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In `F33f`_`[mathematics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Mathematics]`_`f, `!generalized functions`! are objects extending the notion of `F33f`_`[functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Function_(mathematics)]`_`f on real or complex numbers. There is more than one recognized theory, for example the theory of `F33f`_`[distributions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Distribution_(mathematics)]`_`f. Generalized functions are especially useful for treating `F33f`_`[discontinuous functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Discontinuous_function]`_`f more like `F33f`_`[smooth functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Smooth_function]`_`f, and describing discrete physical phenomena such as `F33f`_`[point charges`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Point_charge]`_`f. They are applied extensively, especially in `F33f`_`[physics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Physics]`_`f and `F33f`_`[engineering`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Engineering]`_`f. Important motivations have been the technical requirements of theories of `F33f`_`[partial differential equations`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Partial_differential_equation]`_`f and `F33f`_`[group representations`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Group_representation]`_`f.

A common feature of some of the approaches is that they build on `F33f`_`[operator`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Operator_(mathematics)]`_`f aspects of everyday, numerical functions. The early history is connected with some ideas on `F33f`_`[operational calculus`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Operational_calculus]`_`f, and some contemporary developments are closely related to `F33f`_`[Mikio Sato`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Mikio_Sato]`_`f's `F33f`_`[algebraic analysis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Algebraic_analysis]`_`f.

>>Contents

• `F0af`_`[Some early history`#some-early-history]`_`f

• `F0af`_`[Schwartz distributions`#schwartz-distributions]`_`f

• `F0af`_`[Algebras of generalized functions`#algebras-of-generalized-functions]`_`f

• `F0af`_`[Non-commutative algebra of generalized functions`#non-commutative-algebra-of-generalized-functions]`_`f

• `F0af`_`[Multiplication of distributions`#multiplication-of-distributions]`_`f

• `F0af`_`[Example: Colombeau algebra`#example-colombeau-algebra]`_`f

• `F0af`_`[Injection of Schwartz distributions`#injection-of-schwartz-distributions]`_`f

• `F0af`_`[Sheaf structure`#sheaf-structure]`_`f

• `F0af`_`[Microlocal analysis`#microlocal-analysis]`_`f

• `F0af`_`[Other theories`#other-theories]`_`f

• `F0af`_`[Topological groups`#topological-groups]`_`f

• `F0af`_`[Generalized section`#generalized-section]`_`f

• `F0af`_`[See also`#see-also]`_`f

• `F0af`_`[Books`#books]`_`f

• `F0af`_`[References`#references]`_`f

-─

>>Some early history

In the mathematics of the nineteenth century, aspects of generalized function theory appeared, for example in the definition of the `F33f`_`[Green's function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Green's_function]`_`f, in the `F33f`_`[Laplace transform`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Laplace_transform]`_`f, and in `F33f`_`[Riemann`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Riemann]`_`f's theory of `F33f`_`[trigonometric series`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Trigonometric_series]`_`f, which were not necessarily the `F33f`_`[Fourier series`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Fourier_series]`_`f of an `F33f`_`[integrable function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Integrable_function]`_`f. These were disconnected aspects of `F33f`_`[mathematical analysis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Mathematical_analysis]`_`f at the time.

The intensive use of the Laplace transform in engineering led to the `F33f`_`[heuristic`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Heuristic]`_`f use of symbolic methods, called `F33f`_`[operational calculus`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Operational_calculus]`_`f. Since justifications were given that used `F33f`_`[divergent series`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Divergent_series]`_`f, these methods were questionable from the point of view of `F33f`_`[pure mathematics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Pure_mathematics]`_`f. They are typical of later application of generalized function methods. An influential book on operational calculus was `F33f`_`[Oliver Heaviside`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Oliver_Heaviside]`_`f's `*Electromagnetic Theory`* of 1899.

