Gordon’s Conjecture 3: Fourier transforms in the hyperfinite setting

Using methods of nonstandard analysis and building upon the results of our previous paper in which we proved Gordon’s Conjectures 1 and 2 we will show that for any locally compact abelian group G the Fourier transform F : L(G) → C0 ( Ĝ ) , the Fourier–Stieltjes transform F : M(G) → Cbu ( Ĝ ) , as well as all the generalized Fourier transforms F : L(G) → L ( Ĝ ) for any pair of adjoint exponents p ∈ (1, 2], q ∈ [2,∞) can be approximated in a fairly natural way by the discrete Fourier transform F : CG → CĜ on a (hyper)finite abelian group G . In particular, we will prove Gordon’s Conjecture 3, originally stated for the Fourier–Plancherel transform F : L(G) → L ( Ĝ ) , and generalize it to all the above mentioned cases. Some standard consequences will be considered, as well. 2010 Mathematics Subject Classification 43A25 (primary); 26E35, 28E05, 46S20, 43A10, 43A15, 43A20 (secondary)


Introduction
This paper is a direct continuation of our previous work [19], dealing with approximation of locally compact abelian groups (briefly LCA groups) by (hyper)finite abelian groups and using the concepts and methods of nonstandard analysis (NSA). We will assume that the reader is acquainted with the notions and results which were introduced or proved there, and use them freely throughout the present article, frequently without any reference. Some motivational accounts as well as a more detailed outline of the results of the paper follow.
For a finite abelian group G its dual group G = Hom(G, T), where T denotes the compact multiplicative group of complex units, is isomorphic (though not canonically) to G, the |G|-dimensional vector space C G is endowed with the Hermitian inner product ⟨ f , g⟩ d = d x∈G f (x) g(x) P Zlatoš where d > 0 is some normalizing coefficient, the characters γ ∈ G form an orthogonal basis in C G and the discrete Fourier transform (DFT) F : C G → C G is defined as the inner (or scalar) product Once the inner product ⟨φ, ψ⟩ d on C G is defined using the adjoint normalizing coefficient d = (d |G|) − Using the density of the intersection L 1 (G) ∩ L p (G) in the Lebesgue space L p (G) with respect to its norm ∥·∥ p , the Fourier transform can be extended to a bounded linear operator F : L p (G) → L q G for p ∈ (1, 2] and the adjoint exponent q = p/( p − 1) ∈ [ 2, ∞). Under a proper normalization of the Haar measure n n n on the dual group G we have the Fourier inversion formula f f f = f f f (γ γ γ) γ γ γ d n n n (both with respect to the supremum norm ∥·∥ ∞ and the L p -norm ∥·∥ p ) just for functions f f f ∈ L p (G) ∩ F M G ⊆ L p (G) ∩ C bu (G), with F denoting the Fourier-Stieltjes transform M G → C bu (G).
The main goal of this paper is to provide arguments that nonstandard analysis offers a solution to this question which is not only reasonable and satisfactory but also fairly elegant, as well as several additional insights. Our approach is based on some ideas, introduced by Gordon in [6] and further advanced in Gordon [7] and Zlatoš [19]. Namely, in [19] we elaborated concepts and techniques enabling us to approximate simultaneously, with infinitesimal precision on finite elements, any pair consisting of a (Hausdorff) LCA group G and its dual group G by a pair of hyperfinite abelian groups G, G and approximating maps η : G → * G, ϕ : G → * G in such a way that the pairing function (x x x, γ γ γ) → γ γ γ(x x x) on * G × * G is approximated by the pairing function (x, γ) → γ(x) on G × G , and we proved Gordon's Conjectures 1 and 2. From this we derived analogous standard results on arbitrarily precise simultaneous approximation of the pair of LCA groups G, G by a finite abelian group G and its dual group G , as well as of the pairing function on G × G by the pairing function on G × G .
In the present paper we are going to carry out the third and final of the three steps announced in the introduction to [19], namely to approximate the Fourier transform F : L 1 (G) → C 0 G , the Fourier-Stieltjes transform F : M(G) → C bu G , as well as the generalized Fourier transforms F : L p (G) → L q G , for any pair of adjoint exponents p ∈ (1, 2], q ∈ [2, ∞), by the hyperfinite dimensional DFT F : * C G → * C G . In particular, we will prove Gordon's Conjecture 3, originally stated for the Fourier-Plancherel transform F : L 2 (G) → L 2 G in Gordon [6,7], and generalize it to all the cases mentioned above.

