Approximations in $L^1$ with convergent Fourier series

For a separable finite diffuse measure space $\mathcal{M}$ and an orthonormal basis $\{\varphi_n\}$ of $L^2(\mathcal{M})$ consisting of bounded functions $\varphi_n\in L^\infty(\mathcal{M})$, we find a measurable subset $E\subset\mathcal{M}$ of arbitrarily small complement $|\mathcal{M}\setminus E|<\epsilon$, such that every measurable function $f\in L^1(\mathcal{M})$ has an approximant $g\in L^1(\mathcal{M})$ with $g=f$ on $E$ and the Fourier series of $g$ converges to $g$, and a few further properties. The subset $E$ is universal in the sense that it does not depend on the function $f$ to be approximated. Further in the paper this result is adapted to the case of $\mathcal{M}=G/H$ being a homogeneous space of an infinite compact second countable Hausdorff group. As a useful illustration the case of $n$-spheres with spherical harmonics is discussed. The construction of the subset $E$ and approximant $g$ is sketched briefly at the end of the paper.


Introduction
In the present paper we work with finite measure spaces (M, Σ, µ). For efficiency of nomenclature we will write M = (M, Σ, µ) and |A| = |A| µ = µ(A) for every A ∈ Σ, where the σ-algebra Σ and the measure µ are clear from the context. Consider a separable finite measure space (M, Σ, µ).
Separability here simply means that all spaces L p (M) for 1 ≤ p < ∞ are separable. Let {ϕ n } ∞ n=1 be an orthonormal basis of L 2 (M) with ϕ n ∈ L ∞ (M) for all n ∈ N. For a function f ∈ L 1 (M) we denote its Fourier components by where ϕ * n denotes the complex conjugate of ϕ n . The possibly divergent Fourier series of f will be ∞ n=1 Y n (x; f ).
Note that already for the trigonometric system on the interval there exists an integrable function of which the Fourier series diverges in L 1 ( [1], Chapter VIII, §22). We will often make use of Fourier polynomials and orthogonal series of the form without reference to a particular function for which these may be the Fourier components. Denote by the spectrum of a function f .
Before stating our main theorem let us recall the notion of diffuseness for a measure space. The main result of this paper is the following Theorem 1 Let M be a separable finite diffuse measure space, and let {ϕ n } ∞ n=1 be an orthonormal system in L 2 (M) consisting of bounded functions ϕ n ∈ L ∞ (M), n ∈ N. For every ǫ, δ > 0 there exists a measurable subset E ∈ Σ with measure |E| > |M| − δ and with the following property; for each function f ∈ L 1 (M) with f 1 > 0 there exists an approximating function g ∈ L 1 (M) that satisfies: 3. the Fourier series of g converges in L 1 (M),

Theorem 2
Let M be a separable finite diffuse measure space, and let {ϕ n } ∞ n=1 be an orthonormal system in L 2 (M) consisting of bounded functions ϕ n ∈ L ∞ (M), n ∈ N. There exists an increasing , so that the following statements hold: 3. the Fourier series of g m converges in L 1 (M), ∀m ∈ N, Remark 1 Not for every orthonormal system {ϕ n } ∞ n=1 does an arbitrary integrable function f ∈ L 1 (M) have an orthogonal series ∞ n=1 Y n of the form (1) that converges to f in L 1 (M), and if that happens then ∞ n=1 Y n is necessarily the Fourier series of f , i.e., Y n = Y n (f ).
For instance, in case of spherical harmonics this is guaranteed only in L 2 (S 2 ) [2]. However, the following weaker statement is a corollary of Theorem 2 and holds true for all integrable functions.
Corollary 1 Under the assumptions of Theorem 2, there exists an increasing sequence of subsets

