Abstract
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 nspheres with spherical harmonics is discussed. The construction of the subset E and approximant g is sketched briefly at the end of the paper.
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1 Introduction
In the present paper we work with finite measure spaces \(({\mathcal {M}},\Sigma ,\mu )\). For efficiency of nomenclature we will write \({\mathcal {M}}=({\mathcal {M}},\Sigma ,\mu )\) and \(A=A_\mu =\mu (A)\) for every \(A\in \Sigma \), where the \(\sigma \)algebra \(\Sigma \) and the measure \(\mu \) are clear from the context. Consider a separable finite measure space \(({\mathcal {M}},\Sigma ,\mu )\). Separability here simply means that all spaces \(L^p({\mathcal {M}})\) for \(1\le p<\infty \) are separable. Let \(\{\varphi _n\}_{n=1}^\infty \) be an orthonormal basis of \(L^2({\mathcal {M}})\) with \(\varphi _n\in L^\infty ({\mathcal {M}})\) for all \(n\in {\mathbb {N}}\). For a function \(f\in L^1({\mathcal {M}})\) we denote its Fourier components by
where \(\varphi _n^*\) denotes the complex conjugate of \(\varphi _n\). The possibly divergent Fourier series of f will be
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.
Definition 1
In a measure space \(({\mathcal {M}},\Sigma ,\mu )\), a measurable subset \(A\in \Sigma \) is called an atom if \(A>0\) and for every \(B\in \Sigma \) with \(B\subseteq A\) either \(B=A\) or \(B=0\). The measure space \(({\mathcal {M}},\Sigma ,\mu )\) is called diffuse or nonatomic if it has no atoms.
The main result of this paper is the following
Theorem 1
Let \({\mathcal {M}}\) be a separable finite diffuse measure space, and let \(\{\varphi _n\}_{n=1}^\infty \) be an orthonormal system in \(L^2({\mathcal {M}})\) consisting of bounded functions \(\varphi _n\in L^\infty ({\mathcal {M}})\), \(n\in {\mathbb {N}}\). For every \(\epsilon ,\delta >0\) there exists a measurable subset \(E\in \Sigma \) with measure \(E>{\mathcal {M}}\delta \) and with the following property; for each function \(f\in L^1({\mathcal {M}})\) with \(\Vert f\Vert _1>0\) there exists an approximating function \(g\in L^1({\mathcal {M}})\) that satisfies:

1.
\(\Vert fg\Vert _1<\epsilon \),

2.
\(f=g\) on E,

3.
the Fourier series of g converges in \(L^1({\mathcal {M}})\),

4.
we have
$$\begin{aligned} \sup _m\left\ \sum _{n=1}^m Y_n(g)\right\ _1<2\min \left\{ \Vert f\Vert _1,\Vert g\Vert _1\right\} . \end{aligned}$$
Luzin proved that every almost everywhere finite function f on [0, 1] can be modified on a subset of arbitrarily small positive measure so that it becomes continuous. Further results in this direction were obtained by Menshov and others. See [4,5,6,7,8,9,10,11, 13,14,15,16,17,18,19] for earlier results in this direction for classical orthonormal systems. Let us note that if \({\mathcal {M}}\) is not diffuse (i.e., it has atoms) then Statement 4 of this theorem may not hold with any coefficient on the right hand side. This is illustrated in the next
Example 1
For every natural \(N\in {\mathbb {N}}\), let \(({\mathcal {M}},\Sigma ,\mu )=({\mathbb {N}}_2,2^{{\mathbb {N}}_2},P)\) be the probability space with orthonormal basis \(\{\varphi _1,\varphi _2\}\) of \(L^2({\mathcal {M}})\), where
Take \(\delta =1/35\) and \(f=(1,0)\). Then \(E>1\delta \) forces \(E={\mathcal {M}}\), and therefore \(f=g\) on E implies \(f=g\) on \({\mathcal {M}}\). Now
Theorem 1 is equivalent to the following theorem, which can be obtained by repeatedly applying Theorem 1 with fixed \(f\in L^1({\mathcal {M}})\) and \(\epsilon _m=\frac{1}{m}\), \(\delta _m=\frac{1}{m}\), \(m=1, 2, \ldots \).
Theorem 2
Let \({\mathcal {M}}\) be a separable finite diffuse measure space, and let \(\{\varphi _n\}_{n=1}^\infty \) be an orthonormal system in \(L^2({\mathcal {M}})\) consisting of bounded functions \(\varphi _n\in L^\infty ({\mathcal {M}})\), \(n\in {\mathbb {N}}\). There exists an increasing sequence of subsets \(\{E_m\}_{m=1}^\infty \), \(E_m\subset E_{m+1}\subset {\mathcal {M}}\), with \(\lim \limits _{m\rightarrow \infty }E_m={\mathcal {M}}\), such that for every integrable function \(f\in L^1({\mathcal {M}})\) with \(\Vert f\Vert _1>0\) there exists a sequence of approximating functions \(\{g_m\}_{m=1}^\infty \), \(g_m\in L^1({\mathcal {M}})\), so that the following statements hold:

1.
\(g_m\xrightarrow [m\rightarrow \infty ]{}f\) in \(L^1({\mathcal {M}})\),

2.
\(f=g_m\) on \(E_m\), \(\forall m\in {\mathbb {N}}\),

3.
the Fourier series of \(g_m\) converges in \(L^1({\mathcal {M}})\), \(\forall m\in {\mathbb {N}}\),

