In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numbers of modulus one), the finite abelian groups (with the discrete topology), and the additive group of the integers (also with the discrete topology), the real numbers, and every finite-dimensional vector space over the reals or a p-adic field. The Pontryagin dual of a locally compact abelian group is the locally compact abelian topological group, consisting of the continuous group homomorphisms from the group to the circle group, with the operation of pointwise multiplication and the topology of uniform convergence on compact sets. The Pontryagin duality theorem establishes Pontryagin duality by stating that any locally compact abelian group is naturally isomorphic with its bidual (the dual of its dual). The Fourier inversion theorem is a special case of this theorem. The subject is named after Lev Pontryagin, who laid down the foundations for the theory of locally compact abelian groups and their duality during his early mathematical works in 1934. Pontryagin's treatment relied on the groups being second-countable and either compact or discrete. This was improved to cover the general locally compact abelian groups by Egbert van Kampen in 1935 and André Weil in 1940.
Introduction Pontryagin duality places in a unified context a number of observations about functions on the real line or on finite abelian groups:
Suitably regular complex-valued periodic functions on the real line have Fourier series and these functions can be recovered from their Fourier series; Suitably regular complex-valued functions on the real line have Fourier transforms that are also functions on the real line and, just as for periodic functions, these functions can be recovered from their Fourier transforms; and Complex-valued functions on a finite abelian group have discrete Fourier transforms, which are functions on the dual group, which is a (non-canonically) isomorphic group. Moreover, any function on a finite abelian group can be recovered from its discrete Fourier transform. The theory, introduced by Lev Pontryagin and combined with the Haar measure introduced by John von Neumann, André Weil and others depends on the theory of the dual group of a locally compact abelian group. It is analogous to the dual vector space of a vector space: a finite-dimensional vector space V {\displaystyle V} and its dual vector space V ∗ {\displaystyle V^{*}} are not naturally isomorphic, but the endomorphism algebra (matrix algebra) of one is isomorphic to the opposite of the endomorphism algebra of the other: End ( V ) ≅ End ( V ∗ ) op , {\displaystyle {\text{End}}(V)\cong {{\text{End}}(V^{*})}^{\text{op}},} via the transpose. Similarly, a group G {\displaystyle G} and its dual group G ^ {\displaystyle {\widehat {G}}} are not in general isomorphic, but their endomorphism rings are opposite to each other: End ( G ) ≅ End ( G ^ ) op {\displaystyle {\text{End}}(G)\cong {\text{End}}({\widehat {G}})^{\text{op}}} . More categorically, this is not just an isomorphism of endomorphism algebras, but a contravariant equivalence of categories – see § Categorical considerations.
Definition
A locally compact group is a topological group whose underlying space is locally compact and Hausdorff, and it is called abelian if the underlying group is abelian. Examples of locally compact abelian groups include finite abelian groups, the integers (both for the discrete topology, which is also induced by the usual metric), the real numbers, the circle group T (both with their usual metric topology), and also the p-adic numbers (with their usual p-adic topology). For a locally compact abelian group G {\displaystyle G} , the Pontryagin dual is the group G ^ {\displaystyle {\widehat {G}}} of continuous group homomorphisms from G {\displaystyle G} to the circle group T {\displaystyle T} . That is,
G ^ := Hom ( G , T ) . {\displaystyle {\widehat {G}}:=\operatorname {Hom} (G,T).}
The Pontryagin dual G ^ {\displaystyle {\widehat {G}}} is usually endowed with the topology given by uniform convergence on compact sets (that is, the topology induced by the compact-open topology on the space of all continuous functions from G {\displaystyle G} to T {\displaystyle T} ).
