Let <i>G</i> be a locally compact Lie group and <img src="https://html.scirp.org//file/5302366-rId13.svg?20240402112528" /> its Lie algebra. We consider a fuzzy analogue of <i>G,</i> denoted by <img src="https://html.scirp.org//file/5302366-rId15.svg?20240402112528" /> called a fuzzy Lie group. Spherical functions on <img src="https://html.scirp.org//file/5302366-rId17.svg?20240402112528" /> are constructed and a version of the existence result of the Helgason-spherical function on <i>G</i> is then established on <img src="https://html.scirp.org//file/5302366-rId19.svg?20240402112528" />.
One of the basic tools for classical computation, modeling and reasoning is crisp, which is exact in nature. A crisp is dichogamous, indicating Yes or No type rather than more or less type. In this case, a membership function, is often used to assign binary values to each element of the universal X. Fuzzy set theory gives sufficient mathematical configuration in which vague conceptual facts can be precisely and rigorously examined. This has found application in many fields, including, computer science, biomedical engineering, telecommunication, decision making, differential equations, rings, semirings, group, automation and robotics, networking, discrete mathematics, etc.
This originated from the novel work of Zadeh in 1965 [
In this section, we give some basis definitions that will be needed in the sequel following [
Definition 2.1 [
A ˜ = { ( s , μ A ˜ ( s ) ) | s ∈ S } ,
of ordered pairs where μ A ˜ ( s ) is called the membership function.
The fuzzy empty set is denoted by 0 S and defined as 0 S ( s ) = 0 ∀ s ∈ S while the entire set in a set S is denoted 1 S and defined as 1 S ( s ) = 1 for all s ∈ S . The basic operations on fuzzy sets and their standard results can be seen in [
Definition 2.2 Let the universe of discourse be S and U a fuzzy set on a S. Given t ∈ [ 0,1 ] , we define a t-cut set (t-level set) of U as
U t = { s ∈ S | U ( s ) ≥ t } .
Definition 2.3 Let E be a vector space and A 1 ⋯ A n be fuzzy sets in E. We define A 1 × ⋯ × A n to be the fuzzy set A in E n whose membership function is given by
μ A ( s 1 , ⋯ , s n ) = min { μ A 1 ( s 1 ) , ⋯ , μ A n ( s n ) }
Let f : E n → E , f ( s 1 , ⋯ , s n ) = s 1 + ⋯ + s n . We define A 1 + ⋯ + A n = f ( A ) .
For λ ∈ K and D a fuzzy set in E, we define λ D = g ( D ) , where g : E → E , g ( s ) = λ ( s ) .
Definition 2.4 Let S ≠ ∅ and I = [ 0,1 ] . Define the set I S by I S : = { γ | γ : S → I } . Then, γ ∈ I S is known as fuzzy subset of S. Let γ ∈ I S . A collection σ of fuzzy subsets of γ satisfying the following:
κ ∩ γ ∈ σ ∀ κ ∈ I (1)
U i ∈ σ ∀ i ∈ N ⇒ ∪ { U i : i ∈ N } ∈ σ (2)
U , V ∈ σ ⇒ U ∩ V ∈ σ (3)
is called a fuzzy topology on γ and the pair ( γ , σ ) is called a fuzzy topological space. Members of σ are called fuzzy open sets and their complements with respect to γ are known as closed sets of ( γ , σ ) .
If B be a collection of fuzzy subset of γ , then the family of arbitrary unions and finite intersections of the member of B and the family { γ ∩ κ ; κ ∈ I } forms a fuzzy topology on γ denoted by σ ( B ) .
Definition 2.5 B ∈ σ is referred to as open base of σ if every member of σ can be expressed uniquely as the union of certain members of B .
Definition 2.6 A fuzzy topological space ( γ , σ ) is said to be Hausdorff if ∀ x p , y p ∈ γ ( x ≠ y ), ∃ U , V ∈ σ such that x p ∈ U , y p ∈ V and U ∩ V = ∅ .
Definition 2.7 A fuzzy topological space ( γ , σ ) is said to be fuzzy compact if ∀ β ∈ σ with ∪ { U : U ∈ β } = γ and ∀ ε > 0 , ∃ a finite subcollection β 0 of β such that ∪ { U , U ∈ β 0 } ≥ γ ε where γ ε is defined by γ ε ( x ) = γ ( x ) − ε or 0 according as γ ( x ) > ε or γ ( x ) ≤ ε .
Definition 2.8 A fuzzy subset U of γ is called fuzzy separated if ∃ v , δ ∈ σ such that U = v ∪ δ , v ≠ δ and v ∩ δ = ∅ .
