Approximations in $L^1$ with convergent Fourier series

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 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.

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

$$\begin{aligned} Y_n(x;f)=c_n(f)\varphi _n(x),\quad c_n(f)=(f,\varphi _n)_2=\int _{\mathcal {M}}f(x)\varphi _n^*(x)d\mu (x),\quad \forall n\in {\mathbb {N}}, \end{aligned}$$

where $\varphi _n^*$ denotes the complex conjugate of $\varphi _n$. The possibly divergent Fourier series of f will be

$$\begin{aligned} \sum _{n=1}^\infty Y_n(x;f). \end{aligned}$$

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

$$\begin{aligned} Q(x)=\sum _nY_n(x)=\sum _nc_n\varphi _n(x),\quad c_n\in {\mathbb {C}}, \end{aligned}$$

(1)

without reference to a particular function for which these may be the Fourier components. Denote by

$$\begin{aligned} \sigma (f)=\left\{ n\in {\mathbb {N}}\,\big |\quad c_n(f)\ne 0\right\} ,\quad f\in L^1({\mathcal {M}}), \end{aligned}$$

(2)

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 non-atomic 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. $\Vert f-g\Vert _1<\epsilon$,

• 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

$$\begin{aligned} {\mathbb {N}}_2=\{1,2\},\quad P(\{1\})=\frac{3}{16N^2-1},\quad \varphi _1=\left( 2N,\frac{1}{2}\right) . \end{aligned}$$

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

$$\begin{aligned} \left\| Y_1(g)\right\| _1=\left\| Y_1(f)\right\| _1=|c_1(f)|\Vert \varphi _1\Vert _1 =\frac{3N[(8N+3)^2-25]}{4(16N^2-1)}>N\Vert f\Vert _1=\frac{3N}{16N^2-1}. \end{aligned}$$

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. $\Vert f-g\Vert _1<\epsilon$,

• 2. $(f-g)\chi _E=0$,

• 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

$$\begin{aligned} {\mathcal {B}}({\mathcal {M}})\simeq {\mathcal {B}}\left( [0,a]\right) , \end{aligned}$$

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

$$\begin{aligned} {\mathcal {M}}=[0,1]\otimes {\mathcal {N}}. \end{aligned}$$

(3)

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 $(b-a)$-periodic,

$$\begin{aligned} \lim _{\lambda \rightarrow +\infty }\int _a^bf(t)g(\lambda t)dt=\frac{1}{b-a}\int _a^bf(t)dt\int _a^bg(t)dt. \end{aligned}$$

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

$$\begin{aligned} Q(x)=\sum _{n=N}^MY_n(x),\quad N\le M\in {\mathbb {N}}, \end{aligned}$$

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. $\Vert Q-g\Vert _1<\epsilon$,

• 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

$$\begin{aligned} \delta _*\doteq \frac{\delta }{1+\delta }\in \left( 0,\frac{1}{2}\right) . \end{aligned}$$

(4)

Define the 1-periodic function $I:{\mathbb {R}}\rightarrow {\mathbb {R}}$ by setting

$$\begin{aligned} I(t)=1-\frac{1}{\delta _*}\chi _{[0,\delta _*)}(t)={\left\{ \begin{array}{ll} 1\quad \text{ if }\quad t\in [\delta _*,1),\\ 1-\frac{1}{\delta _*}\quad \text{ if }\quad t\in [0,\delta _*) \end{array}\right. }, \end{aligned}$$

(5)

for $t\in [0,1)$ and continuing periodically. Then obviously

$$\begin{aligned} \int _0^1I(t)dt=0. \end{aligned}$$

(6)

By Féjér’s lemma

$$\begin{aligned}&\lim _{s\rightarrow +\infty }\int _\Delta I(st)\varphi _n^*(t,y)dtdy=\lim _{s\rightarrow +\infty }\int _a^bI(st)\int _{\Delta _0}\varphi _n^*(t,y)dydt\nonumber \\&\quad =\lim _{s\rightarrow +\infty }\int _0^1I(st)\left[ \chi _{[a,b]}(t)\int _{\Delta _0} \varphi _n^*(t,y)dy\right] dt=\int _0^1I(t)dt\int _\Delta \varphi _n^*(x)dx=0. \end{aligned}$$

(7)

Choose a natural number $s_0\in {\mathbb {N}}$ sufficiently large so that

$$\begin{aligned} s_0>\frac{(1-\delta _*)^2}{\delta _*^2(b-a)}\quad \text{ and }\quad \left| \int _\Delta I(s_0t)\varphi _n^*(t,y)dtdy\right| <\frac{\epsilon }{2N|\gamma |},\quad 1\le n\le N. \end{aligned}$$

(8)

Set

$$\begin{aligned} g(x)\doteq \gamma I(s_0t)\chi _{\Delta }(x),\quad E\doteq \left\{ x\in \Delta \,\big |\quad g(x)=\gamma \right\} . \end{aligned}$$

