temperedness. A broader multiplier class consists of smooth functions whose derivatives grow no faster than suitable polynomials. Such a function (g) acts through
[ \langle gT,\varphi\rangle
\langle T,g\varphi\rangle, ]
because multiplication by (g) then maps the Schwartz space continuously into itself.
Convolution requires additional restrictions. The convolution of a tempered distribution with a Schwartz function is well defined and produces a smooth function with at most polynomial growth. Convolution between two arbitrary tempered distributions is not generally defined, although it becomes available when one factor has compact support or when their Fourier transforms admit a suitable product.
Topological interpretation
The Schwartz space is a Fréchet space: it is complete, metrizable, and locally convex under its defining seminorms. It is also a nuclear space, a property that governs tensor products and distribution kernels. Its continuous dual can be equipped with either the weak dual topology of pointwise convergence on test functions or the strong dual topology of uniform convergence on bounded subsets of (\mathcal{S}).
Convergence in the usual distributional sense is expressed by
[ T_j\longrightarrow T \quad\Longleftrightarrow\quad \langle T_j,\varphi\rangle \longrightarrow \langle T,\varphi\rangle ]
for every Schwartz function (\varphi). Strong convergence imposes uniformity over bounded families of test functions and consequently carries more topological information.
Israel Gelfand and Georgy Shilov subsequently placed tempered distributions within a wider hierarchy of generalized-function spaces organized by test-function growth and analyticity conditions. Their treatment related the Schwartz dual to nuclear-space methods and to kernel constructions used in functional analysis.
Differential equations
A constant-coefficient linear differential operator has the form
[ P(D)=\sum_{|\alpha|\leq m}a_\alpha\partial^\alpha. ]
On applying the Fourier transform, the equation
[ P(D)u=f ]
becomes the distributional multiplier equation
[ P(2\pi i\xi)\widehat{u}(\xi)=\widehat{f}(\xi). ]
This correspondence turns differentiation in physical space into algebraic multiplication in frequency space. Zeros of the polynomial symbol (P(2\pi i\xi)) remain distributionally significant because division by the symbol can generate singularities supported on its zero set.
Tempered distributions also support the definition of fundamental solutions whose growth is controlled at infinity. The general existence theorem for constant-coefficient operators is formulated in the broader distribution space, while tempered fundamental solutions arise under additional Fourier-analytic constructions.