4.
we have
$$\begin{aligned} \sup _{N}\left\ \sum _{n=1}^N Y_n(g_m)\right\ _1<2\min \left\{ \Vert f\Vert _1,\Vert g_m\Vert _1\right\} , \quad \forall m\in {\mathbb {N}}. \end{aligned}$$
Remark 1
Not for every orthonormal system \(\{\varphi _n\}_{n=1}^\infty \) does an arbitrary integrable function \(f\in L^1({\mathcal {M}})\) have an orthogonal series \(\sum _{n=1}^\infty Y_n\) of the form (1) that converges to f in \(L^1({\mathcal {M}})\), and if that happens then \(\sum _{n=1}^\infty Y_n\) 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 \(L^2({\mathbb {S}}^2)\) [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 \(\{E_m\}_{m=1}^\infty \), \(E_m\subset E_{m+1}\subset {\mathcal {M}}\), such that \(\lim \limits _{m\rightarrow \infty }E_m={\mathcal {M}}\) with the following property. For any fixed integrable function \(f\in L^1({\mathcal {M}})\) with \(\Vert f\Vert _1>0\) and for every natural number \(m\in {\mathbb {N}}\) there is an orthogonal series \(\sum _{n=1}^\infty Y_n^{(m)}\) of which the restriction \(\sum _{n=1}^\infty Y_n^{(m)}_{E_m}\) to the subset \(E_m\) converges to the restriction \(f_{E_m}\) in \(L^1(E_m)\). In \(L^1({\mathcal {M}})\) the series \(\sum _{n=1}^\infty Y_n^{(m)}\) converges to a function \(g_m\in L^1({\mathcal {M}})\). The sequence of these functions \(\{g_m\}_{m=1}^\infty \) converges to f in \(L^1({\mathcal {M}})\).
2 The general case
Theorem 1 is true for every finite separable diffuse measure space \({\mathcal {M}}\), 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 \({\mathcal {M}}\) but only to its measure algebra \({\mathcal {B}}({\mathcal {M}})\). We note that if we replace the set E produced by Theorem 1 by another measurable set \(E'\in \Sigma \) such that the symmetric difference is null, \(E\bigtriangleup E'=0\), 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 \({\mathcal {M}}\) be a finite separable diffuse measure space, and let \(\{\varphi _n\}_{n=1}^\infty \) be an orthonormal system in \(L^2({\mathcal {M}})\) consisting of bounded functions \(\varphi _n\in L^\infty ({\mathcal {M}})\), \(n\in {\mathbb {N}}\). For every \(\epsilon ,\delta >0\) there exists a function \(\chi _E\in L^\infty ({\mathcal {M}})\) with \(\chi _E^2=\chi _E\) and \(\Vert \chi _E\Vert _1>M\delta \), with the following properties; for each function \(f\in L^1({\mathcal {M}})\) with \(\Vert f\Vert _1>0\) there exists an approximating function \(g\in L^1({\mathcal {M}})\) that satisfies:

1.
\(\Vert fg\Vert _1<\epsilon \),

2.
\((fg)\chi _E=0\),

3.
the Fourier series of g converges in \(L^1({\mathcal {M}})\),

4.
we have
$$\begin{aligned} \sup _m\left\ \sum _{n=1}^m Y_n(g)\right\ _1<2\min \left\{ \Vert f\Vert _1,\Vert g\Vert _1\right\} . \end{aligned}$$
In this form the theorem relies only upon spaces \(L^p({\mathcal {M}})\), \(p=1,2,\infty \), which can be constructed purely out of the measure algebra \({\mathcal {B}}({\mathcal {M}})\) with no recourse to the underlying measure space \({\mathcal {M}}\). In particular, if two measure spaces have isomorphic measure algebras then the statements of Theorem 1 on these two spaces are equivalent.
Remark 2
It is known in measure theory that every finite separable diffuse measure space \({\mathcal {M}}\) satisfies
where \(a>0\) is a positive real number.
Thus, without loss of generality, we can restrict ourselves to measure spaces \({\mathcal {M}}=[0,a]\). The next reduction comes from the following observation.
Remark 3
If Theorem 3 is true for the finite separable measure space \(({\mathcal {M}},\mu )\) then it is true also for \(({\mathcal {M}},\lambda \mu )\) for every \(\lambda >0\).
Indeed, for every \(p\in [1,\infty ]\) the operator \({\text {T}}_pf\doteq \lambda ^{\frac{1}{p}}f\) defines an isometric isomorphism
\({{\text {T}}_p:L^p({\mathcal {M}},\mu )\rightarrow L^p({\mathcal {M}},\lambda \mu )}\). It is now straightforward to check that if the statements of Theorem 3 hold on \(({\mathcal {M}},\mu )\) with data \(\{\varphi _n\}_{n=1}^\infty \), \(\epsilon \), \(\delta \), \(\chi _E\), f, g, then they hold on \(({\mathcal {M}},\lambda \mu )\) with data \(\{{\text {T}}_2\varphi _n\}_{n=1}^\infty \), \(\epsilon \), \(\lambda \delta \), \({\text {T}}_\infty \chi _E\), \({\text {T}}_1f\), \({\text {T}}_1g\).
Thus we established that without loss of generality we are allowed to prove the theorem just for the unit interval \({\mathcal {M}}=[0,1]\). In fact, in the next sections we will prove Theorem 1 on separable cylindric probability spaces, i.e., separable probability spaces of the form \({\mathcal {M}}=[0,1]\otimes {\mathcal {N}}\), where \({\mathcal {N}}\) is another probability space. The unit interval is trivially cylindric, \([0,1]\simeq [0,1]\otimes \{1\}\), and it may seem an unnecessary effort to prove the theorem for a cylindric space instead of [0, 1]. But note 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.
3 The particular case
In this section we will prove the main theorem for the particular case where \({\mathcal {M}}=({\mathcal {M}},\Sigma ,\mu )\) is a separable cylindric probability space
Here \({\mathcal {N}}=({\mathcal {N}},\Sigma _0,\nu )\) is any separable probability space. We will write \({\mathcal {M}}\ni x=(t,y)\in [0,1]\times {\mathcal {N}}\).
3.1 The core lemmata
First let us state a variant of Féjér’s lemma.
Lemma 1
Let \(a,b\in {\mathbb {R}}\), \(a<b\). For every \(f\in L^1[a,b]\) and \(g\in L^\infty ({\mathbb {R}})\), g being \((ba)\)periodic,
This lemma is given in [1, page 77] with \([a,b]=[\pi ,\pi ]\), but the proof for arbitrary a and b follows with only trivial modifications.
We proceed to our first critical lemma.
Lemma 2
Let \(\Delta =[a,b]\times \Delta _0\in \Sigma \) with \([a,b]\subset [0,1]\) and \(\Delta _0\in \Sigma _0\), \(0\ne \gamma \in {\mathbb {R}}\), \(\epsilon ,\delta \in (0,1)\) and \(N\in {\mathbb {N}}\) be given. Then there exists a function \(g\in L^\infty ({\mathcal {M}})\), a measurable set \(\Sigma \ni E\subset \Delta \) and a Fourier polynomial of the form
such that