(9)

Then it can be seen that

$$\begin{aligned} |E|\ge |\Delta |\frac{\lfloor s_0(b-a)\rfloor (1-\delta _*)}{s_0(b-a)}>|\Delta |(1-\delta _*)\left( 1-\frac{1}{s_0(b-a)}\right) >|\Delta |(1-\delta ), \end{aligned}$$

(10)

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

$$\begin{aligned} \int _\Delta |I(s_0t)|dx=\int _Edx+\int _{\Delta {\setminus } E}\left| 1-\frac{1}{\delta _*}\right| dx=|E|+\frac{1}{\delta }(|\Delta |-|E|), \end{aligned}$$

(11)

and then by $(1-\delta )|\Delta |<|E|<|\Delta |$ we find that

$$\begin{aligned} |\Delta |<|E|+\frac{1}{\delta }(|\Delta |-|E|)<2|\Delta |, \end{aligned}$$

(12)

which together entail

$$\begin{aligned} |\Delta |<\int _\Delta |I(s_0t)|dx<2|\Delta |. \end{aligned}$$

(13)

Similarly,

$$\begin{aligned} \int _\Delta |I(s_0t)|^2dx= & {} \int _Edx+\int _{\Delta {\setminus } E}\left( 1-\frac{1}{\delta _*}\right) ^2dx\nonumber \\= & {} |E|+\frac{1}{\delta ^2}(|\Delta |-|E|)<\left( 1+\frac{1}{\delta }\right) |\Delta |. \end{aligned}$$

(14)

Formulae (9) and (13) imply that

$$\begin{aligned} |\gamma ||\Delta |<\Vert g\Vert _1=\int _{\mathcal {M}}|g(x)|dx=|\gamma |\int _\Delta |I(s_0t)|dx<2|\gamma ||\Delta |, \end{aligned}$$

(15)

which proves Statement 3. In a similar fashion we obtain

$$\begin{aligned} \Vert g\Vert _2^2=\int _{\mathcal {M}}|g(x)|^2dx=\gamma ^2\int _\Delta |I(s_0t)|^2dx<\left( 1+\frac{1}{\delta }\right) \gamma ^2|\Delta |. \end{aligned}$$

(16)

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

$$\begin{aligned} \left\| \sum _{n=1}^M Y_n(g)-g\right\| _2<\frac{\epsilon }{2}. \end{aligned}$$

(17)

Further, from formula (8) we estimate the magnitude of the first N Fourier coefficients of g as

$$\begin{aligned} |c_n(g)|=\left| \int _{\mathcal {M}}g(x)\varphi _n^*(x)dx\right| =|\gamma |\left| \int _\Delta I(s_0t)\varphi _n^*(x)dx\right| <\frac{\epsilon }{2N},\quad 1\le n\le N. \end{aligned}$$

(18)

Finally, set

$$\begin{aligned} Q(x)\doteq \sum _{n=N}^MY_n(x;g),\quad \forall x\in {\mathcal {M}}. \end{aligned}$$

(19)

In order to prove Statement 4 we write

$$\begin{aligned} \left\| Q-g\right\| _1\le & {} \left\| Q-g\right\| _2=\left\| \sum _{n=N}^MY_n(g) -g\right\| _2\le \left\| \sum _{n=1}^MY_n(g)-g\right\| _2+\left\| \sum _{n=1}^{N-1}Y_n(g)\right\| _2\nonumber \\\le & {} \left\| \sum _{n=1}^MY_n(g)-g\right\| _2+\sum _{n=1}^{N-1}|c_n(g)|\Vert \varphi _n\Vert _2<\epsilon , \end{aligned}$$

(20)

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

$$\begin{aligned} \left\| \sum _{n=N}^mY_n(g)\right\| _2^2=\sum _{n=N}^m\left\| Y_n(g)\right\| _2^2 \le \sum _{n=1}^\infty \left\| Y_n(g)\right\| _2^2=\Vert g\Vert _2^2<\left( 1+\frac{1}{\delta }\right) \gamma ^2|\Delta |, \end{aligned}$$

(21)

which immediately yields

$$\begin{aligned} \left\| \sum _{n=N}^mY_n(g)\right\| _1\le \left\| \sum _{n=N}^mY_n(g)\right\| _2 <\frac{|\gamma |\sqrt{|\Delta |(1+\delta )}}{\sqrt{\delta }},\quad m>N, \end{aligned}$$

(22)

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

$$\begin{aligned} Q(x)=\sum _{n=N_0}^NY_n(x),\quad N\in {\mathbb {N}}, \end{aligned}$$

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. $\Vert g-Q\Vert _1<\epsilon$,