P Zlatoš
Should we encapsulate the moral of these results in a single sentence, the best we can do seems to be to phrase it as a response to the question formulated in the title of the paper by Epstein [5]: How well does the finite Fourier transform approximate the Fourier transform?
The response in the abstract of [5] is "very well indeed"; and, in the conclusion: "as well as it possibly could". We aim to convince the reader to agree finally with the following: Even better than one could ever hope. 1 Intuitively, we can unfold the above slogan in the following rather imprecise way: any of the Fourier transforms L 1 (G) → C 0 G , M(G) → C bu G , L p (G) → L q G , for adjoint exponents 1 < p ≤ 2 ≤ q < ∞, can be "arbitrarily well" approximated by the discrete Fourier transform C G → C G , based on some "sufficiently good" simultaneous approximations of G, G by finite abelian groups G, G , respectively.
The plan of the paper is as follows. Section 1 consists mainly of some preliminary material concerning hyperfinite dimensional representations of some Banach spaces, namely various spaces of continuous functions, the Lebesgue spaces L p and certain space of complex measures. Thus it is sufficient to highlight some central points from the particular sections of Section 2. All missing notions used here will be defined later in the paper. In the rest of this introduction, (G, G 0 , G f ) denotes a condensing IMG group triplet with a hyperfinite abelian ambient group G and a normalizing multiplier d , and G = G f /G 0 is its observable trace.
The main result of Section 2.1 is the characterization of internal functions which are liftings (ie, certain kinds of infinitesimally precise approximations) of functions f f f ∈ L p (G) in terms of certain natural continuity condition for the shift f → f a on the linear space * C G (cf Theorem 2.1.4). Given an internal norm N on * C G , an A Characterization of Liftings Let p ∈ [1, ∞) and f : G → * C be an internal function such that ∥ f ∥ p < ∞ and ∥ f · 1 X ∥ p ≈ 0 for any internal set X ⊆ G ∖ G f . Then f is a lifting of a function f f f ∈ L p (G) if and only if f is S p -continuous. 1 In fact, in [5] that question is asked only for the Fourier transform of periodic functions R → C. In our response we include the Fourier transforms on arbitrary LCA groups. Essentially the same characterization is valid for liftings of functions f f f ∈ C 0 (G) and the norm ∥ f ∥ ∞ = max x∈G | f (x)| (cf Proposition 1.1.1(c)). The sets of all liftings f of functions f f f from L p (G) or C 0 (G) satisfying the above conditions will be denoted by L p (G, G 0 , G f ) or C 0 (G, G 0 , G f ), respectively. Additionally, C bu (G, G 0 ) denotes the set of all internal functions f : G → * C satisfying ∥f ∥ ∞ < ∞ which are S-continuous with respect to the equivalence relation corresponding to the monadic subgroup G 0 ⊆ G and M(G, The view through the lens of the IMG group triplet (G, G 0 , G f ) and its dual triplet G , G ‹ f , G ‹ 0 offers an intuitively appealing explanation of the Smoothness-and-Decay Principle for internal functions f : G → * C, based on the formulas for the discrete Fourier transform and its inverse: Intuitively, if f is smooth or continuous in some sense then the contribution of the non-S-continuous characters to the expansion of f in the second formula must be somehow negligible. This condition becomes manifest as a kind of quick decay of f . The other way around, viewing the elements x ∈ G as characters of the dual group G , the Fourier transform of f can be expressed as their linear combination given by the first formula. If f decays quickly, ie if the values of f on the infinite elements x ∈ G ∖ G f are somehow negligible, then the values of its Fourier transform are essentially determined by the values of f on the finite elements x ∈ G f , which happen to coincide with the S-continuous characters of G . If neither the coefficients f (x) for x ∈ G f are too big then we can reasonably expect f to be smooth or continuous in some sense. This result is proved as our Theorem 2.2.1.
The Smoothness-and-Decay Principle Let N, M be absolute internal norms on the linear spaces * C G , * C G , respectively, such that N( f ) < ∞ implies M f < ∞ for every function f ∈ * C G . Then for every f ∈ * C G the following implications hold: As a consequence of the Hausdorff-Young inequality ∥ f ∥ q ≤ ∥ f ∥ p and the fact that all the p-norms are absolute, the Smoothness-and-Decay Principle applies to every pair of norms ∥·∥ p , ∥·∥ q where 1 ≤ p ≤ 2 ≤ q ≤ ∞ are adjoint exponents.

P Zlatoš
The main results of the paper are the following three theorems from Section 2.3, dealing with approximations of the particular classical Fourier transforms by the discrete Fourier transform on the hyperfinite dimensional (HFD) linear space * C G (cf The HFD Fourier Transform Approximation Theorem Let the internal function f ∈ L 1 (G, G 0 , G f ) be a lifting of a function f f f ∈ L 1 G . Then the internal function The HFD Generalized Fourier Transform Approximation Theorem Let p ∈ (1, 2] and let q ∈ [2, ∞) be its dual exponent. Let the internal function f ∈ L p (G, The special Hilbert space case p = q = 2 solves Gordon's Conjecture 3. The HFD Fourier-Stieltjes Transform Approximation Theorem Let the internal function g ∈ M(G, G 0 , G f ) be a weak lifting of a complex regular Borel measure µ µ µ ∈ M(G). Then the internal function F(g) = g ∈ C bu G , G ‹ f is a lifting of the function F(µ µ µ) = µ µ µ ∈ C bu G .
The final section, Section 2.4, addresses some standard analogues of the results of Section 2.3, ie with simultaneous approximations of certain functions or measures on G and their Fourier transforms by functions f : G → C and their discrete Fourier transforms f : G → C, respectively, based on simultaneous approximation of the pair of LCA groups G, G by a pair of finite abelian groups G, G .

Nonstandard counterparts of some Banach spaces
In this section we briefly review some facts about certain specific nonstandard approaches to Banach spaces, with particular emphasis on some spaces of continuous functions, the Lebesgue spaces L p (X) for 1 ≤ p < ∞ and the space M(X) of complex-valued regular Borel measures with finite variation on a Hausdorff locally compact topological space X. Each of these Banach spaces will be associated with a nonstandard counterpart closely related to some hyperfinite dimensional space * C X . The canonical reference for nonstandard Banach space theory is the paper Henson and Moore [9]; additionally, the reader can consult Albeverio, Fenstad, Høegh-Krohn and Lindstrøm [1] and Davis [3].
Throughout the section, X denotes a Hausdorff locally compact topological space, whose topology is induced by a uniformity U , represented as the observable trace X ∼ = X ♭ of a condensing IMG triplet (X, E, X f ) with a hyperfinite ambient set X by means of a (not necessarily injective) HFI approximation η : X → * X. Then X f = η −1 Ns( * X) , and we can assume without loss of generality that the equivalence x ≈ y ⇔ η(x) ≈ η(y) holds for all x, y ∈ X and not just for those in X f . Identifying the observable trace X ♭ = X f /E with X ∼ = Ns( * X)/≈ via the homeomorphism η ♭ , we regard each point η ♭ x ♭ = • η(x) ∈ X as the observable trace x ♭ of x ∈ X f (see Zlatoš [19,Section 1.2]).
The hyperfiniteness of X enables us to represent various Banach spaces of functions X → R and X → C by means of the hyperfinite dimensional linear spaces * R X and * C X of all internal functions X → * R and X → * C, respectively. We will systematically exploit the advantage of such an approach. At the same time, we will focus entirely on spaces of complex functions, leaving the formulation of the real-valued version to the reader.