Historical development Before 1938, several developments contributed to what became known as Pontryagin duality. Pavel Alexandrov (late 1920s) developed the theory of inverse systems (projective limits) in topology, including so-called projection spectra. While not directly part of duality theory for topological groups, these methods introduced systematic use of limiting processes that later became important in the study of non-finitely generated groups and duality phenomena. Alfréd Haar (1933) introduced Haar measure in Der Maßbegriff in der Theorie der kontinuierlichen Gruppen, providing a translation-invariant integration theory on locally compact groups. This became a fundamental tool for harmonic analysis and for the later formulation of Pontryagin duality in full generality. Raymond Paley and Norbert Wiener (1933–1934), in work culminating in Fourier Transforms in the Complex Domain (1934), systematically studied Fourier transforms and characters of abelian groups (primarily discrete or Euclidean). They emphasized a duality between a group and its character group, though their framework did not yet extend to general locally compact abelian groups. James Waddell Alexander II (1934–1935) clarified duality phenomena in algebraic topology and introduced early cohomological ideas, notably in On the Chains of a Complex and Their Duals. While not directly part of Pontryagin duality, this work contributed to the broader mathematical notion of duality. John von Neumann (1934) studied almost periodic functions on groups and extended harmonic analysis beyond countable settings, removing certain separability assumptions and anticipating the need for a general locally compact framework. Egbert van Kampen (1935), in Locally Bicompact Abelian Groups and Their Character Groups, refined the topology on character groups and removed countability restrictions, bringing the theory close to its modern formulation. Claude Chevalley (1936) applied duality ideas in class field theory, demonstrating their arithmetic significance. Finally, Lev Pontryagin (1931–1934) established the core results: the definition of the dual group as the group of continuous characters, the natural pairing between a locally compact abelian group and its dual, and the duality theorem identifying a group with its double dual in the compact and discrete cases. These results were soon extended to all locally compact abelian groups.
Examples The Pontryagin dual of a finite cyclic group is isomorphic to itself.
Z / n Z ^ ≅ Z / n Z {\displaystyle {\widehat {\mathbb {Z} /n\mathbb {Z} }}\cong \mathbb {Z} /n\mathbb {Z} }
The Pontryagin dual of the group of integers is the circle group, and the Pontryagin dual of the circle group is the group of integers.
Z ^ ≅ T , T ^ ≅ Z {\displaystyle {\widehat {\mathbb {Z} }}\cong T,~~{\widehat {T}}\cong \mathbb {Z} }
The Pontryagin dual of the group of real numbers is itself.
R ^ ≅ R {\displaystyle {\widehat {\mathbb {R} }}\cong \mathbb {R} }
The Pontryagin dual of the group of p-adic integers Z p {\displaystyle \mathbb {Z} _{p}} is the Prüfer p-group Z ( p ∞ ) {\displaystyle \mathbb {Z} (p^{\infty })} , and the Pontryagin dual of the Prüfer p-group is the group of p-adic integers.
Z p ^ ≅ Z ( p ∞ ) , Z ( p ∞ ) ^ ≅ Z p {\displaystyle {\widehat {\mathbb {Z} _{p}}}\cong \mathbb {Z} (p^{\infty }),~~{\widehat {\mathbb {Z} (p^{\infty })}}\cong \mathbb {Z} _{p}}
Pontryagin duality theorem
Canonical means that there is a naturally defined map ev G : G → G ^ ^ {\displaystyle \operatorname {ev} _{G}\colon G\to {\widehat {\widehat {G}}}} ; more importantly, the map should be functorial in G {\displaystyle G} . For the multiplicative character χ {\displaystyle \chi } of the group G {\displaystyle G} , the canonical isomorphism ev G {\displaystyle \operatorname {ev} _{G}} is defined on x ∈ G {\displaystyle x\in G} as follows:
ev G ( x ) ( χ ) = χ ( x ) ∈ T . {\displaystyle \operatorname {ev} _{G}(x)(\chi )=\chi (x)\in \mathbb {T} .}
That is, ev G ( x ) : ( χ ↦ χ ( x ) ) . {\displaystyle \operatorname {ev} _{G}(x):(\chi \mapsto \chi (x)).}
In other words, each group element x {\displaystyle x} is identified to the evaluation character on the dual. This is strongly analogous to the canonical isomorphism between a finite-dimensional vector space and its double dual, V ≅ V ∗ ∗ {\displaystyle V\cong V^{**}} , and it is worth mentioning that any vector space V {\displaystyle V} is an abelian group. If G {\displaystyle G} is a finite abelian group, then G ≅ G ^ {\displaystyle G\cong {\widehat {G}}} but this isomorphism is not canonical. Making this statement precise (in general) requires thinking about dualizing not only on groups, but also on maps between the groups, in order to treat dualization as a functor and prove the identity functor and the dualization functor are not naturally equivalent. Also the duality theorem implies that for any group (not necessarily finite) the dualization functor is an exact functor.