• 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 ${\mathcal {N}}$

$$\begin{aligned} {\mathcal {N}}=\bigsqcup _{i=1}^{{\tilde{\nu }}_0}{\tilde{\Delta }}_i,\quad {\tilde{\Delta }}_i\in \Sigma _0,\quad i\ne j\quad \Rightarrow \quad \left| {\tilde{\Delta }}_i\cap {\tilde{\Delta }}_j\right| =0,\quad \forall i,j=1,\ldots {\tilde{\nu }}_0,\quad {\tilde{\nu }}_0\in {\mathbb {N}},\nonumber \\ \end{aligned}$$

(23)

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

$$\begin{aligned}&\Delta _k=[x_{l-1},x_l]\times {\tilde{\Delta }}_i,\nonumber \\&k={\bar{\nu }}_0\cdot i+l-1=1,\ldots ,\nu _0\doteq {\bar{\nu }}_0\cdot {\tilde{\nu }}_0,\quad l=1,\ldots ,{\bar{\nu }}_0,\quad i=1,\ldots ,{\tilde{\nu }}_0, \end{aligned}$$

(24)

is a measurable partition of ${\mathcal {M}}$ with the property that

$$\begin{aligned} \max _{1\le k\le \nu _0}|\Delta _k|\le \max _{1\le l\le {\bar{\nu }}_0}|x_l-x_{l-1}|. \end{aligned}$$

(25)

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

$$\begin{aligned} \Lambda (x)=\sum _{k=1}^{\nu _0}\gamma _k\chi _{\Delta _k}(x),\quad \forall x\in {\mathcal {M}}. \end{aligned}$$

(26)

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

$$\begin{aligned} \Vert \Lambda -f\Vert _1<\min \left\{ \frac{1}{2}\epsilon ,\frac{1}{3}\Vert f\Vert _1\right\} . \end{aligned}$$

(27)

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

$$\begin{aligned} 144\gamma _k^2|\Delta _k|(1+\delta )<\delta \Vert f\Vert _1^2,\quad k=1,\ldots ,\nu _0. \end{aligned}$$

(28)

Now we apply Lemma 2 iteratively with

$$\begin{aligned} \Delta \leftarrow \Delta _k,\quad \gamma \leftarrow \gamma _k,\quad \epsilon \leftarrow \frac{1}{2^{\nu _0+2}}\min \left\{ \epsilon ,\Vert f\Vert _1\right\} ,\quad \delta \leftarrow \delta ,\quad N\leftarrow N_{k-1} \end{aligned}$$

(29)

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 $N_{k-1}\le N_k\in {\mathbb {N}}$ and a Fourier polynomial

$$\begin{aligned} Q_k(x)=\sum _{n=N_{k-1}}^{N_k-1}Y_n(x) \end{aligned}$$

(30)

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 $x\notin \Delta _k$,

$3^\circ$.:

$|\gamma _k||\Delta _k|<\Vert g_k\Vert _1<2|\gamma _k||\Delta _k|$,

$4^\circ$.:

$\Vert Q_k-g_k\Vert _1<\frac{1}{2^{\nu _0+2}}\min \{\epsilon ,\Vert f\Vert _1\}$,

$5^\circ$.:

and

$$\begin{aligned} \max _{N_{k-1}\le m<N_k}\left\| \sum _{n=N_{k-1}}^mY_n\right\| _1\le \frac{|\gamma _k|\sqrt{|\Delta _k|(1+\delta )}}{\sqrt{\delta }}. \end{aligned}$$

Set

$$\begin{aligned}&E\doteq \bigcup _{k=1}^{\nu _0}E_k,\quad g(x)\doteq f(x)-\left[ \Lambda (x)-\sum _{k=1}^{\nu _0}g_k(x)\right] , \end{aligned}$$

(31)

$$\begin{aligned}&N\doteq N_{\nu _0}-1,\quad Q(x)\doteq \sum _{k=1}^{\nu _0}Q_k(x)=\sum _{n=N_0}^NY_n(x),\quad \forall x\in {\mathcal {M}}. \end{aligned}$$

(32)

First we check that from (26), ($1^\circ$), ($2^\circ$) and (31) it follows that

$$\begin{aligned}&|E|=\sum _{k=1}^{\nu _0}|E_k|>\sum _{k=1}^{\nu _0}|\Delta _k|\left( 1-\delta \right) =1-\delta , \end{aligned}$$

(33)

$$\begin{aligned}&x\in E\quad \Longrightarrow \quad x\in E_k\Longrightarrow \Lambda (x)=\gamma _k=g_k(x),\,g_l(x)=0,\,l\ne k\Longrightarrow g(x)=f(x),\nonumber \\ \end{aligned}$$