Spaces of continuous functions
The hyperfinite dimensional (HFD) linear space * C X admits several internal norms. For any internal norm N on * C X we denote by the F * C-linear subspaces (more precisely, F * C-submodules) of * C X , consisting of functions which are infinitesimal or finite, respectively, with respect to the norm N. For any "reasonable" norm N, the associated IMG triplet * C X , I N * C X , F N * C X , with I N * C standing in place of the indistinguishability equivalence relation is condensing if and only if X is finite. Its observable trace (nonstandard hull) becomes a (standard) Banach space under the norm N ♭ given by An internal function f : [19,Section 1.2]). Its observable trace is the function f ♭ : X ♭ ∼ = X → C given by for x ∈ X f . However, unless X f = X , the monadic equivalence relation on * C X of infinitesimal nearness on finite elements corresponding to the compact-open topology on the space C(X) of all continuous functions X → C, is not of the form ≈ N for any internal norm N on * C X .
When dealing with S-continuous functions, the maximum norm (or max-norm) where f ∈ * C X , becomes important. Denoting by the F * C-linear subspaces of * C X , consisting of internal functions which are infinitesimal or finite, respectively, with respect to the max-norm we get the IMG triplet We are particularly interested in the Banach spaces C b (X), C bu (X), and C 0 (X) of all bounded continuous, bounded uniformly continuous, and continuous vanishing at infinity functions X → C, respectively, and also in the (non-Banach) normed linear space C c (X) of all continuous functions X → C with compact support, all with the supremum norm denoted by ∥·∥ ∞ , as well. Let us denote by for f f f ∈ C X , f ∈ * C X . Now, (a), (b), (c) and (d) follow from the first quadruple of implications. (The additional property E = {(x, y) ∈ X × X : η(x) ≈ η(y)} is needed for the second implication.) The "conversely part" follows from the second quadruple of implications.
We emphasize that in general the observable traces f ♭ and f ♭ ∞ of S-continuous functions should not be confused. For f , g ∈ C 0 (X, E, X f ) (and therefore f , g ∈ C c (X, E, X f )) we do have the equivalence: However, for f , g ∈ C bu (X, E) (and therefore f , g ∈ C b (X, E, X f )) we only have the implication: Summing up we have the following. (c) a Banach space isomorphism of the subspace C 0 (X, E, X f )/ I ∞ * C X of the nonstandard hull F ∞ * C X / I ∞ * C X onto the Banach space C 0 (X) (d) a normed space isomorphism of the subspace C c (X, E, X f )/ I ∞ * C X of the nonstandard hull F ∞ * C X / I ∞ * C X onto the normed space C c (X)

Lebesgue spaces and spaces of measures
The natural way of getting functions f f f : X → C as observable traces f f f = f ♭ of internal functions f : X → * C works only under the assumption that f is finite and S-continuous on X f . In this way, however, only continuous functions f f f : X → C can be obtained. Therefore, many internal functions f : X → * C do not represent classical functions X → C. On the other hand, they can be used to represent various objects of a different nature: measures, distributions, etc. The class of functions f f f on X = X ♭ representable that way can be extended to include the Lebesgue spaces L p (X) by relaxing the equality f f f x ♭ = • f (x) on X f to the equality almost everywhere on X f with respect to some measure. Strictly speaking, the elements of L p (X) themselves are not genuine functions but certain equivalence classes of functions. We are going to make this point more precise. However, instead of functions, we will start with representation of certain complex valued measures Similarly, as in Zlatoš [19, Section 1.2] every internal function g : X → * C such that x∈X |g(x)| is finite gives rise to the finite complex Loeb measure λ g : P(X) → C, where P(X) is a σ -algebra extending the algebra of internal subsets of X , such that for each internal set A ⊆ X , and to a complex regular Borel measure θ θ θ g on X given by for Borel Y ⊆ X. Moreover, θ θ θ g has finite variation: Proposition 1.2.1 Every complex regular Borel measure µ µ µ on X with finite variation has the form µ µ µ = θ θ θ g for some internal function g : X → * C, such that Sketch of proof Essentially the same construction as used to obtain the function d from the nonnegative measure m m m in [19, Proposition 1.2.6] works for every complex regular Borel measure µ µ µ with finite variation to give the function g. One just has to take care that the hyperfinite * Borel partition of K formed by all the sets of the Since this is true for any (standard) X-raster (K, U) and a finite (K, U) approximation η : X → X, the existence of such a * X-raster (K, U) follows from the saturation assumption. A complete proof of Proposition 1.2.1, following a slightly different chain of arguments, can be found in Ziman and Zlatoš [18,Section 4] (cf also the construction of the functions g π π π µ µ µ in Section 2.4).