Definition 2.9 A fuzzy topological space ( γ , σ ) is said to be connected in the fuzzy sense if for any β 0 fuzzy closed subset of ( γ , σ ) can be fuzzy separated.
We shall take for granted that all information and ideas needed on fuzzy Lie algebras follow from [
Definition 2.10 Let W be a vector space over K . A fuzzy subset U of W satisfying the following conditions
U ( s + t ) ≥ min { U ( s ) , U ( t ) } for all s , t ∈ W (1)
U ( α s ) ≥ U ( s ) for all s ∈ W , α ∈ K (2)
is called a fuzzy subspace of W.
Definition 2.11 A fuzzy set U : L → [ 0,1 ] is called a fuzzy Lie subalgebra of L over a field K if it is a fuzzy subspace of L such that
each non empty U ( L , t ) is a subspace of L (1)
U ( [ s , t ] ) ≥ min { L ( s ) , U ( t ) } . (2)
hold for all s , t ∈ L and α ∈ K .
Definition 2.12 A fuzzy set U : L → [ 0,1 ] is called a fuzzy Lie ideal of L if
U ( s + t ) ≥ min { U ( s ) , U ( t ) } (1)
U ( α s ) ≥ U ( s ) (2)
U ( [ s , t ] ) ≥ U ( s ) (3)
hold for all s , t ∈ L and α ∈ K .
Definition 2.13 Let L 1 and L 2 be two Lie algebras and φ a function from L 1 to L 2 . If U is a fuzzy set in L 2 , then the pre-image of U under φ is the fuzzy set in L 1 defined by
φ − 1 ( U ) ( s ) = U ( φ ( s ) ) ∀ s ∈ L 1
We now introduce notion of fuzzy topological group and its corresponding differentiable manifold in what follows.
Definition 2.14 A fuzzy subset F of X × Y is said to be a fuzzy proper function from λ to U if
F ( x , y ) ≤ min { λ ( x ) , U ( y ) } ∀ ( x , y ) ∈ X × Y (1)
∀ x ∈ X , ∃ y 0 ∈ Y such that F ( x , y 0 ) = λ ( x ) and F ( x , y ) = 0 if y ≠ y 0 (2)
Definition 2.15 A proper function F : ( λ , τ ) → ( U , σ ) , is said to be
1. Fuzzy continuous is F − 1 ( U ) ∈ τ , ∀ U ∈ σ ;
2. Fuzzy open if F ( δ ) ∈ σ ∀ δ ∈ τ ;
3. Fuzzy homeomorphism if F be bijective, fuzzy continuous and open.
Definition 2.16 [
m : ( G × G , σ × σ ) → ( G , σ ) : m ( x , y ) ↦ x y
i : ( G , σ ) → ( G , σ ) : i ( x ) ↦ x
are fuzzy continuous. A group G equipped with a compatible fuzzy topology σ on G is called a fuzzy topological group.
Definition 2.17 A fuzzy topological vector space (ftvs) is a vector space Ξ over the field K , Ξ equipped with a fuzzy topology σ and K equipped with the usual topology κ , such that the two mappings
( s , t ) → s + t of ( Ξ , σ ) × ( Ξ , σ ) into ( Ξ , σ )
( α , s ) → α s of ( K , κ ) × ( Ξ , σ ) into ( Ξ , σ )
are fuzzy continuous.
Definition 2.18 [
ϕ [ t V ] ⊂ ø ( t ) W
for some function ø ( t ) .
Definition 2.19 Let Ξ and Ϝ be two fuzzy topological vector space, each equipped with a fuzzy topology (possibly T 1 ). Let φ : Ξ → Ϝ be a fuzzy continuous mapping. Then, φ is called fuzzy differentiable at s ∈ Ξ if ∃ a continuous linear fuzzy map U : Ξ → Ϝ such that
φ ( s + t ) = φ ( s ) + U ( t ) + ρ ( t ) , t ∈ Ξ .
where ρ is tangent to 0. This mapping U is known as the fuzzy derivative of φ at s ∘ . The fuzzy derivative of φ at s ∘ is denoted by φ ′ ( s ∘ ) ; it is an element of L ( Ξ , Ϝ ) : = { φ | φ : Ξ → Ϝ , φ is fuzzy linear and continuous } . φ is differentiable in the fuzzy sense if it is differentiable at every point in Ξ in the fuzzy sense.
Definition 2.20 Let Ξ , Ϝ be ftvs. A bijection φ : Ξ → Ϝ is called a fuzzy diffeomorphism of class C 1 if φ and its inverse φ − 1 are differentiable in the fuzzy sense and φ ′ and ( φ − 1 ) ′ are fuzzy continuous.