1.
\(E>\Delta (1\delta )\),

2.
\(g(x)=\gamma \) for \(x\in E\) and \(g(x)=0\) for \(x\notin \Delta \),

3.
\(\gamma \Delta <\Vert g\Vert _1<2\gamma \Delta \),

4.
\(\Vert Qg\Vert _1<\epsilon \),

5.
and
$$\begin{aligned} \max _{N\le m\le M}\left\ \sum _{n=N}^mY_n\right\ _1\le \frac{\gamma \sqrt{\Delta (1+\delta )}}{\sqrt{\delta }}. \end{aligned}$$
Proof
Set
Define the 1periodic function \(I:{\mathbb {R}}\rightarrow {\mathbb {R}}\) by setting
for \(t\in [0,1)\) and continuing periodically. Then obviously
By Féjér’s lemma
Choose a natural number \(s_0\in {\mathbb {N}}\) 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, \(g\in L^\infty ({\mathcal {M}})\) and thus we have proven Statements 1 and 2. Next we note using (4) that
and then by \((1\delta )\Delta <E<\Delta \) 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 \(g\in L^\infty ({\mathcal {M}})\subset L^2({\mathcal {M}})\), and therefore the Fourier series \(\sum Y_n(g)\) converges to g in \(L^2({\mathcal {M}})\). Thus we can choose the natural number \(M\in {\mathbb {N}}\) 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 \(\Vert \varphi _n\Vert _2=1\). Using the pairwise orthogonality of the Fourier components \(Y_n(g)\) and formula (16) we can obtain the coarse estimate
which immediately yields
thus proving Statement 5. Note that the inequality \(\Vert .\Vert _1\le \Vert .\Vert _2\) used above holds thanks to the convenient assumption that we are in a probability space. \(\square \)
Lemma 3
Let \(f\in L^1({\mathcal {M}})\) with \(\Vert f\Vert _1>0\), \(\epsilon ,\delta \in (0,1)\), \(N_0\in {\mathbb {N}}\). Then \(\exists E\in \Sigma \), \(g\in L^\infty ({\mathcal {M}})\) and
such that

1.
\(E>1\delta \),

2.
\(x\in E\) implies \(g(x)=f(x)\),

3.
\(\frac{1}{3}\Vert f\Vert _1<\Vert g\Vert _1<3\Vert f\Vert _1\),

4.
\(\Vert gQ\Vert _1<\epsilon \),

5.
and
$$\begin{aligned} \sup _{N_0\le m\le N}\left\ \sum _{n=N_0}^mY_n\right\ _1<3\Vert f\Vert _1. \end{aligned}$$
Proof
For every measurable partition of \({\mathcal {N}}\)
and every partition \(0=x_0<x_1<\cdots <x_{{\bar{\nu }}_0}=1\), \({\bar{\nu }}_0\in {\mathbb {N}}\), of the unit interval, the product partition
is a measurable partition of \({\mathcal {M}}\) with the property that
For every product partition \(\{\Delta _k\}_{k=1}^{\nu _0}\) as above and every tuple of real numbers \(\{\gamma _k\}_{k=1}^{\nu _0}\) consider the step function
By the assumption of separability of \({\mathcal {M}}\) we know that step functions of the form (26) subordinate to product partitions are dense in all spaces \(L^p({\mathcal {M}})\) for \(1\le p<\infty \). Choose a product partition and a subordinate step function such that
Note that the numbers \(\gamma _k\) are not assumed to be distinct, thus we can refine the given partition without changing \(\gamma _k\) and the function \(\Lambda (x)\). We use the property (25) to refine the product partition \(\{\Delta _k\}_{k=1}^{\nu _0}\) until it satisfies
Now we apply Lemma 2 iteratively with
for \(k=1,\dots ,\nu _0\), obtaining at each k a function \(g_k\in L^\infty ({\mathcal {M}})\), a set \(\Sigma \ni E_k\subset \Delta _k\), a number \(N_{k1}\le N_k\in {\mathbb {N}}\) and a Fourier polynomial
with the following properties:
 \(1^\circ \).:

\(E_k>\Delta _k(1\delta )\),
 \(2^\circ \).:

\(g_k(x)=\gamma _k\) for \(x\in E_k\) and \(g_k(x)=0\) for \(x\notin \Delta _k\),
 \(3^\circ \).:

\(\gamma _k\Delta _k<\Vert g_k\Vert _1<2\gamma _k\Delta _k\),
 \(4^\circ \).:

\(\Vert Q_kg_k\Vert _1<\frac{1}{2^{\nu _0+2}}\min \{\epsilon ,\Vert f\Vert _1\}\),
 \(5^\circ \).:

and
Set
First we check that from (26), (\(1^\circ \)), (\(2^\circ \)) and (31) it follows that
so that Statements 1 and 2 are proven. Next we observe using (\(4^\circ \)), (27), (31) and (32) that
which proves Statement 4. Further, from (26), (27), (\(2^\circ \)) and (\(3^\circ \)) we deduce that
Moreover, the same formulae also imply
i.e., \(\Vert f\Vert _1<3\Vert g\Vert _1\), thus proving Statement 3. In order to prove Statement 5 let us fix an \(N_0\le m\le N\). Then there is a \(1\le k_0\le \nu _0\) such that \(N_{k_01}\le m<N_{k_0}\), and thus by (30) and (32) we have
Finally we use this along with formulae (\(3^\circ \)), (\(4^\circ \)), (\(5^\circ \)) and (28) to obtain
and this completes the proof. \(\square \)
Lemma 4
Let \(\{R_k\}_{k=1}^\infty \) be any fixed ordering of the set of all nonzero Fourier polynomials with rational coefficients into a sequence. Then for every \(f\in L^1({\mathcal {M}})\) and sequence \(\{b_s\}_{s=1}^\infty \) of positive numbers \(b_s>0\) there exists subsequence \(\{R_{k_s}\}_{s=0}^\infty \) such that

1.
\(\Vert R_{k_0}f\Vert _1\le \frac{1}{2}\Vert f\Vert _1\)

2.
\(\Vert R_{k_s}\Vert _1<b_s\) for \(s\ge 1\)

3.
\(\sum _{s=0}^\infty R_{k_s}=f\) in \(L^1({\mathcal {M}})\).
Proof
Let us first convince ourselves that Fourier polynomials with rational coefficients are dense in \(L^1({\mathcal {M}})\). Indeed, by the assumption of separability, step functions are dense in \(L^1({\mathcal {M}})\), but they all belong also to the separable Hilbert space \(L^2({\mathcal {M}})\). On the other hand, Fourier polynomials are clearly dense in \(L^2({\mathcal {M}})\). And finally, an arbitrary Fourier polynomial can be approximated in \(L^2({\mathcal {M}})\) by a Fourier polynomial with rational coefficients (this amounts to approximating the Fourier coefficients by rational numbers). A threeepsilon argument together with \(\Vert .\Vert _1\le \Vert .\Vert _2\) then yields the assertion.
Using the denseness of \(\{R_k\}_{k=1}^\infty \) let us choose a natural number \(k_0\in {\mathbb {N}}\) such that
Then we can choose further natural numbers \(k_s\in {\mathbb {N}}\) iteratively as follows. For every \(s\in {\mathbb {N}}\), again by using the denseness argument, choose a number \(k_s\) so that \(k_s>k_{s1}\) and
Statements 1 and 3 are clearly satisfied. For Statement 2 we have
The lemma is proven. \(\square \)
3.2 The main theorem
Here we will prove Theorem 1 for the particular case of \(({\mathcal {M}},\Sigma ,\mu )\) being a separable cylindric probability space as in (3).
Proof
Recall that \(\epsilon ,\delta >0\) and \(f\in L^1({\mathcal {M}})\) with \(\Vert f\Vert _1>0\) are given. Denote
Let \(\{R_k\}_{k=1}^\infty \) be any ordering of the set of all nonzero Fourier polynomials with rational coefficients into a sequence. Iteratively applying Lemma 3 with
for \(k=1,2,\dots \) we obtain for each \(k\in {\mathbb {N}}\) a subset \({\tilde{E}}_k\in \Sigma \), a function \({\tilde{g}}_k\in L^\infty ({\mathcal {M}})\), a number \(N_{k1}\le N_k\in {\mathbb {N}}\) (set \(N_0=0\)) and a Fourier polynomial
with the following properties:
 \(1^\dagger \).:

\({\tilde{E}}_k>1\frac{\delta }{2^k}\),
 \(2^\dagger \).:

\(x\in {\tilde{E}}_k\) implies \({\tilde{g}}_k(x)=R_k(x)\),
 \(3^\dagger \).:

\(\frac{1}{3}\Vert R_k\Vert _1<\Vert {\tilde{g}}_k\Vert _1<3\Vert R_k\Vert _1\),
 \(4^\dagger \).:

\(\Vert {\tilde{g}}_k{\tilde{Q}}_k\Vert _1<\epsilon _02^{k7}\),
 \(5^\dagger \).:

and
Define the desired set E as
Observe from (\(1^\dagger \)) that
Note that E is universal, i.e., independent of f.
Let \(\{R_{k_s}\}_{s=0}^\infty \) be the subsequence of Fourier polynomials provided by Lemma 4 applied with
It satisfies
 \(1^\circ \).:

\(\Vert R_{k_0}f\Vert _1\le \frac{1}{2}\Vert f\Vert _1\),
 \(2^\circ \).:

\(\Vert R_{k_s}\Vert _1<\epsilon _02^{s6}\) for \(s\ge 1\),
 \(3^\circ \).:

\(\sum _{s=0}^\infty R_{k_s}=f\) in \(L^1({\mathcal {M}})\).
We want to use mathematical induction in order to define a sequence of natural numbers \({1<\nu _1<\nu _2<\cdots }\) and a sequence of functions \(\{g_s\}_{s=1}^\infty \), \(g_s\in L^1({\mathcal {M}})\), such that for all \(s\in {\mathbb {N}}\) we have
 \(1^*.\):

\(x\in {\tilde{E}}_{\nu _s}\) implies \(g_s(x)=R_{k_s}(x)\),
 \(2^*.\):

\(\Vert g_s\Vert _1<\epsilon _02^{s2}\),
 \(3^*.\):

$$\begin{aligned} \left\ \sum _{j=1}^s[\tilde{Q}_{\nu _j}g_j]\right\ _1<\frac{\epsilon _0}{2^{s+6}}, \end{aligned}$$
 \(4^*.\):

$$\begin{aligned} \max _{N_{\nu _s1}\le m<N_{\nu _s}}\left\ \sum _{n=N_{\nu _s1}}^m\tilde{Y}_n\right\ _1<\frac{\epsilon _0}{2^s}. \end{aligned}$$
Assume that for some \(s\in {\mathbb {N}}\), the choice of \(1<\nu _1<\nu _2<\cdots <\nu _{s1}\) and \(g_1, g_2,\ldots ,g_{s1}\) satisfying (\(3^*\)) has been already made (for \(s=1\) this is trivially correct). Remember that by \(\sigma (h)\) we have denoted the \(\{\varphi _n\}\)spectrum of a function \(h\in L^1({\mathcal {M}})\), i.e., the support of its Fourier series. Using the denseness of \(\{R_k\}_{k=1}^\infty \) (see Lemma 4) choose a natural number \(\nu _s\in {\mathbb {N}}\) such that \(N_{\nu _11}>\max \sigma (R_{k_0})\) and \(\nu _s>\nu _{s1}\) for \(s>1\), and
Then by (\(2^\circ \)) and (\(3^*\)) we have for all \(s\in {\mathbb {N}}\) that
which combined with (49) implies
Set
Condition (\(1^*\)) is easily satisfied thanks to (\(2^\dagger \)) with \(k=\nu _s\). For condition (\(2^*\)) we write
where we used (49), (\(3^\dagger \)), (\(3^*_{s1}\)) and (51) in the last step. To show that condition (\(3^*\)) is satisfied we observe that
where (\(4^\dagger \)), (49) and \(\nu _s>s\) were used in the second inequality. Finally we satisfy condition (\(4^*\)) using (\(5^\dagger \)) and (51),
The iteration is thus complete, and by mathematical induction we construct the sequences \(\{\nu _s\}_{s=1}^\infty \) and \(\{g_s\}_{s=1}^\infty \) satisfying conditions (\(1^*\)) through (\(4^*\)) for all \(s\in {\mathbb {N}}\). Define
From (53) it follows that
thus \(g\in L^1({\mathcal {M}})\). 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 \(x\in E\) means \(x\in {\tilde{E}}_{\nu _s}\), and hence by (\(2^\dagger \)), \(g_s(x)=R_{k_s}(x)\) for all \(s\in {\mathbb {N}}\). It then follows from (\(3^\circ \)) that
Let \(\{Y_n\}_{n=1}^\infty \) be the series of \(Y_n=c_n\varphi _n\) such that
Let \(m\in {\mathbb {N}}\), and let \(r\in {\mathbb {N}}\) be the largest natural number such that \(N_{\nu _{r}1}\le m\) (if \(m<N_{\nu _11}\) set \(r=1\)). Set \({m_*\doteq \min \{m,N_{\nu _{r}}1\}}\). Then by (56), (\(3^*\)), (\(4^*\)) and (53) we get
Now as \(m\rightarrow \infty \) obviously \(r\rightarrow \infty \) as well, thus making the above expression vanish, which proves that \(\sum Y_n\) 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 (\(2^\circ \)), (\(3^\circ \)), (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 \(25\Vert f\Vert _1<32\Vert g\Vert _1\). This together with (63) yields