The general case
Theorem 1 is true for every finite separable diffuse measure space M, but it will be more convenient to reduce the problem to that for a smaller class of measure spaces and then to prove the theorem for that class. First let us show that Theorem 1 is invariant under isomorphisms of measure algebras.
For that purpose we will reformulate Theorem 1 in a way that makes no reference to the actual measure space M but only to its measure algebra B(M). We note that if we replace the set E produced by Theorem 1 by another measurable set E ′ ∈ Σ such that the symmetric difference is null, |E △ E ′ | = 0, then all statements of the theorem remain valid with E ′ instead of E. This brings us to the following equivalent formulation of Theorem 1.
Theorem 3 Let M be a finite separable diffuse measure space, and let {ϕ n } ∞ n=1 be an orthonormal system in L 2 (M) consisting of bounded functions ϕ n ∈ L ∞ (M), n ∈ N. For every ǫ, δ > 0 there exists a function χ E ∈ L ∞ (M) with χ 2 E = χ E and χ E 1 > |M | − δ, with the following properties; for each function f ∈ L 1 (M) with f 1 > 0 there exists an approximating function g ∈ L 1 (M) that satisfies: 3. the Fourier series of g converges in L 1 (M), In this form the theorem relies only upon spaces L p (M), p = 1, 2, ∞, which can be constructed purely out of the measure algebra B(M) with no recourse to the underlying measure space M.
In particular, if two measure spaces have isomorphic measure algebras then the statements of Theorem 1 on these two spaces are equivalent.  that the result cited in Remark 2 is very abstract and the produced isomorphisms are in general far from being geometrically natural. Our proof of Theorem 1 is constructive, and the construction of the set E highly depends on the cylindric structure. If the space at hand has a natural cylindric structure then this approach gives a geometrically more sensible set E than what we would expect had we identified the cylinder with the unit interval through a wild measure algebra isomorphism.

The particular case
In this section we will prove the main theorem for the particular case where M = (M, Σ, µ) is a separable cylindric probability space Here N = (N , Σ 0 , ν) is any separable probability space. We will write M ∋ x = (t, y) ∈ [0, 1] × N .

The core lemmata
First let us state a variant of Féjér's lemma.
This lemma is given in [1, page 77] with [a, b] = [−π, π], but the proof for arbitrary a and b follows with only trivial modifications.
We proceed to our first critical lemma.
N ∈ N be given. Then there exists a function g ∈ L ∞ (M), a measurable set Σ ∋ E ⊂ ∆ and a Fourier polynomial of the form Proof: Set Define the 1-periodic function I : R → R by setting for t ∈ [0, 1) and continuing periodically. Then obviously By Féjér's lemma Choose a natural number s 0 ∈ N sufficiently large so that Set Then it can be seen that where the first inequality of formula (8) and then formula (4) were used in the last step. Clearly, g ∈ L ∞ (M) and thus we have proven Statements 1 and 2. Next we note using (4) that and then by (1 − δ)|∆| < |E| < |∆| we find that which together entail Similarly, Formulae (9) and (13) imply that which proves Statement 3. In a similar fashion we obtain We have g ∈ L ∞ (M) ⊂ L 2 (M), and therefore the Fourier series Y n (g) converges to g in L 2 (M).
Thus we can choose the natural number M ∈ N so large that Further, from formula (8) we estimate the magnitude of the first N Fourier coefficients of g as Finally, set In order to prove Statement 4 we write where formulae (17) and (18) were used in the last step along with the normalization ϕ n 2 = 1.
Using the pairwise orthogonality of the Fourier components Y n (g) and formula (16) we can obtain which immediately yields thus proving Statement 5. Note that the inequality . 1 ≤ . 2 used above hold thanks to the convenient assumption that we are in a probability space.
such that and every partition 0 = x 0 < x 1 < . . . < xν 0 = 1,ν 0 ∈ N, of the unit interval, the product partition is a measurable partition of M with the property that For every product partition {∆ k } ν0 k=1 as above and every tuple of real numbers {γ k } ν0 k=1 consider the step function By the assumption of separability of M we know that step functions of the form (26) subordinate to product partitions are dense in all spaces L p (M) for 1 ≤ p < ∞. Choose a product partition and a subordinate step function such that Note that the numbers γ k are not assumed to be distinct, thus we can refine the given partition without changing γ k and the function Λ(x). We use the property (25) to refine the product partition Now we apply Lemma 2 iteratively with with the following properties: First we check that from (26), (1 • ), (2 • ) and (31) it follows that so that Statements 1 and 2 are proven. Next we observe using (4 • ), (27), (31) and (32) that which proves Statement 4. Further, from (26), (27), (2 • ) and (3 • ) we deduce that Moreover, the same formulae also imply i.e., f 1 < 3 g 1 , thus proving Statement 3. In order to prove Statement 5 let us fix an N 0 ≤ m ≤ N . Then there is a 1 ≤ k 0 ≤ ν 0 such that N k0−1 ≤ m < N k0 , and thus by (30) and (32) Finally we use this along with formulae (3 • ), (4 • ), (5 • ) and (28) to obtain m n=N0 Y n (x) and this completes the proof.
Lemma 4 Let {R k } ∞ k=1 be any fixed ordering of the set of all nonzero Fourier polynomials with rational coefficients into a sequence. Then for every f ∈ L 1 (M) and sequence {b s } ∞ s=1 of positive Proof: Let us first convince ourselves that Fourier polynomials with rational coefficients are dense in L 1 (M). Indeed, by the assumption of separability, step functions are dense in L 1 (M), but they all belong also to the separable Hilbert space L 2 (M). On the other hand, Fourier polynomials are clearly dense in L 2 (M). And finally, an arbitrary Fourier polynomial can be approximated in L 2 (M) by a Fourier polynomial with rational coefficients (this amounts to approximating the Fourier coefficients by rational numbers). A three-epsilon argument together with . 1 ≤ . 2 then yields the assertion.
Using the denseness of {R k } ∞ k=1 let us choose a natural number k 0 ∈ N such that Then we can choose further natural numbers k s ∈ N iteratively as follows. For every s ∈ N, again by using the denseness argument, choose a number k s so that k s > k s−1 and Statements 1 and 3 are clearly satisfied. For Statement 2 we have Lemma is proven.