The whole idea of fuzzy C 1 -manifold and atlas is well-known (see [
Definition 2.21 A fuzzy Lie group, G f is a C 1 -fuzzy manifold G f which is also a group, such that the mappings
m : ( G f × G f , σ × σ ) → ( G f , σ )
i : ( G f , σ ) → ( G f , σ )
are fuzzy differentiable.
In this section, we first introduce the general construction of the popular spherical functions as can be seen in [
Definition 3.1 Let G f be a fuzzy Lie group and K f a closed fuzzy Lie subgroup of G f . Suppose ρ be a complex-valued function on G f / K f (where G f is a fuzzy Lie group and K f is a compact subgroup) of class C ∞ which satisfies ρ ( π ( e ) ) = 1 . Then, ρ is referred as spherical function if
ρ σ ( k ) = ρ ∀ k ∈ K f (1)
D ρ = λ D ρ for each D ∈ D ( G f / K f ) . (2)
Here, λ D is a complex number and D ( G f / K f ) is the algebra of differential operators on G f / K f invariant under all the translations
σ ( g ) : x K f → g x K f of G f / K f (T)
In what follows, we shall construct spherical function on a locally compact fuzzy Lie group G f following the [
Let M be a C 1 -fuzzy manifold and ρ : M → M a C 1 -fuzzy diffeomorphism of M onto itself. We put
f ρ = f ∘ ρ − 1 , f ∈ ε ( M )
and if D is a differential operator on M, we define D ρ by
D ρ : f → ( D f ρ − 1 ) ρ = ( D ( f ∘ ρ ) ) ∘ ρ − 1 , f ∈ ε ( M )
where D ρ is another differential operator. The operator D is said to be invariant under ρ if D ρ = D i.e. D ( f ∘ ρ ) = D f ∘ ρ for all f. Note that ( D f ) ρ = D ρ f ρ .
If T is a distribution on M, we put T ρ for the distribution T ρ ( f ) = T ( f ρ − 1 ) , f ∈ D ( M ) .
Let G f be a fuzzy Lie group, ℌ f ⊂ G f a closed fuzzy subgroup of G f . G f / ℌ f the fuzzy manifold of left cosets g ℌ f ( g ∈ G f ) and D ( G f / ℌ f ) the algebra of all differential operators on G f / ℌ f which are invariant under the usual transformations (T).
Given a coset space G f / ℌ f , we intend to define the operator in D ( G f / ℌ f ) . We consider a case when ℌ f = { e } and put D ( G f ) for D ( G f / { e } ) , the set of left-invariant differential operators on G f .
If ò is a fuzzy vector spaces over ℝ , the symmetric fuzzy algebra S (ò) over ò is defined as the fuzzy algebra of complex-valued polynomial functions on the dual space òå. If X 1 , ⋯ , X n is the basis for ò, then S (ò) can be identified with the commutative fuzzy algebra of polynomials ∑ ( k ) a k 1 , ⋯ , k n X 1 k 1 ⋯ X n k n . Let g denote the fuzzy Lie algebra of G f (the tangent space to G f at e) and exp : g ↦ G f the exponential mapping which maps a line through 0 in g onto a one parameter subgroup t → exp t x of G f . If X ∈ g , let X ˜ denote the fuzzy vector field on g given by
( X ˜ f ) ( g ) = X ( f ∘ L g ) = { d d t f ( g exp t X ) } t = 0 for f ∈ E ( G f ) (1)
where L g denote the left translation x → g x of G f onto itself. Then X ˜ is a differential operator on G f if h ∈ G f then
( X ˜ L h f ) ( g ) = ( X ˜ ( f ∘ L h ) ) ( h − 1 g ) = ( X ˜ f ) ( g )
So X ˜ ∈ D ( G f ) . Moreover, the bracket on g is by definition given by
[ X , Y ] ∼ = X ˜ Y ˜ − Y ˜ X ˜ , X , Y ∈ g
the multiplication on the right-hand side being composition of operators.
Definition 3.2 [
Theorem 3.3 Let G f be fuzzy Lie group with fuzzy algebra g . Let S ( g ) denote the symmetric fuzzy algebra over the fuzzy vector space g . Then there exists a unique linear bijection
λ : S ( g ) → D ( g )
such that
λ ( X m ) = X ˜ m , X ∈ g , m ∈ [ 0,1 ] (2)
If X 1 ⋯ X n is any basis of g and p ∈ S ( g ) , then
( λ ( p ) f ) ( g ) = { p ( ∂ 1 , ⋯ , ∂ n ) f ( g exp ( t 1 X 1 + ⋯ + t n X n ) ) } t = 0 , (3)
where f ∈ E ( G f ) , ∂ i = ∂ ∂ t i and t = ( t 1 , ⋯ , t n ) .