P Zlatoš
In the rest of this section d : X → * R denotes a fixed internal function such that The strict positivity of d entails that the formula defines an internal norm on the linear space * C X for each real number p ≥ 1. In particular, for each internal set A ⊆ X we have Suppressing d in our notation, we denote by consisting of all internal functions which are infinitesimal or finite, respectively, with respect to the p-norm ∥·∥ p . Thus we get the IMG triplet * C X , We also fix the notation m m m = m m m d for the nonnegative (and not identically 0) regular Borel measure on X = X ♭ induced by d , and L p (X) = L p (X, m m m) for the corresponding Lebesgue spaces with the norms: We will relate them to some subspaces of F p * C X and of the observable trace (nonstandard Let M(X) denote the Banach space of all complex regular Borel measures µ µ µ on X with finite total variation ∥µ µ µ∥. According to the Riesz representation theorem, M(X) is isomorphic to the dual C 0 (X) ⋆ of the Banach space C 0 (X). By the Radon-Nikodym Theorem, L 1 (X) can be identified with the closed subspace of all measures µ µ µ ∈ M(X) absolutely continuous with respect to m m m.
This, together with the representation of functions f f f ∈ C 0 (X) by their liftings which are internal functions f ∈ C 0 (X, E, X f ) (see Proposition 1.1.1), justifies the following notion. An internal function g ∈ * C X is called a weak lifting of the measure µ µ µ ∈ M(X), if Fourier transforms in the hyperfinite setting 13 for every function f ∈ C 0 (X, E, X f ). This is obviously equivalent to the condition: If µ µ µ is absolutely continuous with respect to m m m and dµ µ µ = g g g dm m m, where g g g ∈ L 1 (X), then g ∈ * C X is called a weak lifting of g g g if g is a weak lifting of the measure µ µ µ, ie, if and only if for every function f ∈ C 0 (X, E, X f ).
Before formulating what we have just proved, let us introduce some notation and terminology. Showing explicitly the weight function d we denote by M(X, X f , d) the F * C-linear subspace of * C X consisting of all internal functions g : X → * C satisfying The last condition simply says that the Loeb measure λ |g|d is concentrated on the galaxy of accessible elements X f . Therefore, if g ∈ M(X, X f , d) then holds even for all f ∈ C b (X, E, X f ). Now, the F * C-linear subspace S(X, X f , d) of * C X consists of all functions g ∈ M(X, X f , d) satisfying the absolute S-continuity condition Obviously, the last condition is equivalent to absolute continuity of the Loeb measure λ |g|d with respect to the Loeb measure λ d , as well as to absolute continuity of θ θ θ |g|d with respect to m m m. Summing up, we have: such that ∥µ µ µ∥ = • ∥g∥ 1 . Conversely, every function g ∈ F 1 * C X , in particular, every g ∈ M(X, X f , d), is a weak lifting of the measure θ θ θ gd ∈ M(X, X f , d).
(b) A measure µ µ µ ∈ M(X) has a weak lifting g ∈ S(X, X f , d) if and only if µ µ µ is absolutely continuous with respect to the measure m m m. Conversely, every function g ∈ S(X, X f , d) is a weak lifting of the measure θ θ θ gd ∈ M(X, X f , d) (which is absolutely continuous with respect to m m m).

P Zlatoš
Now, there arises a natural question, namely what is the relation between the weak lifting g ∈ S(X, X f , d) of an absolutely continuous measure µ µ µ ∈ M(X) and the (unique) function g g g ∈ L 1 (X) such that dµ µ µ = g g g dm m m, ie, between g g g and its weak lifting g. Unless g g g is continuous, we cannot have g g g = g ♭ , and, unless g is S-continuous on X f , the formula for g ♭ doesn't make sense. Nevertheless, we can still generalize the original notion of lifting of continuous functions in the following sense. An internal function g : X → * C is called a lifting of a function g g g : holds for almost all x ∈ X f with respect to the Loeb measure λ d . As the function g g g ∈ L 1 (X) is only determined up to the equality almost everywhere with respect to the measure m m m = m m m d , this is the best one can expect.
Proposition 1.2.3 (a) Let g g g ∈ L 1 (X) and g ∈ F 1 * C X . Then g is a weak lifting of g g g if and only if g is a lifting of g g g.
(b) Let g g g : X → C. Then the following conditions are equivalent: (i) g g g ∈ L 1 (X) (ii) g g g has a weak lifting g ∈ S(X, X f , d) (iii) g g g has a lifting g ∈ S(X, X f , d) Sketch of proof (a) If g ∈ F 1 * C X is a lifting of g g g then it is obviously a weak lifting of g g g. The reverse implication follows from Proposition 1.1.1(c) and the uniqueness part of the Radon-Nikodym Theorem.
(b) As the implications (iii) ⇒ (ii) ⇒ (i) are obvious, it suffices to prove (i) ⇒ (iii). To this end denote by µ µ µ ∈ M(X) the measure satisfying dµ µ µ = g g g dm m m and by g ∈ * C X the internal function guaranteed to µ µ µ in Proposition 1.2.1. Then the function g = g/d has all the required properties and • g(x) = g g g x ♭ for λ d -almost all x ∈ X f , again due to the uniqueness part of the Radon-Nikodym Theorem.
Remark It is worthwhile to notice that, in the notation from the proof of Zlatoš [19, Proposition 1.2.6], for a "typical" x ∈ X f we have where the right hand term is the mean value of the function * g g g = * (dµ µ µ/dm m m) on the set . This is in accord with the intuition that the Radon-Nikodym derivative g g g( Unfortunately, not every S-integrable function is a lifting of some function g g g ∈ L 1 (X). For instance, every function g ∈ F ∞ * C X with internal support contained in X f is S-integrable; however, unless E is internal, such a function need not be lifting of any function g g g ∈ L 1 (X). One can naturally expect that, in order to lift a function g g g ∈ L 1 (X), the function g ∈ * C X has to display some "reasonable amount" of continuity, which is not clear for the moment. This leads us to define the external subspace L 1 (X, E, X f ) ⊆ * C X as the space of all internal functions g ∈ M(X, X f , d) which are liftings of functions g g g ∈ L 1 (X). Further we put , and the subspaces L p (X, E, X f ) are formed by the liftings of functions g g g ∈ L p (X).
Here we do not present a more explicit description of the spaces L p (X, E, X f ). However, in case that (G, G 0 , G f ) is a condensing IMG group triplet with a hyperfinite abelian ambient group G and m m m d is the Haar measure on its observable trace G = G f /G 0 , we will give a characterization of functions in L p (G, G 0 , G f ) as those belonging to M p (G, G f , d) and satisfying a certain natural continuity condition (see Theorem 2.1.4).
From the definition of L p (X, E, X f ) and the last Proposition we readily obtain the following result, justifying our notation.
with respect to a natural weak topology which we need not describe precisely here.