When the `F33f`_`[Lebesgue integral`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Lebesgue_integral]`_`f was introduced, there was for the first time a notion of generalized function central to mathematics. An integrable function, in Lebesgue's theory, is equivalent to any other which is the same `F33f`_`[almost everywhere`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Almost_everywhere]`_`f. That means its value at each point is (in a sense) not its most important feature. In `F33f`_`[functional analysis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Functional_analysis]`_`f a clear formulation is given of the `*essential`* feature of an integrable function, namely the way it defines a `F33f`_`[linear functional`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Linear_functional]`_`f on other functions. This allows a definition of `F33f`_`[weak derivative`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Weak_derivative]`_`f.

During the late 1920s and 1930s further basic steps were taken. The `F33f`_`[Dirac delta function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Dirac_delta_function]`_`f was boldly defined by `F33f`_`[Paul Dirac`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Paul_Dirac]`_`f (an aspect of his `F33f`_`[scientific formalism`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Scientific_formalism]`_`f); this was to treat `F33f`_`[measures`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Measure_(mathematics)]`_`f, thought of as densities (such as `F33f`_`[charge density`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Charge_density]`_`f) like genuine functions. `F33f`_`[Sergei Sobolev`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Sergei_Sobolev]`_`f, working in `F33f`_`[partial differential equation theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Partial_differential_equation_theory]`_`f, defined the first rigorous theory of generalized functions in order to define `F33f`_`[weak solutions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Weak_solution]`_`f of partial differential equations (i.e. solutions which are generalized functions, but may not be ordinary functions).`:cite-ref-1[`F5bf`_`[1`#cite-note-1]`_`f] Others proposing related theories at the time were `F33f`_`[Salomon Bochner`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Salomon_Bochner]`_`f and `F33f`_`[Kurt Friedrichs`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Kurt_Friedrichs]`_`f. Sobolev's work was extended by `F33f`_`[Laurent Schwartz`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Laurent_Schwartz]`_`f.`:cite-ref-2[`F5bf`_`[2`#cite-note-2]`_`f]

>>Schwartz distributions

The most definitive development was the theory of `F33f`_`[distributions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Distribution_(mathematics)]`_`f developed by `F33f`_`[Laurent Schwartz`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Laurent_Schwartz]`_`f, systematically working out the principle of `F33f`_`[duality`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Dual_space]`_`f for `F33f`_`[topological vector spaces`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Topological_vector_space]`_`f. Its main rival in `F33f`_`[applied mathematics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Applied_mathematics]`_`f is `F33f`_`[mollifier`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Mollifier]`_`f theory, which uses sequences of smooth approximations (the '`F33f`_`[James Lighthill`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=James_Lighthill]`_`f' explanation).`:cite-ref-3[`F5bf`_`[3`#cite-note-3]`_`f]

This theory was very successful and is still widely used, but suffers from the main drawback that distributions cannot usually be multiplied: unlike most classical `F33f`_`[function spaces`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Function_space]`_`f, they do not form an `F33f`_`[algebra`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Algebra]`_`f. For example, it is meaningless to square the `F33f`_`[Dirac delta function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Dirac_delta_function]`_`f. Work of Schwartz from around 1954 showed this to be an intrinsic difficulty.

>>Algebras of generalized functions

Some solutions to the multiplication problem have been proposed. One is based on a simple definition of by Yu. V. Egorov`:cite-ref-yuvegorov1990-4-0[`F5bf`_`[4`#cite-note-yuvegorov1990-4]`_`f] (see also his article in Demidov's book in the book list below) that allows arbitrary operations on, and between, generalized functions.