(34)

so that Statements 1 and 2 are proven. Next we observe using ($4^\circ$), (27), (31) and (32) that

$$\begin{aligned} \Vert Q-g\Vert _1=\left\| \sum _{k=1}^{\nu _0}\left[ Q_k-g_k\right] +\left[ f-\Lambda \right] \right\| _1\le \sum _{k=1}^{\nu _0}\Vert Q_k-g_k\Vert _1+\Vert f-\Lambda \Vert _1<\epsilon , \end{aligned}$$

(35)

which proves Statement 4. Further, from (26), (27), ($2^\circ$) and ($3^\circ$) we deduce that

$$\begin{aligned} \Vert g\Vert _1\le & {} \sum _{k=1}^{\nu _0}\Vert g_k\Vert _1+\Vert f-\Lambda \Vert _1\le 2\sum _{k=1}^{\nu _0}|\gamma _k||\Delta _k|+\Vert f-\Lambda \Vert _1\nonumber \\= & {} 2\Vert \Lambda \Vert _1+\Vert f-\Lambda \Vert _1\le 3\Vert f-\Lambda \Vert _1+2\Vert f\Vert _1<3\Vert f\Vert _1. \end{aligned}$$

(36)

Moreover, the same formulae also imply

$$\begin{aligned}&\Vert g\Vert _1+\frac{1}{3}\Vert f\Vert _1>\Vert g\Vert _1+\Vert f-\Lambda \Vert _1\ge \Vert g-f+\Lambda \Vert _1\nonumber \\&\quad =\sum _{k=1}^{\nu _0^2}\Vert g_k\Vert _1>\sum _{k=1}^{\nu _0^2}|\gamma _k||\Delta _k| =\Vert \Lambda \Vert _1\ge \bigl |\Vert \Lambda -f\Vert _1-\Vert f\Vert _1\bigr |>\frac{2}{3}\Vert f\Vert _1, \end{aligned}$$

(37)

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 $N_{k_0-1}\le m<N_{k_0}$, and thus by (30) and (32) we have

$$\begin{aligned} \sum _{n=N_0}^mY_n(x)=\sum _{k=1}^{k_0-1}Q_k(x)+\sum _{N_{k_0-1}}^mY_n(x). \end{aligned}$$

(38)

Finally we use this along with formulae ($3^\circ$), ($4^\circ$), ($5^\circ$) and (28) to obtain

$$\begin{aligned}&\left\| \sum _{n=N_0}^mY_n(x)\right\| _1\le \sum _{k=1}^{k_0-1}\left\| Q_k-g_k\right\| _1 +\sum _{k=1}^{k_0-1}\Vert g_k\Vert _1+\left\| \sum _{k=N_{k_0-1}}^mY_n\right\| _1\nonumber \\&\quad<\frac{1}{4}\Vert f\Vert _1+2\Vert \Lambda \Vert _1+\frac{|\gamma _{k_0}|\sqrt{|\Delta _{k_0}|(1+\delta )}}{\sqrt{\delta }}<3\Vert f\Vert _1, \end{aligned}$$

(39)

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 three-epsilon 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

$$\begin{aligned} \Vert R_{k_0}-f\Vert _1\le \frac{1}{2}\min \left\{ \Vert f\Vert _1, b_1\right\} . \end{aligned}$$

(40)

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 $k_s>k_{s-1}$ and

$$\begin{aligned} \left\| f-\sum _{r=0}^sR_{k_r}\right\| _1=\left\| R_{k_s}-\left( f-\sum _{r=0}^{s-1}R_{k_r}\right) \right\| _1<\frac{1}{2}\min \left\{ b_s, b_{s+1},\frac{1}{s}\right\} ,\quad \forall s\in {\mathbb {N}}. \end{aligned}$$

(41)

Statements 1 and 3 are clearly satisfied. For Statement 2 we have

$$\begin{aligned} \left\| R_{k_s}\right\| _1= & {} \left\| R_{k_s}-\left( f-\sum _{r=0}^{s-1}R_{k_r}\right) +\left( f-\sum _{r=0}^{s-1}R_{k_r}\right) \right\| _1\nonumber \\\le & {} \left\| R_{k_s}-\left( f-\sum _{r=0}^{s-1}R_{k_r}\right) \right\| _1 +\left\| f-\sum _{r=0}^{s-1}R_{k_r}\right\| _1\nonumber \\< & {} \frac{1}{2}b_s+\frac{1}{2}b_s=b_s,\quad \forall s\in {\mathbb {N}}. \end{aligned}$$

(42)

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

$$\begin{aligned} \epsilon _0\doteq \min \left\{ \epsilon ,\Vert f\Vert _1\right\} . \end{aligned}$$

(43)

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

$$\begin{aligned} f\leftarrow R_k,\quad \epsilon \leftarrow \frac{\epsilon _0}{2^{k+7}},\quad \delta \leftarrow \frac{\delta }{2^k},\quad N_0\leftarrow N_{k-1} \end{aligned}$$

(44)