The Fourier Transform on hyperfinite abelian groups
This section addresses our main topics, namely (i) the analysis of the discrete Fourier transform on some subspaces of the hyperfinite dimensional linear space * C G , arising P Zlatoš from a condensing IMG group triplet (G, G 0 , G f ) with hyperfinite abelian ambient group G, and (ii) its application to Fourier transforms on various spaces of functions f f f : G → C defined on its observable trace, the Hausdorff LCA group G = G f /G 0 . In particular, we will formulate and prove a generalization of the third of Gordon's Conjectures to approximations of Fourier transforms L 1 (G) → C 0 G , M(G) → C bu G and L p (G) → L q G , for adjoint exponents 1 < p ≤ 2 ≤ q < ∞, by the discrete Fourier Throughout the first three parts of Section 2, (G, G 0 , G f ) denotes a condensing IMG group triplet with hyperfinite abelian ambient group G in a sufficiently saturated nonstandard universe. Its observable trace is denoted by

A characterization of liftings
In Section 1.2 we noted that for a locally compact Hausdorff space X, represented as the observable trace X ∼ = X ♭ = X f /E of an IMG triplet (X, E, X f ) with hyperfinite X , not every S-integrable function f ∈ * C X is lifting of some function f f f ∈ L 1 (X), but were not able to describe these liftings more closely. For Hausdorff LCA groups, however, we can give an intuitively appealing characterization of such liftings in terms of a certain continuity condition.
Let N be an arbitrary internal norm on the vector space * C G . An internal function f : G → * C is called S-continuous with respect to the norm N (or, briefly, S N - ) for a, x ∈ G.) In case of the p-norms we speak about S p -continuous functions. In particular, S ∞ -continuity coincides with the usual notion of S-continuity.
Let us recall from [19, Section 2.1] that a norm N on * C G is called translation invariant if N( f a ) = N( f ) for any f ∈ * C G , a ∈ G. The following lemma is obvious, once we realize that for any functions f , g ∈ * C G , a ∈ G and internal translation invariant norm N.
Lemma 2.1.1 Let N be an internal translation invariant norm on * C G and f , g ∈ * C G .
If f is S N -continuous and ∥g∥ 1 < ∞ then f * g is S N -continuous as well.
In analyzing the structure of S N -continuous functions we will make use of a family of Then each of the functions ϑ ϱr is S-continuous, even, nonnegative and satisfies both ∥ϑ ϱr ∥ 1 = 1 and ∥ϑ ϱr ∥ ∞ < ∞. Moreover, G 0 ⊆ supp ϑ ϱr ⊆ B ϱ (r), thus, in particular, The family of internal functions ϑ ϱr behaves like an approximate unit for the operation of convolution on the set of all S N -continuous functions in the sense of Hewitt and Ross [10,11]. The precise formulation follows.
Lemma 2.1.2 Let N be any internal norm on * C G . Then for every S N -continuous function f ∈ * C G the system of functions ϑ ϱr * f , where ϱ ∈ V , 0 < r ∈ R, converges to the function f with respect to the norm N in the following sense: for each (standard) ε > 0 there is an internal set Q such that G 0 ⊆ Q ⊆ G and for any ϱ, r the inclusion Let us remark that, for each internal set Q ⊇ G 0 , there are indeed ϱ ∈ V and r > 0 such that B ϱ (r) ⊆ Q, hence the situation described in the lemma is not merely hypothetical.
Proof The S N -continuity of f means, in standard terms, that for each ε > 0 there is The last sentence of Lemma 2.1.2 is an immediate consequence of the standard statement just proved.
The last of our lemmas deals with a density condition for certain S N -continuous functions. To this end denote by C N,1 Proof Taking any function f ∈ C N,1 c (G, G 0 , G f ), we know that the system of functions ϑ ϱr * f , where ϱ ∈ V , r > 0, converges to f with respect to N in the sense of Lemma 2.1.2. It remains to show that ϑ * f ∈ C c (G, G 0 , G f ) for each such a function ϑ = ϑ ϱr .
As all the internal norms ∥·∥ p , 1 ≤ p ≤ ∞, on * C G are translation invariant and satisfy the inclusion C c (G, G 0 , G f ) ⊆ C p,1 c (G, G 0 , G f ), Lemmas 2.1.1-2.1.3 apply to them in particular.
Further, let us denote by We are going to characterize the subspaces L p (G, G 0 , G f ) of the internal linear space The following theorem resembles an early theorem by Rudin [15], Proof For brevity let us denote: Then we are to prove that L p (G, G 0 , G f ) = CM p (G, G 0 , G f ). Clearly, they both are subspaces of the internal vector space * C G and contain the subspace C c (G, G 0 , G f ). We divide the proof into three simpler Claims. Putting them together, the Theorem easily follows.
This is almost obvious. If ( f n ) n∈N is a sequence in L p (G, G 0 , G f ) converging to a function f ∈ * C G and each f n is a lifting of some f f f n ∈ L p (G) then the sequence ( f f f n ) n∈N satisfies the Bolzano-Cauchy condition, hence it converges to a function f f f ∈ L p (G). It is routine to check that f is a lifting of f f f , ie f ∈ L p (G, G 0 , G f ).
It suffices to show that each lifting f of a function f f f ∈ L p (G) is S p -continuous. It is known that the shift a a a → f f f a a a is a uniformly continuous mapping G → L p (G) (see Pedersen [13] or Rudin [16]). Translating this condition into the language of infinitesimals one readily obtains the S p -continuity of f .
and we can additionally assume that A n + A n ⊆ A n+1 . Then, for each n, there is an S-continuous function g n ∈ * C G such that g n (x) = 1 for x ∈ A n , g n (x) = 0 for x ∈ G ∖ A n+1 and 0 ≤ g n (x) ≤ 1 for x ∈ A n+1 ∖ A n . We put f n = f · g n . From ( We adopt a similar and equally justified conven- just in replacing the condition of absolute S-continuity by that of S p -continuity. As it follows from 2.1.4 and 2.1.5, L p (G, G 0 , G f ) ⊆ S p (G, G 0 , G f ), ie, for a function f ∈ M p (G, G 0 , G f ), S p -continuity implies absolute S-continuity (but not vice versa). However, one would like to have a more direct proof of this inclusion. Remark 2 Given any condensing IMG triplet (X, E, X f ) with hyperfinite ambient set X and a nonnegative internal function d : X → * R, the observable trace X = X f /E is a Hausdorff locally compact space, so that it still makes sense to ask which internal functions f : X → * C are liftings of functions f f f ∈ L p (X, m m m), where m m m = m m m d is the Lebesgue measure on X obtained by pushing down the Loeb measure λ d . However, as long as no group structure on X is involved, L p (X, E, X f ) cannot be characterized in terms of S p -continuity. It would be nice to have some reasonable intrinsic characterization of L p (X, E, X f ) within such a more general setting, at least for constant d(x) = d such that d |A| ̸ ≈ 0 for some and d |A| < ∞ for each internal set A ⊆ X f .