Another solution allowing multiplication is suggested by the `F33f`_`[path integral formulation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Path_integral_formulation]`_`f of `F33f`_`[quantum mechanics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Quantum_mechanics]`_`f. Since this is required to be equivalent to the `F33f`_`[Schrödinger`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Schrödinger]`_`f theory of `F33f`_`[quantum mechanics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Quantum_mechanics]`_`f which is invariant under coordinate transformations, this property must be shared by path integrals. This fixes all products of generalized functions as shown by `F33f`_`[H. Kleinert`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Hagen_Kleinert]`_`f and A. Chervyakov.`:cite-ref-5[`F5bf`_`[5`#cite-note-5]`_`f] The result is equivalent to what can be derived from `F33f`_`[dimensional regularization`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Dimensional_regularization]`_`f.`:cite-ref-6[`F5bf`_`[6`#cite-note-6]`_`f]

Several constructions of algebras of generalized functions have been proposed, among others those by Yu. M. Shirokov `:cite-ref-shirokovalgebra1dim-7-0[`F5bf`_`[7`#cite-note-shirokovalgebra1dim-7]`_`f] and those by E. Rosinger, Y. Egorov, and R. Robinson. In the first case, the multiplication is determined with some regularization of generalized function. In the second case, the algebra is constructed as `*multiplication of distributions`*. Both cases are discussed below.

>>>Non-commutative algebra of generalized functions

The algebra of generalized functions can be built-up with an appropriate procedure of projection of a function F = F ( x ) {\\displaystyle F=F(x)} to its smooth F s m o o t h {\\displaystyle F_{\\rm {smooth}}} and its singular F s i n g u l a r {\\displaystyle F_{\\rm {singular}}} parts. The product of generalized functions F {\\displaystyle F} and G {\\displaystyle G} appears as

Such a rule applies to both the space of main functions and the space of operators which act on the space of the main functions. The associativity of multiplication is achieved; and the function signum is defined in such a way, that its square is unity everywhere (including the origin of coordinates). Note that the product of singular parts does not appear in the right-hand side of (`!`F33f`_`[1`#math-1]`_`f`!); in particular, δ δ ( x ) 2 = 0 {\\displaystyle \\delta (x)^{2}=0} . Such a formalism includes the conventional theory of generalized functions (without their product) as a special case. However, the resulting algebra is non-commutative: generalized functions signum and delta anticommute.`:cite-ref-shirokovalgebra1dim-7-1[`F5bf`_`[7`#cite-note-shirokovalgebra1dim-7]`_`f] Few applications of the algebra were suggested.`:cite-ref-goriaga-8-0[`F5bf`_`[8`#cite-note-goriaga-8]`_`f]`:cite-ref-tolok-9-0[`F5bf`_`[9`#cite-note-tolok-9]`_`f]

>>>Multiplication of distributions

The problem of `*multiplication of distributions`*, a limitation of the Schwartz distribution theory, becomes serious for `F33f`_`[non-linear`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Non-linear]`_`f problems.

Various approaches are used today. The simplest one is based on the definition of generalized function given by Yu. V. Egorov.`:cite-ref-yuvegorov1990-4-1[`F5bf`_`[4`#cite-note-yuvegorov1990-4]`_`f] Another approach to construct `F33f`_`[associative`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Associative]`_`f `F33f`_`[differential algebras`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Differential_algebra]`_`f is based on J.-F. Colombeau's construction: see `F33f`_`[Colombeau algebra`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Colombeau_algebra]`_`f. These are `F33f`_`[factor spaces`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Factor_space]`_`f

G = M / N {\\displaystyle G=M/N}

of "moderate" modulo "negligible" nets of functions, where "moderateness" and "negligibility" refers to growth with respect to the index of the family.