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 $N_{k-1}\le N_k\in {\mathbb {N}}$ (set $N_0=0$) and a Fourier polynomial

$$\begin{aligned} {\tilde{Q}}_k(x)=\sum _{n=N_{k-1}}^{N_k-1}{\tilde{Y}}_n(x) \end{aligned}$$

(45)

with the following properties:

$1^\dagger$.:

$|{\tilde{E}}_k|>1-\frac{\delta }{2^k}$,

$2^\dagger$.:

$x\in {\tilde{E}}_k$ implies ${\tilde{g}}_k(x)=R_k(x)$,

$3^\dagger$.:

$\frac{1}{3}\Vert R_k\Vert _1<\Vert {\tilde{g}}_k\Vert _1<3\Vert R_k\Vert _1$,

$4^\dagger$.:

$\Vert {\tilde{g}}_k-{\tilde{Q}}_k\Vert _1<\epsilon _02^{-k-7}$,

$5^\dagger$.:

and

$$\begin{aligned} \sup _{N_{k-1}\le m<N_k}\left\| \sum _{n=N_{k-1}}^m\tilde{Y}_n\right\| _1<3\Vert R_k\Vert _1. \end{aligned}$$

Define the desired set E as

$$\begin{aligned} E\doteq \bigcap _{k=1}^\infty {\tilde{E}}_k. \end{aligned}$$

(46)

Observe from ($1^\dagger$) that

$$\begin{aligned} |E|=1-|{\mathcal {M}}{\setminus } E|\ge 1-\sum _{s=1}^\infty |{\mathcal {M}}{\setminus }\tilde{E}_k|>1-\sum _{k=1}^\infty \frac{\delta }{2^k}=1-\delta . \end{aligned}$$

(47)

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

$$\begin{aligned} f\leftarrow f,\quad b_s\leftarrow \frac{\epsilon _0}{2^{s+6}}. \end{aligned}$$

(48)

It satisfies

$1^\circ$.:

$\Vert R_{k_0}-f\Vert _1\le \frac{1}{2}\Vert f\Vert _1$,

$2^\circ$.:

$\Vert R_{k_s}\Vert _1<\epsilon _02^{-s-6}$ 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 $g_s(x)=R_{k_s}(x)$,

$2^*.$:

$\Vert g_s\Vert _1<\epsilon _02^{-s-2}$,

$3^*.$:

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

$4^*.$:

$$\begin{aligned} \max _{N_{\nu _s-1}\le m<N_{\nu _s}}\left\| \sum _{n=N_{\nu _s-1}}^m\tilde{Y}_n\right\| _1<\frac{\epsilon _0}{2^s}. \end{aligned}$$

Assume that for some $s\in {\mathbb {N}}$, the choice of $1<\nu _1<\nu _2<\cdots <\nu _{s-1}$ and $g_1, g_2,\ldots ,g_{s-1}$ 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 $N_{\nu _1-1}>\max \sigma (R_{k_0})$ and $\nu _s>\nu _{s-1}$ for $s>1$, and

$$\begin{aligned} \left\| R_{\nu _s}-\left( R_{k_s}-\sum _{j=1}^{s-1}[\tilde{Q}_{\nu _j}-g_j]\right) \right\| _1<\frac{\epsilon _0}{2^{s+7}}. \end{aligned}$$

(49)

Then by ($2^\circ$) and ($3^*$) we have for all $s\in {\mathbb {N}}$ that

$$\begin{aligned} \left\| R_{k_s}-\sum _{j=1}^{s-1}[\tilde{Q}_{\nu _j}-g_j]\right\| _1\le \Vert R_{k_s}\Vert _1+\left\| \sum _{j=1}^{s-1}[\tilde{Q}_{\nu _j}-g_j]\right\| _1<\frac{3\,\epsilon _0}{2^{s+6}}, \end{aligned}$$

(50)

which combined with (49) implies

$$\begin{aligned} \Vert R_{\nu _s}\Vert _1\le \left\| R_{\nu _s}-\left( R_{k_s}-\sum _{j=1}^{s-1}[\tilde{Q}_{\nu _j}-g_j]\right) \right\| _1+\left\| R_{k_s}-\sum _{j=1}^{s-1}[\tilde{Q}_{\nu _j}-g_j]\right\| _1<\frac{7\,\epsilon _0}{2^{s+7}}.\quad \end{aligned}$$

(51)

Set

$$\begin{aligned} g_s(x)\doteq R_{k_s}(x)+{\tilde{g}}_{\nu _s}(x)-R_{\nu _s}(x). \end{aligned}$$

(52)