Remark 3
The characterizing conditions of L p (G, G 0 , G f ) make sense also for p = ∞. More precisely, the conjunction of the conditions ∥ f ∥ ∞ < ∞, ∥ f · 1 Z ∥ ∞ ≈ 0 for each internal set Z ⊆ G ∖ G f and S ∞ -continuity defines the subspace C 0 (G, G 0 , G f ) of S-continuous internal functions f : G → * C which are finite on the whole G and infinitesimal outside of G f . Thus we could formally write L ∞ (G, G 0 , G f ) = C 0 (G, G 0 , G f ). This, however, would interfere with the adopted standard notation, as such an L ∞ (G, G 0 , G f ) would be formed just by the liftings of functions in C 0 (G) which is a proper closed subspace of the Banach space L ∞ (G) (cf Proposition 1.1.1).

The Smoothness-and-Decay Principle
The more smooth is a function f : R n → C, the more rapidly its Fourier transform f : R n → C decays; conversely, the more rapidly a function f : R n → C decays, the smoother is its Fourier transform f : R n → C. This vague informal statement is known as the Smoothness-and-Decay Principle and-jointly with the Uncertainty Principle to which it is closely related-it belongs to fundamental heuristic principles of the Fourier or time-frequency analysis. It can take the form of various precise mathematical statements, some of which generalize from R or R n to arbitrary LCA groups (see, eg, Gröchenig [8] and Tao [17] for discussion).
The view through the lens of an IMG group triplet (G, G 0 , G f ) with hyperfinite abelian ambient group G and its dual triplet G , G ‹ f , G ‹ 0 offers an intuitively appealing explanation of this principle for internal functions f : G → * C, based on the Fourier inversion formula in which both S-continuous characters γ ∈ G ‹ 0 as well as non-S-continuous characters γ ∈ G ∖ G ‹ 0 occur. If f is smooth or continuous (in some intuitive meaning of these P Zlatoš words) then the contribution of the non-S-continuous characters to the above expansion of f must be negligible in some sense. This condition causes a kind of quick decay of f . The other way around, viewing the elements x ∈ G as characters of the dual group G , the Fourier transform of f can be expressed as their linear combination: The next theorem is a fairly general precise statement of this form of the Smoothnessand-Decay Principle. Both its formulation as well as its proof borrow some ideas from a paper by Pego [14].
A pair of internal norms N on * C G and M on * C G is called Fourier compatible if the Fourier transform F : * C G → * C G is a bounded linear operator with respect to the norms N, M, ie, for each f ∈ * C G . This is equivalent to the S-continuity of F , ie: Theorem 2.2.1 (The Smoothness-and-Decay Principle) Let N, M be Fourier compatible internal norms on the linear spaces * C G , * C G , respectively. Then for every function f ∈ * C G the following implications hold: Proof (a) In this part of proof we will once more make use of the families of internal functions h ϱr and ϑ ϱr (see Lemma 2.1.2 and the text immediately preceding it).
(b) Assume that N( f ) is finite and N( f · 1 X ) ≈ 0 for each internal set X ⊆ G ∖ G f . We are to show that M f γ − f ≈ 0 for any γ ∈ G ‹ f . First notice that: Obviously, ε ≈ 0. As N is absolute: Therefore, M f γ − f ≈ 0 as well.
The last theorem applies to any pair of norms ∥·∥ p on * C G and ∥·∥ q on * C G for 1 ≤ p ≤ 2 and q = p/(p − 1), including p = 1, q = ∞, in which case we have: For every function f ∈ * C G such that ∥ f ∥ 2 < ∞ the following conditions are equivalent: Corollary 2.2.5 generalizes a result by Albeverio, Gordon and Khrennikov [2], where the equivalence of conditions (i), (ii) and (iv) in case there is an internal subgroup K of G such that G 0 ⊆ K ⊆ G f was proved. This assumption is equivalent to the existence of a compact open subgroup of G ♭ . It is also mentioned there without proof that the group of reals R can be represented as well as R ∼ = G ♭ = G f /G 0 for some triplet (G, G 0 , G f ) satisfying their restricted version of Corollary 2.2.5.

Generalized
by Corollary 2.2.2(c). Thus it suffices to prove that for each γ ∈ G ‹ 0 . However, as γ is bounded and S-continuous, ie, γ ∈ C bu (G, G 0 ), it is routine to check that the internal function f γ ∈ L 1 (G, G 0 , G f ) is a lifting of the function f f f γ ♭ ∈ L 1 G . Then by Ziman and Zlatoš [18,Proposition 3.5]: See also the text preceding Proposition 1.2.3.) For 1 < p ≤ 2 and 1/p + 1/q = 1 the Fourier transform F : L p (G) → L q G is defined as the continuous extension (with respect to the norms ∥·∥ p on L p (G) and ∥·∥ q on L q G ) of the restriction of the Fourier transform F : L 1 (G) → C 0 G) to the dense subspace L p (G) ∩ L 1 (G) of L p (G) (see Hewitt and Ross [11], Loomis [12] or Rudin [16]). For functions in this subspace everything works as in the proof above. Thus, by a continuity argument Theorem 2.3.1 together with Corollary 2.2.3(c) give rise to HFD approximations of the classical Fourier transforms F : L p (G) → L q G) in a similar way. The case p = q = 2 of the Fourier-Plancherel transform F : L 2 (G) → L 2 G settles Gordon's Conjecture 3.
The HFD Fourier Transform Approximation Theorem 2.3.1 extends to the Fourier-Stieltjes transform F : M(G) → C bu G), as well. for every internal function In particular, if g g g ∈ L 1 (G), dµ µ µ = g g g dm m m and g ∈ L 1 (G, G 0 , G f ) is a lifting of g g g then for each γ ∈ G ‹ 0 , reproving Theorem 2.3.1. This account indicates that it is Theorem 2.3.3 which is crucial for hyperfinite dimensional approximations of the Fourier transform on LCA groups. Therefore we address the issue raised in the remark closing the introductory part of Zlatoš [19, Section 2.5] primarily for the Fourier-Stieltjes transform.
Assume, for the rest of this section, that (G, G 0 , G f ) is an IMG group triplet with hyperfinite abelian ambient group G, arising from an HFI approximation η : G → * G of the Hausdorff LCA group G. Let us denote F η : * C G → * C * G the internal linear operator given by for f ∈ * C G , χ χ χ ∈ * G. The modified discrete Fourier transform F η , defined by means of the internal inner product on * C G , can be employed for the approximation of the classical Fourier transform on G without the need to mention the adjoint HFI approximation ϕ : G → * G of the dual group G .