>>>Example: Colombeau algebra

A simple example is obtained by using the polynomial scale on `!N`!, s = { a m : N → → R , n ↦ ↦ n m ; m ∈ ∈ Z } {\\displaystyle s=\\{a_{m}:\\mathbb {N} \\to \\mathbb {R} ,n\\mapsto n^{m};~m\\in \\mathbb {Z} \\}} . Then for any semi normed algebra (E,P), the factor space will be

G s ( E , P ) = { f ∈ ∈ E N ∣ ∣ ∀ ∀ p ∈ ∈ P , ∃ ∃ m ∈ ∈ Z : p ( f n ) = o ( n m ) } { f ∈ ∈ E N ∣ ∣ ∀ ∀ p ∈ ∈ P , ∀ ∀ m ∈ ∈ Z : p ( f n ) = o ( n m ) } . {\\displaystyle G_{s}(E,P)={\\frac {\\{f\\in E^{\\mathbb {N} }\\mid \\forall p\\in P,\\exists m\\in \\mathbb {Z} :p(f_{n})=o(n^{m})\\}}{\\{f\\in E^{\\mathbb {N} }\\mid \\forall p\\in P,\\forall m\\in \\mathbb {Z} :p(f_{n})=o(n^{m})\\}}}.}

In particular, for (`*E`*, `*P`*)=(`!C`!,|.|) one gets (Colombeau's) generalized complex numbers (which can be "infinitely large" and "infinitesimally small" and still allow for rigorous arithmetics, very similar to `F33f`_`[nonstandard numbers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Non-standard_analysis]`_`f). For (`*E`*, `*P`*) = (`*C∞`*(`!R`!),{`*pk`*}) (where `*pk`* is the supremum of all derivatives of order less than or equal to `*k`* on the ball of radius `*k`*) one gets `F33f`_`[Colombeau's simplified algebra`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Colombeau_algebra]`_`f.

>>>Injection of Schwartz distributions

This algebra "contains" all distributions `*T`* of `* D' `* via the injection

`*j`*(`*T`*) = (φ`*n`* ∗ `*T`*)`*n`* + `*N`*,

where ∗ is the `F33f`_`[convolution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Convolution]`_`f operation, and

φ`*n`*(`*x`*) = `*n`* φ(`*nx`*).

This injection is `*non-canonical `*in the sense that it depends on the choice of the `F33f`_`[mollifier`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Mollifier]`_`f φ, which should be `*C∞`*, of integral one and have all its derivatives at 0 vanishing. To obtain a canonical injection, the indexing set can be modified to be `!N`! × `*D`*(`!R`!), with a convenient `F33f`_`[filter base`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Filter_base]`_`f on `*D`*(`!R`!) (functions of vanishing `F33f`_`[moments`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Moment_(mathematics)]`_`f up to order `*q`*).

>>>Sheaf structure

If (`*E`*,`*P`*) is a (pre-)`F33f`_`[sheaf`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Sheaf_(mathematics)]`_`f of semi normed algebras on some topological space `*X`*, then `*Gs`*(`*E`*, `*P`*) will also have this property. This means that the notion of `F33f`_`[restriction`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Restriction_(mathematics)]`_`f will be defined, which allows to define the `F33f`_`[support`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Support_(mathematics)]`_`f of a generalized function w.r.t. a subsheaf, in particular:

• For the subsheaf {0}, one gets the usual support (complement of the largest open subset where the function is zero).

• For the subsheaf `*E`* (embedded using the canonical (constant) injection), one gets what is called the `F33f`_`[singular support`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Singular_support]`_`f, i.e., roughly speaking, the closure of the set where the generalized function is not a smooth function (for `*E`* = `*C`*∞).

>>>Microlocal analysis

The `F33f`_`[Fourier transformation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Fourier_transformation]`_`f being (well-)defined for compactly supported generalized functions (component-wise), one can apply the same construction as for distributions, and define `F33f`_`[Lars Hörmander`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Lars_Hörmander]`_`f's `*`F33f`_`[wave front set`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Wave_front_set]`_`f`* also for generalized functions.

This has an especially important application in the analysis of `F33f`_`[propagation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Wave_propagation]`_`f of `F33f`_`[singularities`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Mathematical_singularity]`_`f.