Proof. For any fixed basis X i , ⋯ , X n of g . The mapping
g exp ( t 1 X 1 + ⋯ + t n X n ) → ( t 1 , ⋯ , t n )
is a coordinate system on a neighbourhood of g in G f . By equation (2), define a differential operator λ ( P ) on G f . Clearly λ ( p ) is left invariant, and by (1) λ ( X i ) = X ˜ i , so by linearity λ ( X ) = X ˜ for X ∈ g .
Also we show that λ is one to one.
Suppose λ ( P ) = 0 where P ≠ 0 with respect to a lexicographic ordering. Let a X 1 m ⋯ X n m be the leading term in P. Let f be a smooth function on a neighbourhood of e in G f such that
f ( exp ( t 1 X 1 + ⋯ + t n X n ) ) n = t 1 m 1 ⋯ t n m n
for small t, then ( λ ( p ) f ) ( e ) ≠ 0 contradicting λ ( P ) = 0 .
Finally, λ maps S ( g ) onto D ( g ) . Also if u ∈ D ( g ) , there exist a polynomial P such that
( u f ) ( e ) = { P ( ∂ 1 , ⋯ , ∂ n ) + ( exp t 1 X 1 + ⋯ + t n X n ) } .
Then by the left invariance of u, u = λ ( P ) so λ is surjective.
The mapping λ is usually called symmetrization.
Let G f be a fuzzy Lie group, and K a closed fuzzy subgroup of G f . Let ϖ be the natural mapping of G f onto Ξ = G f / K . Let 0 = ϖ ( e ) and f ˜ = f ∘ ϖ for f any function on Ξ . We denote by D ( G f ) the set of all (left-) invariant differential operators on G f , and D K ( G f ) the subspace of all right invariant differential operators under K , then clearly, D ( G f / K ) denotes the algebra of differential operators on G f / K invariant under the usual translations. We shall prove the existence of the following result in the fuzzy sense.
Theorem 4.1 [
∫ K f ( x k y ) = f ( x ) f ( y ) ∀ x , y ∈ G f (4)
Theorem 4.2 [
L : f → ∫ G f f ( x ) ρ ( x ) d x
is a homomorphism of C c * ( G f ) onto ℂ .
Theorem 4.3 The algebra D ( G f / K ) is commutative.
Proof. Let G f be a fuzzy Lie group and g be a fuzzy Lie algebra of G f . Let K be a closed fuzzy subgroup of G f and the symmetric space M = G f / K be a fuzzy manifold of left cosets g k g ∈ G f where D ( G f / K ) the algebra of all differential operators on G f / K which are invariant under the usual transformations. Let f ∈ C 1 ( M ) and let x 1 , x 2 , ⋯ , x n be a basis in M . Let X ˜ ∈ g and Y ˜ ∈ g , we define
X ˜ = ∑ j = 1 n U j ∂ ∂ x j
and
Y ˜ = ∑ i = 1 n V i ∂ ∂ x i
we have
( X ˜ ∘ Y ˜ ) f = X ˜ ( Y ˜ f ) = X ˜ ( ∑ i = 1 n V i ∂ f ∂ x i ) = ∑ i , j = 1 n U j ∂ V i ∂ x j ∂ f ∂ x i + ∑ i , j = 1 n U j V i ⋅ ∂ 2 f ∂ x j . (5)
Also
( Y ˜ ∘ X ˜ ) f = Y ˜ ( X ˜ f ) = Y ˜ ( ∑ j = 1 n V j ∂ f ∂ x j ) = ∑ i , j = 1 n V i ∂ U j ∂ x i ∂ f ∂ x j + ∑ i , j = 1 n V i U j ∂ 2 f ∂ x j ∂ x i . (6)
By subtraction, we obtain for f ∈ C 2 ( M )
X ˜ ( Y ˜ f ) − Y ˜ ( X ˜ f ) = ∑ i = 1 n ( ∑ j = 1 n ( U j ∂ V i ∂ x j − V i ∂ U j ∂ x j ) ) ∂ f ∂ x i ∑ i , j = 1 n U j ∂ V i ∂ x j ∂ f ∂ x i − ∑ i , j = 1 n V i ∂ U j ∂ x j ∂ f ∂ x i = 0 ( by the invariance property ) . (7)
This implies that X ˜ ( Y ˜ f ) − Y ˜ ( X ˜ f ) = 0 , X ˜ ( Y ˜ f ) = Y ˜ ( X ˜ f ) whenever U = k V for k any scalar.