The main theorem
Here we will prove Theorem 1 for the particular case of (M, Σ, µ) being a separable cylindric probability space as in (3).
Proof: Recall that ǫ, δ > 0 and f ∈ L 1 (M) with f 1 > 0 are given. Denote Let {R k } ∞ k=1 be any ordering of the set of all nonzero Fourier polynomials with rational coefficients into a sequence. Iteratively applying Lemma 3 with for k = 1, 2, . . . we obtain for each k ∈ N a subsetẼ k ∈ Σ, a functiong k ∈ L ∞ (M), a number N k−1 ≤ N k ∈ N (set N 0 = 0) and a Fourier polynomial with the following properties: Define the desired set E as Observe from (1 † ) that Note that E is universal, i.e., independent of f .
Let {R ks } ∞ s=0 be the subsequence of Fourier polynomials provided by Lemma 4 applied with It satisfies We want to use mathematical induction in order to define a sequence of natural numbers 1 < ν 1 < ν 2 < . . . and a sequence of functions {g s } ∞ s=1 , g s ∈ L 1 (M), such that for all s ∈ N we have 1 * . x ∈Ẽ νs implies g s (x) = R ks (x), Assume that for some s ∈ N, the choice of 1 < ν 1 < ν 2 < . . . < ν s−1 and g 1 , g 2 , . . . , g s−1 satisfying (3 * ) has been already made (for s = 1 this is trivially correct). Remember that by σ(h) we have denoted the {ϕ n }-spectrum of a function h ∈ L 1 (M), i.e., the support of its Fourier series. Using the denseness of {R k } ∞ k=1 (see Lemma 4) choose a natural number ν s ∈ N such that N ν1−1 > max σ(R k0 ) and ν s > ν s−1 for s > 1, and Then by (2 • ) and (3 * ) we have for all s ∈ N that which combined with (49) implies Set Condition (1 * ) is easily satisfied thanks to (2 † ) with k = ν s . For condition (2 * ) we write where we used (49), (3 † ), (3 * | s−1 ) and (51) in the last step. To show that condition (3 * ) is satisfied where (4 † ), (49) and ν s > s were used in the second inequality. Finally we satisfy condition (4 * ) using (5 † ) and (51), The iteration is thus complete, and by mathematical induction we construct the sequences {ν s } ∞ s=1 and {g s } ∞ s=1 satisfying conditions (1 * ) through (4 * ) for all s ∈ N. Define From (53) it follows that thus g ∈ L 1 (M). The construction is now complete, and it remains to verify the statements of the theorem.
To prove Statement 2 of the theorem we note that x ∈ E means x ∈Ẽ νs , and hence by (2 † ) g s (x) = R ks (x) for all s ∈ N. It then follows from (3 • ) that Let {Y n } ∞ n=1 be the series of Y n = c n ϕ n such that Let m ∈ N, and let r ∈ N be the largest natural number such that N νr −1 ≤ m (if m < N ν1−1 set r = 1). Set m * . = min{m, N νr − 1}. Then by (56), (3 * ), (4 * ) and (53) we get Now as m → ∞ obviously r → ∞ as well, thus making the above expression vanish, which proves that Y n is the Fourier series of g, i.e., Y n = Y n (g), and it converges to g as required in Statement 3. Further, from (2 • ), (3 • ), (56) and (57) we have that which in view of (43) proves Statement 1. Finally, using (60) and (61) we establish that but also Note that by (61) and therefore 25 f 1 < 32 g 1 . This together with (63) yields m n=1 Y n 1 < 23 25 which establishes Statement 4. The proof of the theorem is accomplished.