which establishes Statement 4. The proof of the theorem is accomplished. \(\square \)
4 Compact groups
Let G be a compact Hausdorff topological group, \(\Sigma \) the Borel \(\sigma \)algebra and \(d\mu (x)=dx\) the normalized Haar measure. Then \(\mu \) is diffuse if and only if G is infinite, which we will assume here. Now by [12, Theorem 28.2] we have that \(\dim L^2(G)={\text {w}}(G)\), therefore \((G,\Sigma ,\mu )\) is separable if and only if G is second countable. This will also be assumed in what follows. This implies in particular, through PeterWeyl theorem, that the dual \({\hat{G}}\) is countable. For a detailed exposition of harmonic analysis on compact groups consult, e.g., [20] or [3].
For every irreducible unitary representation \(\rho \in {\hat{G}}\) (or rather \([\rho ]\in {\hat{G}}\)), let \({\mathcal {H}}_\rho \) be the representation Hilbert space of dimension \(\dim {\mathcal {H}}_\rho \doteq d_\rho \in {\mathbb {N}}\). Choose an arbitrary orthonormal basis \(\{e^\rho _i\}_{i=1}^{d_\rho }\) of \({\mathcal {H}}_\rho \), and denote
By the PeterWeyl theorem \(\{\varphi _{\rho ,i,j}\}\) is an orthonormal basis in \(L^2(G)\). Moreover, since \(\rho (x)\) is unitary for all \(x\in G\), we get
so that \(\varphi _{\rho ,i,j}\in L^\infty (G)\). Therefore, if we put an arbitrary (total) order on \({\hat{G}}\) then Theorem 1 is directly applicable to \((G,\Sigma ,\mu )\) with \(\{\varphi _{\rho ,i,j}\}\).
But the arbitrary choice of the bases \(\{e^\rho _i\}_{i=1}^{d_\rho }\) 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 \({\mathcal {M}}\simeq G/H\) of a compact group G as above with (closed) isotropy subgroup \(H\subset G\), then we define multiplicities
(this reduces to \(d^H_\rho =d_\rho \) if \(H=\{{{\mathbf {1}}}\}\) as before). Moreover, we restrict to
A point \(x\in G/H\) is a coset \(x=\mathrm {x}H\), \(\mathrm {x}\in G\). If dh is the normalized Haar measure on H then there is a unique normalized left Ginvariant measure \(\mu \) on G/H (the pullback of \(d\mathrm {x}\) through the quotient map) such that
Denote
Note that \(d^H_\rho =\dim {\mathbb {P}}_H{\mathcal {H}}_\rho \). Let \(\{e^\rho _i\}_{i=1}^{d_\rho }\) be an orthonormal basis in \({\mathcal {H}}_\rho \) as before, and choose an orthonormal basis \(\{e^\rho _\alpha \}_{\alpha =1}^{d^H_\rho }\) in \({\mathbb {P}}_H{\mathcal {H}}_\rho \). Now the Fourier transform of a function \(f\in L^1(G/H)\) becomes
and the corresponding block Fourier series is
If the homogeneous space G/H is infinite then the invariant measure \(\mu \) is diffuse. Applying Theorem 1 to \((G/H,\Sigma _H,\mu )\) (\(\Sigma _H\) is the Borel \(\sigma \)algebra on G/H) and the system \(\{\varphi _{\rho ,i,\alpha }\}\) with \(\widehat{G/H}\) ordered arbitrarily, we obtain the following modification.
Theorem 4
Let \({\mathcal {M}}=G/H\) be an infinite homogeneous space of a compact second countable Hausdorff group G with closed isotropy subgroup \(H\subset G\). For every \(\epsilon ,\delta >0\) there exists a measurable subset \(E\in \Sigma _H\) with measure \(E>1\delta \) and with the following property; for each function \(f\in L^1(G/H)\) with \(\Vert f\Vert _1>0\) there exists an approximating function \(g\in L^1(G/H)\) that satisfies:

1.
\(\Vert fg\Vert _1<\epsilon \),

2.
\(f=g\) on E,

3.
the block Fourier series (75) of g converges in \(L^1(G/H)\),

4.
we have
$$\begin{aligned} \sup _{\rho \in \widehat{G/H}}\left\ \sum _{\varrho \le \rho }d_\varrho {\text {tr}}\left[ {\hat{g}}(\varrho )\varrho (x)\right] \right\ _1<2\min \left\{ \Vert f\Vert _1,\Vert g\Vert _1\right\} . \end{aligned}$$
Note that E depends on the chosen order in \(\widehat{G/H}\).
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 \(K\times K\backslash G/H\) for certain infinite closed subgroups \(K\subset G\), which will provide an obvious identification of all spaces \(L^p(G/H)\) and \(L^p(K\times K\backslash G/H)\). 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 \(\varphi :(\Omega _1,\mu _1)\rightarrow (\Omega _2,\mu _2)\) is an almost isomorphism of measure spaces if there exist full measure subspaces \(X\subset \Omega _1\) and \(Y\subset \Omega _2\), \(\Omega _1{\setminus } X=\Omega _2{\setminus } Y=0\), such that the restriction of \(\varphi \) is an isomorphism of measure spaces \(\varphi _X:(X,\mu _1)\rightarrow (Y,\mu _2)\).
If \(K\subset G\) is a closed subgroup of G then denote by \(G/H^{(K)}\subset G/H\) the set of those points \(x\in G/H\) with a nontrivial stabilizer within K,
Let us now fix an infinite closed subgroup \(K\subset G\), and let \({\text {q}}:G/H\rightarrow K\backslash G/H\) be the natural quotient map. Then the pullback measure \(\nu =\mu \circ {\text {q}}^{1}={\text {q}}^*\mu \) is the natural probability measure on \(K\backslash G/H\). Provided that the subset \(G/H^{(K)}\) in G/H is \(\mu \)null, we obtain a natural product structure in the following way.
Proposition 1
If \(\left G/H^{(K)}\right _\mu =0\) then there exists a Borel almost isomorphism \(\varphi :K\times K\backslash G/H\rightarrow G/H\) such that
Proof
Both G/H and \(K\backslash G/H\) are compact Hausdorff second countable, hence metrizable by Urysohn’s metrization theorem. The canonical quotient map \({\text {q}}:G/H\rightarrow K\backslash G/H\) is a continuous surjection between compact metrizable spaces. By FedererMorse theorem there exists a Borel subset \(Z\subset G/H\) such that the restriction \({\text {q}}_Z:Z\rightarrow K\backslash G/H\) is a Borel isomorphism. Let \(W\doteq Z{\setminus }\, G/H^{(K)}\) and \(X={\text {q}}(W)\), so that \({\text {q}}_W:W\rightarrow X\) is a Borel isomorphism. Define \(\varphi :K\times X\rightarrow G/H\) by setting
\(\varphi \) is Borel bimeasurable, since it is the composition of bimeasurable maps \((k,x)\mapsto k\cdot x\) and \({\text {q}}_W^{1}\). The properties (78) are easily implied by the definition of \(\varphi \). The map \(\varphi \) is also injective. Indeed, if \(\varphi (k_1,Kx_1)=\varphi (k_2,Kx_2)\) then
and
so that \(k_2^{1}k_1\varphi ({{\mathbf {1}}},Kx_1)=\varphi ({{\mathbf {1}}},Kx_1)\). If \(k_1\ne k_2\) then \(\varphi ({{\mathbf {1}}},Kx_1)\) has a nontrivial stabilizer, that is, \(\varphi ({{\mathbf {1}}},Kx_1)\in G/H^{(K)}\). But \(\varphi ({{\mathbf {1}}},Kx_1)={\text {q}}_W^{1}(Kx_1)\in W\) and \(W\cap G/H^{(K)}=\emptyset \), which is a contradiction. Thus \(k_1=k_2\) and the injectivity is proven. Denoting \(Y\doteq \varphi (X)\subset G/H\) we see that \(\varphi :X\rightarrow Y\) is a Borel isomorphism.
For every \(f\in C(G/H)\), by measure disintegration theorem, we have
Let \(\chi _X\) and \(\chi _Y\) be the indicator functions of the subsets X and Y, respectively. Since \(K\cdot Y\subset Y\) we have that \(\chi _Y(x)=\chi _X(Kx)\). It follows that
which shows that \(\varphi :(X,dk\otimes \nu )\rightarrow (Y,\mu )\) is a measure space isomorphism.
Finally, let us note that \(K\cdot Z=G/H\). Indeed, for every \(x\in G/H\) we have that \(z={\text {q}}_Z^{1}({\text {q}}(x))\in Z\), and since \({\text {q}}(x)={\text {q}}(z)\) we have that \(\exists k\in K\) such that \(kz=x\). On the other hand, it is easy to see that the subset \(G/H^{(K)}\subset G/H\) is left Kinvariant, for if \(x\in G/H^{(K)}\) with \(k_x\in K\) such that \(k_xx=x\) then for every \(k\in K\) it follows that \(k_yy=y\), where \(y=kx\) and \(k_y=k_0k_xk_0^{1}\), which means that \(y\in G/H^{(K)}\). Therefore
so that \(Y_\mu =X_{dk\otimes \nu }=1\). This completes the proof. \(\square \)
4.1 Spheres
As an instructive illustration of the above constructions we will consider spheres \({\mathbb {S}}^d\), \(2\le d\in {\mathbb {N}}\), 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 \({\mathbb {S}}^d\) can be considered as the homogeneous space G/H with \(G=\mathrm {SO}(d+1)\) and \(H=\mathrm {SO}(d)\). Harmonic analysis in these homogeneous spaces is a classical subject widely available in the literature (see e.g. [21]). The dual space \(\widehat{G/H}\) consists of irreducible representations by harmonic polynomials of fixed degree \(\rho \in {\mathbb {N}}_0\), and it is conveniently ordered according to that degree, \(\rho \in \widehat{G/H}\simeq {\mathbb {N}}_0\). The dimension of the representation \(\rho \) is
whereas the multiplicities are all \(d^H_\rho =1\). We choose standard spherical coordinates \(x=(\theta _1,\ldots ,\theta _{d1},\phi )\), where \(\theta _j\in [0,\pi ]\), \(j=1,\ldots ,d1\), and \(\phi \in [0,2\pi )\). The orthonormal system \(\{\varphi _{\rho ,i,\alpha }\}\) in this case consists of spherical harmonics
The block Fourier series of a function \(f\in L^1({\mathbb {S}}^d)\) is
A natural cylindric structure is obtained by choosing \(K=\mathrm {SO}(2)\), the circle subgroup responsible for rotation in the longitudinal variable \(\phi \). The subset \(G/H^{(K)}\) here contains only the two poles  \(\theta _j=0\), \(j=1,\ldots ,d1\), and \(\theta _j=\pi \), \(j=1,\ldots ,d1\), respectively. Thus indeed \(\left G/H^{(K)}\right _\mu =0\), and Proposition 1 applies. The section \(Z\subset G/H\) appearing in the proof of Proposition 1 can be chosen to correspond to the meridian \(\phi =0\) in \(\mathbb {S^d}\), which is Borel isomorphic to \(\mathrm {SO}(2)\backslash \mathrm {SO}(d+1)/\mathrm {SO}(d)\) through the quotient map \({\text {q}}\). In this way we have the almost isomorphism
given by \(\varphi (\phi ,(\theta _1,\ldots ,\theta _{d1}))=(\theta _1,\ldots ,\theta _{d1},\phi )\), i.e., simply by separation of the variable \(\phi \). As a final step we parameterize \(\mathrm {SO}(2)={\mathbb {S}}^1\) by \(t=\phi /2\pi \) to obtain an almost isomorphism
This is the cylindric structure used implicitly in [4].
5 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 \(\{R_k\}_{k=1}^\infty \) of all Fourier polynomials with rational coefficients.