>>Other theories

These include: the `*convolution quotient`* theory of `F33f`_`[Jan Mikusinski`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Jan_Mikusinski]`_`f, based on the `F33f`_`[field of fractions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Field_of_fractions]`_`f of `F33f`_`[convolution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Convolution]`_`f algebras that are `F33f`_`[integral domains`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Integral_domain]`_`f; and the theories of `F33f`_`[hyperfunctions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Hyperfunction]`_`f, based (in their initial conception) on boundary values of `F33f`_`[analytic functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Analytic_function]`_`f, and now making use of `F33f`_`[sheaf theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Sheaf_theory]`_`f.

>>Topological groups

Bruhat introduced a class of `F33f`_`[test functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Test_function]`_`f, the `F33f`_`[Schwartz–Bruhat functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Schwartz–Bruhat_function]`_`f, on a class of `F33f`_`[locally compact groups`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Locally_compact_group]`_`f that goes beyond the `F33f`_`[manifolds`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Manifold]`_`f that are the typical `F33f`_`[function domains`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Function_domain]`_`f. The applications are mostly in `F33f`_`[number theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Number_theory]`_`f, particularly to `F33f`_`[adelic algebraic groups`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Adelic_algebraic_group]`_`f. `F33f`_`[André Weil`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=André_Weil]`_`f rewrote `F33f`_`[Tate's thesis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Tate's_thesis]`_`f in this language, characterizing the zeta distribution on the `F33f`_`[idele group`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Idele_group]`_`f; and has also applied it to the `F33f`_`[explicit formula of an L-function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Explicit_formula_of_an_L-function]`_`f.

>>Generalized section

A further way in which the theory has been extended is as `!generalized sections`! of a smooth `F33f`_`[vector bundle`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Vector_bundle]`_`f. This is on the Schwartz pattern, constructing objects dual to the test objects, smooth sections of a bundle that have `F33f`_`[compact support`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Compact_support]`_`f. The most developed theory is that of `F33f`_`[De Rham currents`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=De_Rham_current]`_`f, dual to `F33f`_`[differential forms`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Differential_form]`_`f. These are homological in nature, in the way that differential forms give rise to `F33f`_`[De Rham cohomology`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=De_Rham_cohomology]`_`f. They can be used to formulate a very general `F33f`_`[Stokes' theorem`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Stokes'_theorem]`_`f.

>>See also

• `F33f`_`[Beppo-Levi space`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Beppo-Levi_space]`_`f

• `F33f`_`[Dirac delta function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Dirac_delta_function]`_`f

• `F33f`_`[Generalized eigenfunction`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Generalized_eigenfunction]`_`f

• `F33f`_`[Distribution (mathematics)`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Distribution_(mathematics)]`_`f

• `F33f`_`[Hyperfunction`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Hyperfunction]`_`f

• `F33f`_`[Laplacian of the indicator`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Laplacian_of_the_indicator]`_`f

• `F33f`_`[Rigged Hilbert space`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Rigged_Hilbert_space]`_`f

• `F33f`_`[Limit of a distribution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Limit_of_a_distribution]`_`f

• `F33f`_`[Generalized space`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Generalized_space]`_`f

• `F33f`_`[Ultradistribution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Ultradistribution]`_`f

>>Books

• `:citerefschwartz1950`aSchwartz, L. (1950). `*Théorie des distributions`*. Vol. 1. Paris: Hermann. `F33f`_`[OCLC`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=OCLC_(identifier)]`_`f 889264730. Vol. 2. `F33f`_`[OCLC`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=OCLC_(identifier)]`_`f 889391733

• `:citerefbeurling1961`aBeurling, A. (1961). `*On quasianalyticity and general distributions`* (multigraphed lectures). Summer Institute, Stanford University. `F33f`_`[OCLC`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=OCLC_(identifier)]`_`f 679033904.