P Zlatoš
Due to saturation there is an internal set X such that G f ⊆ X ⊆ G and * γ γ γ(η x) ≈ γ(x) holds for all x ∈ X . Denoting Y = G ∖ X we have Then for each γ γ γ ∈ G. Corollary 2.3.6 Let 1 < p ≤ 2 ≤ q < ∞ be dual exponents and F : L p (G) → L q G be the Fourier transform on G.
Let further f f f ∈ L p (G) and f ∈ L p (G, G 0 , G f ) be a lifting of f f f . Then for almost all γ γ γ ∈ G with respect to the Haar measure on G.

Some standard analogues
The reader might naturally expect that we will derive some standard analogues of the nonstandard hyperfinite dimensional Fourier approximation theorems from Section 2.3. Then, as usual in such cases, these standard results would be "highly existential" and giving no explicit bounds for the precision of the approximations. Therefore, it is surprising that, given a Hausdorff LCA group G and a function f f f ∈ L 1 (G) or a measure µ µ µ ∈ M(G), we can explicitly describe some functions f , g : G → C defined on some finite abelian group G approximating f f f or µ µ µ, such that their discrete Fourier transforms f = F( f ), g = F(g) approximate the Fourier transforms f f f = F( f f f ), µ µ µ = F(µ µ µ), respectively. Moreover, we are able to give some explicit bounds for the precision of the approximations of the Fourier transforms F( f f f ) or F( µ µ µ) in terms of the norm ∥ f f f ∥ 1 or the total variation ∥µ µ µ∥ and a parameter ε > 0, given in advance, describing the precision of the approximation of f f f or µ µ µ, respectively, by f or g. These results (Theorems 2.4.4 and 2.4.5) depend just on the Adjoint Approximation Scheme from Zlatoš [19,Theorem 2.5.5], so that the only nonstandard ingredient they are based on is the Adjoint Hyperfinite LCA Group Approximation Theorem [19,Corollary 2.5.2] from which Theorem 2.5.5 follows. For a compactly supported continuous function f f f : G → C (or, more generally, for f f f ∈ C 0 (G) ∩ L 1 (G) or f f f ∈ C 0 (G) ∩ L p (G)) even "nicer" discrete approximations are available. The corresponding estimates (Theorems 2.4.7 and 2.4.8), however, rest on an additional nonstandard result, namely on the fact that the Haar measure on G can be obtained by pushing down the Loeb on an hyperfinite approximating group G constructed from a properly normalized counting measure on G.
Taking advantage of its generality, we will start with the approximation of the Fourier-Stieltjes transform F : M(G) → C bu G .
Assume that X is a Hausdorff locally compact space whose topology is induced by some uniformity U , (K, U) is an X-raster and η : X → X is a finite (K, U) approximation of X. An η -tagged U-fine Borel partition of K, briefly a tagged Borel (U, η) partition of K, is a finite family π π π = {(P 1 , x 1 ), . . . , (P n , x n )} where P i are nonempty pairwise disjoint Borel sets such that K = P 1 ∪ . . . ∪ P n and the elements x i ∈ X satisfy P i ⊆ U[η(x i )] as well as η(x) = η(x i ) for all i ≤ n and x ∈ η −1 [P i ]. (The reader should notice that η(x i ) / ∈ P i , in which case η −1 [P i ] = ∅, may still happen.) Lemma 2.4.1 For every X-raster (K, U) there exist a finite (K, U) approximation η : X → X of X and an η -tagged U-fine Borel partition π π π = {(P 1 , x 1 ), . . . , (P n , x n )} of K.
Proof There exists an entourage V ⊆ U which is an open subset of X × X. Let η : X → X be a finite (K, V) approximation of X (thus η is a (K, U) approximation, of X, as well). Let Y be a minimal subset of X with the property that, for any x ∈ X ,  Let µ µ µ be a complex regular Borel measure on X with finite total variation. Depending on some finite (K, U) approximation η : X → X and a tagged Borel (U, η) partition π π π = {(P 1 , x 1 ), . . . , (P n , x n )} of K, we will define a function g π π π µ µ µ : X → C enabling to approximate the integration with respect to the measure µ µ µ in a sense to be made precise shortly. We denote P i = η −1 [η(x i )] = {x ∈ X : η(x) = η(x i )} for i ≤ n and put: Then the function g π π π µ µ µ is an "approximate weak lifting" of µ µ µ, in the following sense.
Proposition 2.4.2 Let µ µ µ ∈ M(X), h h h ∈ C b (X) and ε > 0. Assume that (K, U) is an X-raster such that |µ µ µ|(X ∖ K) ≤ ε and |h h h(x x x) − h h h(y y y)| ≤ ε for any x x x, y y y ∈ X whenever x x x ∈ K and (x x x, y y y) ∈ U. Finally, let η : X → X be a finite (K, U) approximation of X and π π π = {(P 1 , x 1 ), . . . , (P n , x n )} be a tagged Borel (U, η) partition of K. Then: h h h dµ µ µ − x∈X h h h(ηx) g π π π µ µ µ (x) ≤ ε ∥h h h∥ ∞ + ∥µ µ µ∥ Before passing to the proof itself, the reader should realize that, since the variation ∥µ µ µ∥ = |µ µ µ|(X) is finite, there is indeed a compact set K ⊆ X such that |µ µ µ|(X ∖ K) ≤ ε, and, due to the continuity of h h h, the compactness of K and local compactness of X, there is an entourage U such that |h h h(x x x) − h h h(y y y)| ≤ ε for any x x x ∈ K, y y y ∈ U[x x x].