Pontryagin duality and the Fourier transform
Haar measure
One of the most remarkable facts about a locally compact group G {\displaystyle G} is that it carries an essentially unique natural measure, the Haar measure, which allows one to consistently measure the "size" of sufficiently regular subsets of G {\displaystyle G} . "Sufficiently regular subset" here means a Borel set; that is, an element of the σ-algebra generated by the compact sets. More precisely, a right Haar measure on a locally compact group G {\displaystyle G} is a countably additive measure μ defined on the Borel sets of G {\displaystyle G} which is right invariant in the sense that μ ( A x ) = μ ( A ) {\displaystyle \mu (Ax)=\mu (A)} for x {\displaystyle x} an element of G {\displaystyle G} and A {\displaystyle A} a Borel subset of G {\displaystyle G} and also satisfies some regularity conditions (spelled out in detail in the article on Haar measure). Except for positive scaling factors, a Haar measure on G {\displaystyle G} is unique. The Haar measure on G {\displaystyle G} allows us to define the notion of integral for (complex-valued) Borel functions defined on the group. In particular, one may consider various Lp spaces associated to the Haar measure μ {\displaystyle \mu } . Specifically,
L μ p ( G ) = { ( f : G → C ) | ∫ G | f ( x ) | p d μ ( x ) < ∞ } . {\displaystyle {\mathcal {L}}_{\mu }^{p}(G)=\left\{(f:G\to \mathbb {C} )\ {\Big |}\ \int _{G}|f(x)|^{p}\ d\mu (x)<\infty \right\}.}
Note that, since any two Haar measures on G {\displaystyle G} are equal up to a scaling factor, this L p {\displaystyle L^{p}} -space is independent of the choice of Haar measure and thus perhaps could be written as L p ( G ) {\displaystyle L^{p}(G)} . However, the L p {\displaystyle L^{p}} -norm on this space depends on the choice of Haar measure, so if one wants to talk about isometries it is important to keep track of the Haar measure being used.
Fourier transform and Fourier inversion formula for L1-functions The dual group of a locally compact abelian group is used as the underlying space for an abstract version of the Fourier transform. If f ∈ L 1 ( G ) {\displaystyle f\in L^{1}(G)} , then the Fourier transform is the function f ^ {\displaystyle {\widehat {f}}} on G ^ {\displaystyle {\widehat {G}}} defined by
f ^ ( χ ) = ∫ G f ( x ) χ ( x ) ¯ d μ ( x ) , {\displaystyle {\widehat {f}}(\chi )=\int _{G}f(x){\overline {\chi (x)}}\ d\mu (x),}
where the integral is relative to Haar measure μ {\displaystyle \mu } on G {\displaystyle G} . This is also denoted ( F f ) ( χ ) {\displaystyle ({\mathcal {F}}f)(\chi )} . Note the Fourier transform depends on the choice of Haar measure. It is not too difficult to show that the Fourier transform of an L 1 {\displaystyle L^{1}} function on G {\displaystyle G} is a bounded continuous function on G ^ {\displaystyle {\widehat {G}}} which vanishes at infinity.