• `:citerefgel-fandvilenkin1964`a`F33f`_`[Gelʹfand, Izrailʹ Moiseevič`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=I.M._Gel'fand]`_`f; Vilenkin, Naum Jakovlevič (1964). `*Generalized Functions`*. Vol. I–VI. Academic Press. `F33f`_`[OCLC`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=OCLC_(identifier)]`_`f 728079644.

• `:citerefh-rmander2015`aHörmander, L. (2015) [1990]. `*The Analysis of Linear Partial Differential Operators`* (2nd ed.). Springer. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-3-642-61497-2.

• H. Komatsu, Introduction to the theory of distributions, Second edition, Iwanami Shoten, Tokyo, 1983.

• `:citerefcolombeau2000`a`F33f`_`[Colombeau, J.-F.`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Colombeau_algebra]`_`f (2000) [1983]. `*New Generalized Functions and Multiplication of Distributions`*. Elsevier. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-0-08-087195-0.

• `:citerefvladimirovdrozhzhinovzav-yalov2012`aVladimirov, V.S.; Drozhzhinov, Yu. N.; Zav’yalov, B.I. (2012) [1988]. `*Tauberian theorems for generalized functions`*. Springer. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-94-009-2831-2.

• `:citerefoberguggenberger1992`aOberguggenberger, M. (1992). `*Multiplication of distributions and applications to partial differential equations`*. Longman. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-0-582-08733-0. `F33f`_`[OCLC`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=OCLC_(identifier)]`_`f 682138968.

• `:citerefmorimoto1993`aMorimoto, M. (1993). `*An introduction to Sato's hyperfunctions`*. American Mathematical Society. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-0-8218-8767-7.

• `:citerefdemidov2001`aDemidov, A.S. (2001). `*Generalized Functions in Mathematical Physics: Main Ideas and Concepts`*. Nova Science. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 9781560729051.

• `:citerefgrosserkunzingeroberguggenbergersteinbauer2013`aGrosser, M.; Kunzinger, M.; Oberguggenberger, Michael; Steinbauer, R. (2013) [2001]. `*Geometric theory of generalized functions with applications to general relativity`*. Springer. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-94-015-9845-3.

• `:citerefestradakanwal2012`aEstrada, R.; Kanwal, R. (2012). `*A distributional approach to asymptotics. Theory and applications`* (2nd ed.). Birkhäuser Boston. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-0-8176-8130-2.

• `:citerefvladimirov2002`aVladimirov, V.S. (2002). `*Methods of the theory of generalized functions`*. Taylor & Francis. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-0-415-27356-5.

• `:citerefkleinert2009`a`F33f`_`[Kleinert, H.`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Hagen_Kleinert]`_`f (2009). `*Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets`* (5th ed.). World Scientific. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 9789814273572. (online here). See Chapter 11 for products of generalized functions.

• `:citerefpilipovistankovicvindas2012`aPilipovi, S.; Stankovic, B.; Vindas, J. (2012). `*Asymptotic behavior of generalized functions`*. World Scientific. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 9789814366847.

>>References

`:cite-note-1`!1.`! `F0af`_`[↑`#cite-ref-1]`_`f `:citerefkolmogorovfomin1999`aKolmogorov, A. N.; Fomin, S. V. (1999) [1957]. `*Elements of the theory of functions and functional analysis`*. Mineola, N.Y.: Dover. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 0-486-40683-0. `F33f`_`[OCLC`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=OCLC_(identifier)]`_`f 44675353.

`:cite-note-2`!2.`! `F0af`_`[↑`#cite-ref-2]`_`f `:citerefschwartz1952`aSchwartz, L (1952). "Théorie des distributions". `*Bull. Amer. Math. Soc`*. `!58`!: 78–85. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1090/S0002-9904-1952-09555-0.