We are now ready to use (6) and (7) to define spherical function.
Let G f be a fuzzy Lie group and K a maximal compact subgroup. Let C c ( K \ G f / K ) denote the space of all continuous functions with compact support on G f which satisfy f ( k 1 g k 2 ) = f ( g ) for all k 1 , k 2 ∈ K . Such spaces are called spherical or bi-invariant. Then, C c ( K \ G f / K ) forms a commutative Banach algebra under convolution [
Let G f be a fuzzy Lie group and c a closed fuzzy subgroup of G f . Let M = G f / K be a symmetric space. For any function ρ ∈ G f ∃ a function ρ ∈ C c ( K \ G f / K ) which satisfies ρ ( k g k ′ ) = ρ ( g ) and are integrable on G f for a normed algebra A under the convolution product of G f , g ∈ G f , k , k ′ ∈ K . The functions ρ are continuous, positive-definite, invariant under g → k g k ′ and the linear representation
f → ρ ( f ) = ∫ f ( g ) ρ ( g ) d g
must be a homomorphism of A onto [ 0,1 ] .
We next prove the existence of Helgason-functions theorem in the fuzzy sense.
Proposition 4.4 [
∫ K f ( x k y ) d k = f ( x ) f ( y )
Theorem 4.5 Let G f be a fuzzy Lie group and G f a closed, compact fuzzy subgroup of G f . Let f ∈ C c ( K \ G f / K ) then f is a spherical function on G f , f : G f → [ 0,1 ] and f satisfies the function
f : C c ( K \ G f / K ) → G f | [ 0,1 ]
such that
∫ K f ( x k y ) d k = { f ( x ) f ( y ) , if ∃ no x , y ∈ G f such that t 1 < f ( x y ) < t 2 0 , if otherwise
Proof. Let f be a fuzzy set on a fuzzy Lie group G f . Let t ∈ [ 0,1 ] . We define the t-cut set or t-level set of μ , by
f t = { x ∈ G f : f ( x ) ≥ t } .
If x , y ∈ G f and f ( x ) = t 1 and f ( y ) = t 2 .
Then x ∈ f t 1 and y ∈ f t 2 . If t 2 < t 1 , we have
f t 1 ⊆ f t 2 ⇒ x ∈ f t 2
and we have x , y ∈ f t 2 and since f t 2 is a subgroup of G f , by definition we have x y ∈ f t 2 . Therefore
f ( x y ) ≥ t 2
and t 2 = min ( f ( x ) , f ( y ) ) .
Also let x ∈ G f and f ( x ) = t , we have x ∈ f t , since f t is a subgroup. x − 1 ∈ f t . Therefore
f ( x − 1 ) ≥ t .
Then f ( x − 1 ) ≥ f ( x ) . This shows that f is a subgroup of G f .
Since f is a homomorphism, we have
f ( x y ) = f ( x ) f ( y ) .
Now let x , y ∈ G f and k ∈ K , let f ( x ) f ( y ) = t 0 and ∫ K f ( x k y ) d k = t 1 .
Then, x y ∈ f t 0 and x k y ∈ f t 1 for
∫ K f ( x k y ) d k = f ( x ) f ( y ) .
Then ∃ no x y ∈ G f such that
t 0 < f ( x y ) < t 1 .
Let
∫ K f ( x k y ) = f ( x ) f ( y )
and we have
f t 1 = f t 0
If ∃ x , y ∈ G such that, t 0 < f ( x y ) < t 1 then f t 1 ⊂ f t 0 . Since x y ∈ f t 0 but x y ∉ f t 1 then this contradicts our statement.
Conversely, if ∃ no x y ∈ G such that
t 0 < f ( x y ) < t 1 .
If t 0 < t 1 we get f t 1 ⊆ f t 0 . Let x y ∈ f t 0 , we have f ( x y ) ≥ t 0 and f ( x y ) ≥ t 1 . This implies that f ( x , y ) does not lie between t 1 and t 0 . Hence x y ∈ f t 1 and f t 0 ⊆ f t 1 . Hence f t 0 = f t 1 and
∫ K f ( x k y ) d k = f ( x ) f ( y ) .
The authors declare no conflicts of interest regarding the publication of this paper.
Egwe, M.E. and Sangodele, S.S. (2024) Spherical Functions on Fuzzy Lie Group. Advances in Pure Mathematics, 14, 185-195. https://doi.org/10.4236/apm.2024.144010