The inverse Fourier transform of an integrable function on G ^ {\displaystyle {\widehat {G}}} is given by
g ˇ ( x ) = ∫ G ^ g ( χ ) χ ( x ) d ν ( χ ) , {\displaystyle {\check {g}}(x)=\int _{\widehat {G}}g(\chi )\chi (x)\ d\nu (\chi ),}
where the integral is relative to the Haar measure ν {\displaystyle \nu } on the dual group G ^ {\displaystyle {\widehat {G}}} . The measure ν {\displaystyle \nu } on G ^ {\displaystyle {\widehat {G}}} that appears in the Fourier inversion formula is called the dual measure to μ {\displaystyle \mu } and may be denoted μ ^ {\displaystyle {\widehat {\mu }}} . The various Fourier transforms can be classified in terms of their domain and transform domain (the group and dual group) as follows (note that T {\displaystyle \mathbb {T} } is Circle group):
As an example, suppose G = R n {\displaystyle G=\mathbb {R} ^{n}} , so we can think about G ^ {\displaystyle {\widehat {G}}} as R n {\displaystyle \mathbb {R} ^{n}} by the pairing ( v , w ) ↦ e i v ⋅ w . {\displaystyle (\mathbf {v} ,\mathbf {w} )\mapsto e^{i\mathbf {v} \cdot \mathbf {w} }.} If μ {\displaystyle \mu } is the Lebesgue measure on Euclidean space, we obtain the ordinary Fourier transform on R n {\displaystyle \mathbb {R} ^{n}} and the dual measure needed for the Fourier inversion formula is μ ^ = ( 2 π ) − n μ {\displaystyle {\widehat {\mu }}=(2\pi )^{-n}\mu } . If we want to get a Fourier inversion formula with the same measure on both sides (that is, since we can think about R n {\displaystyle \mathbb {R} ^{n}} as its own dual space we can ask for μ ^ {\displaystyle {\widehat {\mu }}} to equal μ {\displaystyle \mu } ) then we need to use
μ = ( 2 π ) − n 2 × Lebesgue measure μ ^ = ( 2 π ) − n 2 × Lebesgue measure {\displaystyle {\begin{aligned}\mu &=(2\pi )^{-{\frac {n}{2}}}\times {\text{Lebesgue measure}}\\{\widehat {\mu }}&=(2\pi )^{-{\frac {n}{2}}}\times {\text{Lebesgue measure}}\end{aligned}}}
However, if we change the way we identify R n {\displaystyle \mathbb {R} ^{n}} with its dual group, by using the pairing
( v , w ) ↦ e 2 π i v ⋅ w , {\displaystyle (\mathbf {v} ,\mathbf {w} )\mapsto e^{2\pi i\mathbf {v} \cdot \mathbf {w} },}
then Lebesgue measure on R n {\displaystyle \mathbb {R} ^{n}} is equal to its own dual measure. This convention minimizes the number of factors of 2 π {\displaystyle 2\pi } that show up in various places when computing Fourier transforms or inverse Fourier transforms on Euclidean space. (In effect it limits the 2 π {\displaystyle 2\pi } only to the exponent rather than as a pre-factor outside the integral sign.) Note that the choice of how to identify R n {\displaystyle \mathbb {R} ^{n}} with its dual group affects the meaning of the term "self-dual function", which is a function on R n {\displaystyle \mathbb {R} ^{n}} equal to its own Fourier transform: using the classical pairing ( v , w ) ↦ e i v ⋅ w {\displaystyle (\mathbf {v} ,\mathbf {w} )\mapsto e^{i\mathbf {v} \cdot \mathbf {w} }} the function e − 1 2 x 2 {\displaystyle e^{-{\frac {1}{2}}x^{2}}} is self-dual. But using the pairing, which keeps the pre-factor as unity, ( v , w ) ↦ e 2 π i v ⋅ w {\displaystyle (\mathbf {v} ,\mathbf {w} )\mapsto e^{2\pi i\mathbf {v} \cdot \mathbf {w} }} makes e − π x 2 {\displaystyle e^{-\pi x^{2}}} self-dual instead. This second definition for the Fourier transform has the advantage that it maps the multiplicative identity to the convolution identity, which is useful as L 1 {\displaystyle L^{1}} is a convolution algebra. See the next section on the group algebra. In addition, this form is also necessarily isometric on L 2 {\displaystyle L^{2}} spaces. See below at Plancherel and L2 Fourier inversion theorems.