Condition ($1^*$) is easily satisfied thanks to ($2^\dagger$) with $k=\nu _s$. For condition ($2^*$) we write

$$\begin{aligned} \Vert g_s\Vert _1= & {} \Vert R_{k_s}+{\tilde{g}}_{\nu _s}-R_{\nu _s}\Vert _1\nonumber \\\le & {} \left\| R_{\nu _s}-R_{k_s}+\sum _{j=1}^{s-1}[\tilde{Q}_{\nu _j}-g_j]\right\| _1+\left\| \sum _{j=1}^{s-1}[\tilde{Q}_{\nu _j}-g_j]\right\| _1+\Vert \tilde{g}_{\nu _s}\Vert _1<\frac{\epsilon _0}{2^{s+2}},\qquad \end{aligned}$$

(53)

where we used (49), ($3^\dagger$), ($3^*|_{s-1}$) and (51) in the last step. To show that condition ($3^*$) is satisfied we observe that

$$\begin{aligned}&\left\| \sum _{j=1}^s[{\tilde{Q}}_{\nu _j}-g_j]\right\| _1=\left\| \tilde{Q}_{\nu _s}-g_s+\sum _{j=1}^{s-1}[\tilde{Q}_{\nu _j}-g_j]\right\| _1\nonumber \\&\quad =\left\| {\tilde{Q}}_{\nu _s}-R_{k_s}-\tilde{g}_{\nu _s}+R_{\nu _s}+\sum _{j=1}^{s-1}[\tilde{Q}_{\nu _j}-g_j]\right\| _1\nonumber \\&\quad \le \left\| {\tilde{Q}}_{\nu _s}-\tilde{g}_{\nu _s}\right\| _1+\left\| R_{\nu _s}-\left( R_{k_s}-\sum _{j=1}^{s-1}[\tilde{Q}_{\nu _j}-g_j]\right) \right\| _1<\frac{\epsilon _0}{2^{s+6}}, \end{aligned}$$

(54)

where ($4^\dagger$), (49) and $\nu _s>s$ were used in the second inequality. Finally we satisfy condition ($4^*$) using ($5^\dagger$) and (51),

$$\begin{aligned} \max _{N_{\nu _s-1}\le m<N_{\nu _s}}\left\| \sum _{n=N_{\nu _s-1}}^m\tilde{Y}_n\right\| _1<3\Vert R_{\nu _s}\Vert _1<\frac{\epsilon _0}{2^s}. \end{aligned}$$

(55)

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

$$\begin{aligned} g(x)\doteq R_{k_0}(x)+\sum _{s=1}^\infty g_s(x),\quad \forall x\in {\mathcal {M}}. \end{aligned}$$

(56)

From (53) it follows that

$$\begin{aligned} \sum _{s=1}^\infty \Vert g_s\Vert _1<\frac{13\,\epsilon _0}{64}<\infty , \end{aligned}$$

(57)

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

$$\begin{aligned} g(x)=R_{k_0}(x)+\sum _{s=1}^\infty g_s(x)=\sum _{s=0}^\infty R_{k_s}(x)=f(x),\quad \forall x\in E. \end{aligned}$$

(58)

Let $\{Y_n\}_{n=1}^\infty$ be the series of $Y_n=c_n\varphi _n$ such that

$$\begin{aligned} \sum _{n=1}^{N_{\nu _s}-1}Y_n=R_{k_0}+\sum _{j=1}^s\tilde{Q}_{\nu _j},\quad \forall s\in {\mathbb {N}}. \end{aligned}$$

(59)

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 $m<N_{\nu _1-1}$ set $r=1$). Set ${m_*\doteq \min \{m,N_{\nu _{r}}-1\}}$. Then by (56), ($3^*$), ($4^*$) and (53) we get

$$\begin{aligned} \left\| \sum _{n=1}^m Y_n-g\right\| _1= & {} \left\| \sum _{n=1}^{m_*} Y_n-g\right\| _1=\left\| \sum _{j=1}^{r-1}\tilde{Q}_{\nu _j}+\sum _{n=N_{\nu _{r}-1}}^{m_*}\tilde{Y}_n-\sum _{j=1}^{r-1}g_j-\sum _{j=r}^\infty g_j\right\| _1\nonumber \\\le & {} \left\| \sum _{j=1}^{r-1}\left[ \tilde{Q}_{\nu _j}-g_j\right] \right\| _1+\left\| \sum _{n=N_{\nu _{r}-1}}^{m_*}\tilde{Y}_n\right\| _1+\left\| \sum _{j=r}^\infty g_j\right\| _1<\frac{23\,\epsilon _0}{2^{r+4}}. \end{aligned}$$

(60)

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

$$\begin{aligned} \left\| f-g\right\| _1=\left\| \sum _{s=1}^\infty R_{k_s}-\sum _{s=1}^\infty g_s\right\| _1\le \sum _{s=1}^\infty \Vert R_{k_s}\Vert _1+\sum _{s=1}^\infty \Vert g_s\Vert _1<\frac{7\,\epsilon _0}{32}, \end{aligned}$$