Compact groups
Let G be a compact Hausdorff topological group, Σ the Borel σ-algebra and dµ(x) = dx the normalized Haar measure. Then µ is diffuse if and only if G is infinite, which we will assume here. Now by [12,Theorem 28.2] we have that dim L 2 (G) = w(G), therefore (G, Σ, µ) is separable if and only if G is second countable. This will also be assumed in what follows. This implies in particular, through Peter-Weyl theorem, that the dualĜ is countable. For a detailed exposition of harmonic analysis on compact groups consult, e.g., [20] or [3].
By the Peter-Weyl theorem {ϕ ρ,i,j } is an orthonormal basis in L 2 (G). Moreover, since ρ(x) is unitary for all x ∈ G, we get so that ϕ ρ,i,j ∈ L ∞ (G). Therefore, if we put an arbitrary (total) order onĜ then Theorem 1 is directly applicable to (G, Σ, µ) with {ϕ ρ,i,j }.

But the arbitrary choice of the bases {e
is artificial from the viewpoint of the group G. The more natural construction is the operator valued Fourier transform, and the corresponding block Fourier series More generally, if we work on a homogeneous space M ≃ G/H of a compact group G as above with (closed) isotropy subgroup H ⊂ G, then we define multiplicities (this reduces to d H ρ = d ρ if H = {1} as before). Moreover, we restrict to A point x ∈ G/H is a coset x = xH, x ∈ G. If dh is the normalized Haar measure on H then there is a unique normalized left G-invariant measure µ on G/H (the pullback of dx through the quotient Denote be an orthonormal basis in H ρ as before, and choose an orthonormal basis {e ρ α } d H ρ α=1 in P H H ρ . Now the Fourier transform of a function f ∈ L 1 (G/H) becomesf and the corresponding block Fourier series is If the homogeneous space G/H is infinite then the invariant measure µ is diffuse. Applying Theorem 1 to (G/H, Σ H , µ) (Σ H is the Borel σ-algebra on G/H) and the system {ϕ ρ,i,α } with G/H ordered arbitrarily, we obtain the following modification.
Theorem 4 Let M = G/H be an infinite homogeneous space of a compact second countable Hausdorff group G with closed isotropy subgroup H ⊂ G. For every ǫ, δ > 0 there exists a measurable subset E ∈ Σ H with measure |E| > 1 − δ and with the following property; for each function f ∈ L 1 (G/H) with f 1 > 0 there exists an approximating function g ∈ L 1 (G/H) that satisfies: 3. the block Fourier series (75) of g converges in L 1 (G/H),