Group algebra
The space of integrable functions on a locally compact abelian group G {\displaystyle G} is an algebra, where multiplication is convolution: the convolution of two integrable functions f {\displaystyle f} and g {\displaystyle g} is defined as
( f ∗ g ) ( x ) = ∫ G f ( x − y ) g ( y ) d μ ( y ) . {\displaystyle (f*g)(x)=\int _{G}f(x-y)g(y)\ d\mu (y).}
This algebra is referred to as the Group Algebra of G {\displaystyle G} . By the Fubini–Tonelli theorem, the convolution is submultiplicative with respect to the L 1 {\displaystyle L^{1}} norm, making L 1 ( G ) {\displaystyle L^{1}(G)} a Banach algebra. The Banach algebra L 1 ( G ) {\displaystyle L^{1}(G)} has a multiplicative identity element if and only if G {\displaystyle G} is a discrete group, namely the function that is 1 at the identity and zero elsewhere. In general, however, it has an approximate identity which is a net (or generalized sequence) { e i } i ∈ I {\displaystyle \{e_{i}\}_{i\in I}} indexed on a directed set I {\displaystyle I} such that f ∗ e i → f . {\displaystyle f*e_{i}\to f.}
The Fourier transform takes convolution to multiplication, i.e. it is a homomorphism of abelian Banach algebras L 1 ( G ) → C 0 ( G ^ ) {\displaystyle L^{1}(G)\to C_{0}\left({\widehat {G}}\right)} (of norm ≤ 1):
F ( f ∗ g ) ( χ ) = F ( f ) ( χ ) ⋅ F ( g ) ( χ ) . {\displaystyle {\mathcal {F}}(f*g)(\chi )={\mathcal {F}}(f)(\chi )\cdot {\mathcal {F}}(g)(\chi ).}
In particular, to every group character on G {\displaystyle G} corresponds a unique multiplicative linear functional on the group algebra defined by
f ↦ f ^ ( χ ) . {\displaystyle f\mapsto {\widehat {f}}(\chi ).}
It is an important property of the group algebra that these exhaust the set of non-trivial (that is, not identically zero) multiplicative linear functionals on the group algebra; see section 34 of Loomis (1953). This means the Fourier transform is a special case of the Gelfand transform.
Plancherel and L2 Fourier inversion theorems
As we have stated, the dual group of a locally compact abelian group is a locally compact abelian group in its own right and thus has a Haar measure, or more precisely a whole family of scale-related Haar measures.
Since the complex-valued continuous functions of compact support on G {\displaystyle G} are L 2 {\displaystyle L^{2}} -dense, there is a unique extension of the Fourier transform from that space to a unitary operator
F : L μ 2 ( G ) → L ν 2 ( G ^ ) . {\displaystyle {\mathcal {F}}:L_{\mu }^{2}(G)\to L_{\nu }^{2}\left({\widehat {G}}\right).}
and we have the formula
∀ f ∈ L 2 ( G ) : ∫ G | f ( x ) | 2 d μ ( x ) = ∫ G ^ | f ^ ( χ ) | 2 d ν ( χ ) . {\displaystyle \forall f\in L^{2}(G):\quad \int _{G}|f(x)|^{2}\ d\mu (x)=\int _{\widehat {G}}\left|{\widehat {f}}(\chi )\right|^{2}\ d\nu (\chi ).}
Note that for non-compact locally compact groups G {\displaystyle G} the space L 1 ( G ) {\displaystyle L^{1}(G)} does not contain L 2 ( G ) {\displaystyle L^{2}(G)} , so the Fourier transform of general L 2 {\displaystyle L^{2}} -functions on G {\displaystyle G} is "not" given by any kind of integration formula (or really any explicit formula). To define the L 2 {\displaystyle L^{2}} Fourier transform one has to resort to some technical trick such as starting on a dense subspace like the continuous functions with compact support and then extending the isometry by continuity to the whole space. This unitary extension of the Fourier transform is what we mean by the Fourier transform on the space of square integrable functions. The dual group also has an inverse Fourier transform in its own right; it can be characterized as the inverse (or adjoint, since it is unitary) of the L 2 {\displaystyle L^{2}} Fourier transform. This is the content of the L 2 {\displaystyle L^{2}} Fourier inversion formula which follows.
In the case G = T {\displaystyle G=\mathbb {T} } the dual group G ^ {\displaystyle {\widehat {G}}} is naturally isomorphic to the group of integers Z {\displaystyle \mathbb {Z} } and the Fourier transform specializes to the computation of coefficients of Fourier series of periodic functions. If G {\displaystyle G} is a finite group, we recover the discrete Fourier transform. Note that this case is very easy to prove directly.
Bohr compactification and almost-periodicity One important application of Pontryagin duality is the following characterization of compact abelian topological groups:
That G {\displaystyle G} being compact implies G ^ {\displaystyle {\widehat {G}}} is discrete or that G {\displaystyle G} being discrete implies that G ^ {\displaystyle {\widehat {G}}} is compact is an elementary consequence of the definition of the compact-open topology on G