(61)

which in view of (43) proves Statement 1. Finally, using (60) and (61) we establish that

$$\begin{aligned} \left\| \sum _{n=1}^m Y_n\right\| _1\le \left\| \sum _{n=1}^m Y_n-g\right\| _1+\left\| f-g\right\| _1+\left\| f\right\| _1<\frac{15\,\epsilon _0}{16}+\Vert f\Vert _1<2\Vert f\Vert _1, \end{aligned}$$

(62)

but also

$$\begin{aligned} \left\| \sum _{n=1}^m Y_n\right\| _1\le \left\| \sum _{n=1}^m Y_n-g\right\| _1+\left\| g\right\| _1<\frac{23}{32}\Vert f\Vert _1+\Vert g\Vert _1. \end{aligned}$$

(63)

Note that by (61)

$$\begin{aligned} \Vert f\Vert _1\le \Vert g\Vert _1+\Vert f-g\Vert _1<\Vert g\Vert _1+\frac{7}{32}\Vert f\Vert _1, \end{aligned}$$

(64)

and therefore $25\Vert f\Vert _1<32\Vert g\Vert _1$. This together with (63) yields

$$\begin{aligned} \left\| \sum _{n=1}^m Y_n\right\| _1<\frac{23}{25}\Vert g\Vert _1+\Vert g\Vert _1<2\Vert g\Vert _1, \end{aligned}$$

(65)

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 Peter-Weyl 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

$$\begin{aligned} \varphi _{\rho ,i,j}(x)\doteq \sqrt{d_\rho }(\rho (x)e^\rho _j, e^\rho _i),\quad \forall x\in G,\quad i,j=1,\dots ,d_\rho ,\quad \forall \rho \in {\hat{G}}. \end{aligned}$$

(66)

By the Peter-Weyl 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

$$\begin{aligned} |\varphi _{\rho ,i,j}(x)|=\sqrt{d_\rho }\,|(\rho (x)e^\rho _j,e^\rho _i)| \le \sqrt{d_\rho }\,\Vert \rho (x)e^\rho _j\Vert \Vert e^\rho _i\Vert \le \sqrt{d_\rho }, \end{aligned}$$

(67)

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,

$$\begin{aligned} {\hat{f}}(\rho )=\int _Gf(x)\rho ^*(x)dx,\quad \forall f\in L^1(G), \end{aligned}$$

(68)

and the corresponding block Fourier series

$$\begin{aligned}&\sum _{\rho \in {\hat{G}}}d_\rho {\text {tr}}\left[ {\hat{f}}(\rho )\rho (x)\right] =\sum _{\rho \in {\hat{G}}}\sum _{i,j=1}^{d_\rho }Y_{\rho ,i,j}(x,f),\nonumber \\&Y_{\rho ,i,j}(x,f)=c_{\rho ,i,j}(f)\varphi _{\rho ,i,j}(x),\quad c_{\rho ,i,j}(f)=(f,\varphi _{\rho ,i,j})_2. \end{aligned}$$

(69)

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

$$\begin{aligned} d^H_\rho ={\text {mult}}({{\mathbf {1}}},\rho \,\big | _H),\quad \forall \rho \in {\hat{G}} \end{aligned}$$

(70)

(this reduces to $d^H_\rho =d_\rho$ if $H=\{{{\mathbf {1}}}\}$ as before). Moreover, we restrict to

$$\begin{aligned} \widehat{G/H}=\left\{ \rho \in {\hat{G}}\,\big |\quad d^H_\rho >0\right\} . \end{aligned}$$

(71)

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 G-invariant measure $\mu$ on G/H (the pullback of $d\mathrm {x}$ through the quotient map) such that

$$\begin{aligned} \int _Gf(\mathrm {x})d\mathrm {x}=\int _{G/H}\left( \int _Hf(\mathrm {x}h)dh\right) d\mu (x),\quad \forall f\in C(G). \end{aligned}$$

(72)

Denote

$$\begin{aligned} {\mathbb {P}}_H\doteq \int _H\rho (h)dh,\quad \rho (x)\doteq \rho (\mathrm {x}){\mathbb {P}}_H,\quad \forall \rho \in \widehat{G/H},\quad \forall x=\mathrm {x}H\in G/H. \end{aligned}$$

(73)

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

$$\begin{aligned} {\hat{f}}(\rho )=\int _{G/H}f(x)\rho ^*(x)d\mu (x),\quad \forall \rho \in \widehat{G/H}, \end{aligned}$$

(74)

and the corresponding block Fourier series is

$$\begin{aligned}&\sum _{\rho \in \widehat{G/H}}d_\rho {\text {tr}}\left[ {\hat{f}}(\rho )\rho (x)\right] =\sum _{\rho \in \widehat{G/H}}\sum _{i=1}^{d_\rho }\sum _{\alpha =1}^{d^H_\rho }Y_{\rho ,i,\alpha }(x;f),\end{aligned}$$