we have
Note that E depends on the chosen order in G/H.
As discussed before, the proof of the above theorem becomes more constructive and transparent if we have a natural cylindric structure in G/H. Then the proof based on the cylindric structure becomes directly applicable (without intermediate measurable transformations), and each step in the proof retains its original interpretation in terms of cylindrical coordinates. To this avail, below we will establish a natural Borel almost isomorphism of measure spaces between G/H and the cylindric space K × K\G/H for certain infinite closed subgroups K ⊂ G, which will provide an obvious identification of all spaces L p (G/H) and L p (K × K\G/H). Note that since K is infinite and compact, the probability space (K, dk) is isomorphic to the unit interval.
Definition 2 A measurable map between two measure spaces ϕ : (Ω 1 , µ 1 ) → (Ω 2 , µ 2 ) is an almost isomorphism of measure spaces if there exist full measure subspaces X ⊂ Ω 1 and Y ⊂ Ω 2 , |Ω 1 \X| = |Ω 2 \ Y | = 0, such that the restriction of ϕ is an isomorphism of measure spaces ϕ| X : (X, If K ⊂ G is a closed subgroup of G then denote by G/H (K) ⊂ G/H the set of those points x ∈ G/H with a non-trivial stabilizer within K, Let us now fix an infinite closed subgroup K ⊂ G, and let q : G/H → K\G/H be the natural quotient map. Then the pullback measure ν = µ • q −1 = q * µ is the natural probability measure on K\G/H. Provided that the subset G/H (K) in G/H is µ-null, we obtain a natural product structure in the following way.
Finally, let us note that K ·Z = G/H. Indeed, for every x ∈ G/H we have that z = q | −1 Z (q(x)) ∈ Z, and since q(x) = q(z) we have that ∃k ∈ K such that kz = x. On the other hand, it is easy to x then for every k ∈ K it follows that k y y = y, where y = kx and k y = k 0 k x k −1 0 , which means that y ∈ G/H (K) . Therefore so that |Y | µ = |X| dk⊗ν = 1. This completes the proof.

Spheres
As an instructive illustration of the above constructions we will consider spheres S d , 2 ≤ d ∈ N, with their Euclidean (Lebesgue) probability measures (surface area normalized to one). For d = 2 the Statements 2 and 3 of Theorem 1 were obtained in [4].
The sphere S d can be considered as the homogeneous space G/H with G = SO(d + 1) and Harmonic analysis in these homogeneous spaces is a classical subject widely available in the literature (see e.g. [21]). The dual space G/H consists of irreducible representations by harmonic polynomials of fixed degree ρ ∈ N 0 , and it is conveniently ordered according to that degree, ρ ∈ G/H ≃ N 0 . The dimension of the representation ρ is whereas the multiplicities are all d H ρ = 1. We choose standard spherical coordinates x = (θ 1 , . . . , θ d−1 , φ), where θ j ∈ [0, π], j = 1, . . . , d − 1, and φ ∈ [0, 2π). The orthonormal system {ϕ ρ,i,α } in this case consists of spherical harmonics The block Fourier series of a function This is the cylindric structure used implicitly in [4].

Construction of E and g
The proof of the main theorem above is constructive, although the construction of the set E and of the approximating function g may be hard to follow due to the complexity of the proof. In this last section we will very briefly sketch that construction step by step.
• Choose an arbitrary ordering {R k } ∞ k=1 of all Fourier polynomials with rational coefficients.
• For every k ∈ N, choose a partition {∆ l (k)} ν0(k) l=1 of the cylindric measure space M = [0, 1]×N of the form ∆ l (k) = [a l (k), b l (k)] ×∆ l (k) such that the measures |∆ l (k)| are small enough, as well as a subordinate real step function Λ(k) = ν0(k) l=1 γ l (k)χ ∆ l (k) (χ X is the indicator function of the subset X), such that Λ(k) − R k 1 is sufficiently small.