`:cite-note-3`!3.`! `F0af`_`[↑`#cite-ref-3]`_`f Halperin, I., & Schwartz, L. (1952). Introduction to the Theory of Distributions. Toronto: University of Toronto Press. (Short lecture by Halperin on Schwartz's theory)

`:cite-note-yuvegorov1990-4`!4.`! `F0af`_`[↑`#cite-ref-yuvegorov1990-4-0]`_`f `:citerefyu-v-egorov1990`aYu. V. Egorov (1990). "A contribution to the theory of generalized functions". `*Russian Math. Surveys`*. `!45`! (5): 1–49. `F33f`_`[Bibcode`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Bibcode_(identifier)]`_`f:1990RuMaS..45....1E. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1070/rm1990v045n05abeh002683. `F33f`_`[S2CID`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=S2CID_(identifier)]`_`f 250877163.

`:cite-note-5`!5.`! `F0af`_`[↑`#cite-ref-5]`_`f `:citerefh-kleinert-and-a-chervyakov2001`aH. Kleinert and A. Chervyakov (2001). "Rules for integrals over products of distributions from coordinate independence of path integrals" (PDF). `*Eur. Phys. J. C`*. `!19`! (4): 743–747. `F33f`_`[arXiv`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ArXiv_(identifier)]`_`f:quant-ph/0002067. `F33f`_`[Bibcode`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Bibcode_(identifier)]`_`f:2001EPJC...19..743K. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1007/s100520100600. `F33f`_`[S2CID`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=S2CID_(identifier)]`_`f 119091100.

`:cite-note-6`!6.`! `F0af`_`[↑`#cite-ref-6]`_`f `:citerefh-kleinert-and-a-chervyakov2000`aH. Kleinert and A. Chervyakov (2000). "Coordinate Independence of Quantum-Mechanical Path Integrals" (PDF). `*Phys. Lett`*. A 269 (1–2): 63. `F33f`_`[arXiv`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ArXiv_(identifier)]`_`f:quant-ph/0003095. `F33f`_`[Bibcode`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Bibcode_(identifier)]`_`f:2000PhLA..273....1K. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1016/S0375-9601(00)00475-8.

`:cite-note-shirokovalgebra1dim-7`!7.`! `F0af`_`[↑`#cite-ref-shirokovalgebra1dim-7-0]`_`f `:citerefyu-m-shirokov1979`aYu. M. Shirokov (1979). "Algebra of one-dimensional generalized functions". `*`F33f`_`[Theoretical and Mathematical Physics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Theoretical_and_Mathematical_Physics]`_`f`*. `!39`! (3): 291–301. `F33f`_`[Bibcode`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Bibcode_(identifier)]`_`f:1979TMP....39..471S. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1007/BF01017992. `F33f`_`[S2CID`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=S2CID_(identifier)]`_`f 189852974.

`:cite-note-goriaga-8`!8.`! `F0af`_`[↑`#cite-ref-goriaga-8-0]`_`f `:citerefo-g-goryagayu-m-shirokov1981`aO. G. Goryaga; Yu. M. Shirokov (1981). "Energy levels of an oscillator with singular concentrated potential". `*`F33f`_`[Theoretical and Mathematical Physics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Theoretical_and_Mathematical_Physics]`_`f`*. `!46`! (3): 321–324. `F33f`_`[Bibcode`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Bibcode_(identifier)]`_`f:1981TMP....46..210G. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1007/BF01032729. `F33f`_`[S2CID`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=S2CID_(identifier)]`_`f 123477107.

`:cite-note-tolok-9`!9.`! `F0af`_`[↑`#cite-ref-tolok-9-0]`_`f `:citerefg-k-tolokonnikov1982`aG. K. Tolokonnikov (1982). "Differential rings used in Shirokov algebras". `*`F33f`_`[Theoretical and Mathematical Physics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Theoretical_and_Mathematical_Physics]`_`f`*. `!53`! (1): 952–954. `F33f`_`[Bibcode`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Bibcode_(identifier)]`_`f:1982TMP....53..952T. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1007/BF01014789. `F33f`_`[S2CID`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=S2CID_(identifier)]`_`f 123078052.

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