Proof A straightforward computation using the notation from the definition of the function g π π π µ µ µ gives: For the first summand on the right we have: Using the fact that η(x) = η(x i ) for x ∈ P i , we obtain the following estimate for each of the remaining right-hand summands: Putting things together we obtain: In the special case when X = G is a Hausdorff LCA group and h h h = χ χ χ where χ χ χ ∈ G is a continuous character of G we obtain the following standard counterpart of Proposition 2.3.4 as a corollary. In the next three results we assume that the inner product on the unitary space C G of functions on the finite abelian group G below is normalized by the coefficient d = 1.
Let m m m be the Haar measure on the Hausdorff LCA group G; the inner product, the p-norms and the Fourier transform f f f = F( f f f ) on the corresponding Lebesgue spaces L p (G) (1 ≤ p ≤ 2) are defined via the Haar integral on G. In the next theorem, nonetheless, the inner product, the corresponding norms and the discrete Fourier transform f = F( f ) on the space C G over a finite abelian group G still use the normalizing multiplier d = 1.
For a function f f f ∈ L 1 (G) we denote by µ µ µ f f f the regular complex Borel measure such that dµ µ µ f f f = f f f dm m m; its total variation is µ µ µ f f f = ∥ f f f ∥ 1 . Given a G-raster (K, U), a finite (K, U) approximation η : G → G of G and a tagged Borel (U, η) partition π π π = {(P 1 , x 1 ), . . . , (P n , x n )} of K, we denote f π π π = g π π π µ µ µ f f f where the function g π π π µ µ µ f f f was defined prior to Proposition 2.4.2. Then, as a special case of the Finite Fourier-Stieltjes Approximation Theorem 2.4.4, we obtain the following result.
Since C c (G) is a dense subspace in L 1 (G), it would be sufficient in some sense to deal with compactly supported continuous functions f f f ∈ L 1 (G) in the last Theorem 2. tables of the function f f f restricted to the set Γ Γ Γ. Before passing to a more precise formulation, we need to introduce some notions and formulate some preliminary results.
Given a Hausdorff LCA group G, a symmetric compact neighborhood K of 0 ∈ G and an ε > 0, a pair (U, η) consisting of a symmetric neighborhood U ⊆ K of 0 and a finite (K, U) approximation η : for every continuous function f f f : G → C such that supp f f f ⊆ K.
Once we realize that every HFI approximation η : G → * G is a ( * K, * U) approximation for any G-raster (K, U) and, due to the fact that the Haar measure on G can be obtained by pushing down the Loeb measure on * G obtained from the normalizing multiplier d = m m m(K) η −1 K , the pair (U, η) is ε-adequate for K and each ε > 0, we readily obtain the following lemma by the transfer principle.
Lemma 2.4.6 Let G be a Hausdorff LCA group, K be a symmetric compact neighborhood of 0 ∈ G and ε > 0. Then there exists a symmetric neighborhood U 0 ⊆ K of 0 such that for any G-raster (K, U) satisfying U ⊆ U 0 and any finite (K, U) approximation η : G → G the pair (U, η) is ε-adequate for K.
In the next theorem the inner product on the finite dimensional linear space C G , the discrete Fourier transform F : C G → C G and all the p-norms on C G use the normalizing coefficient Theorem 2.4.7 (The Finite Fourier Approximation Theorem 2) Let G be a Hausdorff LCA group, f f f ∈ C c (G) and ε ∈ (0, π/3). Assume that a G-raster (K, U) and a G-raster (Γ Γ Γ, Ω Ω Ω) together with a finite abelian group G and maps η : G → G, ϕ : G → G form an ε-adjoint approximation scheme of the pair of groups G, G which is ε-pairing preserving with reserve. Finally, let supp f f f ⊆ K and the pair (U, η) be ε-adequate for K. Then |F( f f f )(χ χ χ) − F( f f f • η)(γ)| ≤ ε 1 + 2m m m(K) ∥ f f f ∥ ∞ for any χ χ χ ∈ Γ Γ Γ, γ ∈ G such that ϕ(γ) ∈ χ χ χ Ω Ω Ω.
Theorem 2.4.8 (The Generalized Finite Fourier Approximation Theorem) Let G be a Hausdorff LCA group, f f f ∈ C c (G), 2 ≤ q ≤ ∞ and ε ∈ (0, π/3). Assume that a G-raster (K, U) and a G-raster (Γ Γ Γ, Ω Ω Ω) together with a finite abelian group G and maps η : G → G, ϕ : G → G form an ε-adjoint, ε-pairing preserving approximation scheme of the pair of groups G, G. Finally, let supp f f f ⊆ K and the pair (U, η) be ε-adequate for K. Then, denoting Γ = ϕ −1 Γ Γ Γ , we have: As already mentioned prior to the theorem, the case p = 1, q = ∞ (under the convention 1/∞ = 0) directly follows from the proof of Theorem 2.4.7. Using this fact, a straightforward computation gives: ( Theorems 2.4.7 and 2.4.8 can easily be generalized, under slightly modified upper bounds, to functions f f f ∈ C 0 (G) ∩ L 1 (G) or f f f ∈ C 0 (G) ∩ L p (G). This is left to the reader. As pointed out by Gordon in [7] for the Fourier-Plancherel transform L 2 (G) → L 2 G , the class of functions f f f ∈ L 2 (G) approximable by the composition f f f • η such that their Fourier transform F( f f f ) is simultaneously approximable by the discrete Fourier transform F( f f f • η) in the sense of Theorem 2.4.8 contains even more general functions, namely such that both f f f and f f f are Riemann integrable, ie, continuous almost everywhere with respect to the Haar measures on G, G, respectively. It is clear that Gordon's remark applies to any pair of adjoint exponents p ∈ [1,2], q ∈ [2, ∞] and not just to the Hilbert space case p = q = 2.