For every \(k\in {\mathbb {N}}\), choose a partition \(\{\Delta _l(k)\}_{l=1}^{\nu _0(k)}\) of the cylindric measure space \({\mathcal {M}}=[0,1]\times {\mathcal {N}}\) of the form \(\Delta _l(k)=[a_l(k),b_l(k)]\times {\tilde{\Delta }}_l(k)\) such that the measures \(\Delta _l(k)\) are small enough, as well as a subordinate real step function \(\Lambda (k)=\sum _{l=1}^{\nu _0(k)}\gamma _l(k)\chi _{\Delta _l(k)}\) (\(\chi _X\) is the indicator function of the subset X), such that \(\Vert \Lambda (k)R_k\Vert _1\) is sufficiently small.

For every \(k\in {\mathbb {N}}\), choose a number \(\delta _*(k)\in (0,\frac{1}{2})\) so that \(\{\delta _*(k)\}_{k=1}^\infty \) decays sufficiently rapidly. Define the periodic step function
$$\begin{aligned} I(t)=1\frac{1}{\delta _*(k)}\chi _{[0,\delta _*(k))}(t\mod 1), \end{aligned}$$and the measurable function \({\hat{g}}_l^k\in L^\infty ({\mathcal {M}})\) by
$$\begin{aligned} {\hat{g}}_l^k(x)=\gamma _l(k)I(s_0(k)t)\chi _{\Delta _l(k)}(x),\quad \forall x\in {\mathcal {M}}, \end{aligned}$$where the positive number \(s_0(k)\) is sufficiently large. Define the measurable subsets \({\hat{E}}_l(k)\subset \Delta _l(k)\) by
$$\begin{aligned} {\hat{E}}_l(k)=\left\{ x\in \Delta _l(k)\,\big \quad {\hat{g}}_l^k(x)=\gamma _l(k)\right\} . \end{aligned}$$Define inductively the natural numbers \({\hat{N}}_l(k)\), \(l=0,\ldots ,\nu _0(k)\) and Fourier polynomials \({\hat{Q}}_l^k\), \(l=1,\ldots ,\nu _0(k)\), by setting \({\hat{N}}_0(1)=1\), \({\hat{N}}_0(k)={\hat{N}}_{\nu _0(k)}(k1)\) for \(k>1\), and \({\hat{Q}}_l^k=\sum _{n={\hat{N}}_{l1}(k)}^{{\hat{N}}_l(k)1}Y_n({\hat{g}}_l^k)\) ( here \(\{Y_n(g)\}_{n=1}^\infty \) is the Fourier series of the function \(g\in L^2({\mathcal {M}})\)), so that the quantities
$$\begin{aligned} \left\ \sum _{n=1}^{{\hat{N}}_l(k)1}Y_n({\hat{g}}_l^k){\hat{g}}_l^k\right\ _2 \end{aligned}$$are sufficiently small.

For every \(k\in {\mathbb {N}}\), define the natural numbers \(N_k={\hat{N}}_{\nu _0(k)}1\), measurable subsets \(\tilde{E}_k=\bigcup _{l=1}^{\nu _0(k)}{\hat{E}}_l(k)\), and measurable functions \({\tilde{g}}_k\in L^\infty ({\mathcal {M}})\) by
$$\begin{aligned} {\tilde{g}}_k=R_k\Lambda (k)+\sum _{l=1}^{\nu _0(k)}{\hat{g}}_l^k, \end{aligned}$$as well as Fourier polynomials
$$\begin{aligned} {\tilde{Q}}_k=\sum _{l=1}^{\nu _0(k)}{\hat{Q}}_l^k=\sum _{n=N_{k1}}^{N_k1}{\tilde{Y}}_n. \end{aligned}$$ 
Set
$$\begin{aligned} E=\bigcap _{k=1}^\infty {\tilde{E}}_k. \end{aligned}$$ 
Choose by Lemma 4 a subsequence \(\{R_{k_s}\}_{s=0}^\infty \) such that \(\Vert R_{k_s}\Vert _1\) decay sufficiently rapidly, and \(\sum _{s=0}^\infty R_{k_s}=f\) in \(L^1({\mathcal {M}})\).

Define inductively the sequence of natural numbers \(\{\nu _s\}_{s=1}^\infty \), \(\nu _s>\nu _{s1}\) for \(s>1\), and measurable functions \(g_s\in L^\infty ({\mathcal {M}})\) by choosing \(\nu _1\) so that \(N_{\nu _11}>\max \sigma (R_{k_0})\) and
$$\begin{aligned} \left\ R_{\nu _s}R_{k_s}+\sum _{j=1}^{s1}\left[ \tilde{Q}_{\nu _j}g_j\right] \right\ _1 \end{aligned}$$is sufficiently small, and setting \(g_s=R_{ks}+\tilde{g}_{\nu _s}R_{\nu _s}\).

Finally, set
$$\begin{aligned} g=R_{k_0}+\sum _{s=1}^\infty g_s. \end{aligned}$$
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Acknowledgements
The first named author thanks Prof. G. Folland for enlightening comments on topological double coset spaces. This work, for the second named author, was supported by the RA MES State Committee of Science, in the frames of the research project No, 18T1A148. The third author was supported in parts by the FWO Odysseus 1 Grant G.0H94.18N: Analysis and Partial Differential Equations, EPSRC Grant EP/R003025/1 and by the Leverhulme Grant RPG2017151.
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Avetisyan, Z., Grigoryan, M. & Ruzhansky, M. Approximations in \(L^1\) with convergent Fourier series. Math. Z. 299, 1907–1927 (2021). https://doi.org/10.1007/s00209021027346
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DOI: https://doi.org/10.1007/s00209021027346