(75)

$$\begin{aligned}&Y_{\rho ,i,\alpha }(x;f)=c_{\rho ,i,\alpha }(f)\varphi _{\rho ,i,\alpha }(x),\quad c_{\rho ,i,\alpha }(f)=(f,\varphi _{\rho ,i,\alpha })_2,\nonumber \\&\quad \varphi _{\rho ,i,\alpha }(x)=\sqrt{d_\rho }(\rho (x)e^\rho _\alpha ,e^\rho _i). \end{aligned}$$

(76)

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. $\Vert f-g\Vert _1<\epsilon$,

• 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 non-trivial stabilizer within K,

$$\begin{aligned} G/H^{(K)}=\left\{ x\in G/H\,\big |\quad \exists \,{{\mathbf {1}}}\ne k\in K\quad \text{ s.t. }\quad kx=x\right\} . \end{aligned}$$

(77)

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

$$\begin{aligned} \varphi (k'k,Kx)=k'\varphi (k,Kx),\quad {\text {q}}\left( \varphi (k,Kx)\right) =Kx,\quad \forall k,k'\in K,\quad \forall Kx\in K\backslash G/H. \end{aligned}$$

(78)

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 Federer-Morse 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

$$\begin{aligned} \varphi (k,Kx)\doteq k\cdot {\text {q}}|_W^{-1}(Kx),\quad \forall k\in K,\quad \forall Kx\in X. \end{aligned}$$

(79)

$\varphi$ is Borel bi-measurable, since it is the composition of bi-measurable 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

$$\begin{aligned} {\text {q}}(\varphi (k_1,Kx_1))=Kx_1={\text {q}}(\varphi (k_2,Kx_2))=Kx_2, \end{aligned}$$

(80)

and

$$\begin{aligned} \varphi (k_1,Kx_1)=k_1\varphi ({{\mathbf {1}}},Kx_1)=\varphi (k_2,Kx_2)=k_2\varphi ({{\mathbf {1}}},Kx_1), \end{aligned}$$

(81)

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 non-trivial 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

$$\begin{aligned} \int _{G/H}f(x)d\mu (x)=\int _{K\backslash G/H}d\nu (Kx)\int _Kf(kx)dk. \end{aligned}$$

(82)

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

$$\begin{aligned} \int _Yf(x)d\mu (x)= & {} \int _{G/H}f(x)\chi _Y(x)d\mu (x)=\int _{K\backslash G/H}d\nu (Kx)\int _Kf(kx)\chi _Y(kx)dk\nonumber \\= & {} \int _{K\backslash G/H}\chi _X(Kx)d\nu (Kx)\int _Kf(kx)dk\nonumber \\= & {} \int _Xd\nu (Kx)\int _Kf(\varphi (k,Kx))dk, \end{aligned}$$

(83)

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 K-invariant, 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

$$\begin{aligned} \left| G/H{\setminus }\, Y\right| _\mu =\left| K\cdot Z{\setminus }\, K\cdot W\right| _\mu =\left| K\cdot (Z{\setminus }\, W)\right| _\mu \le \left| K\cdot G/H^{(K)}\right| _\mu =\left| G/H^{(K)}\right| _\mu =0,\nonumber \\ \end{aligned}$$

(84)

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

$$\begin{aligned} d_\rho =\dim {\mathcal {H}}_\rho ={{d+\rho }\atopwithdelims (){d}}-{{d+\rho -2}\atopwithdelims (){d}},\quad \forall \rho \in {\mathbb {N}}_0, \end{aligned}$$

(85)

whereas the multiplicities are all $d^H_\rho =1$. We choose standard spherical coordinates $x=(\theta _1,\ldots ,\theta _{d-1},\phi )$, where $\theta _j\in [0,\pi ]$, $j=1,\ldots ,d-1$, and $\phi \in [0,2\pi )$. The orthonormal system $\{\varphi _{\rho ,i,\alpha }\}$ in this case consists of spherical harmonics

$$\begin{aligned} \varphi _{\rho ,i,\alpha }(x)=Y_\rho ^i(\theta _1,\ldots ,\theta _{d-1},\phi ),\quad \forall \rho \in {\mathbb {N}}_0,\quad i=1,\ldots ,d_\rho . \end{aligned}$$

(86)

The block Fourier series of a function $f\in L^1({\mathbb {S}}^d)$ is

$$\begin{aligned}&\sum _{\rho =1}^\infty \sum _{i=1}^{d_\rho }{\hat{f}}(\rho ;i)Y_\rho ^i(\theta _1,\ldots ,\theta _{d-1},\phi ),\ \end{aligned}$$

(87)

$$\begin{aligned}&{\hat{f}}(\rho ;i)=\int _0^{2\pi }\int _0^\pi \ldots \int _0^\pi f(\theta _1,\ldots ,\theta _{d-1},\phi )\bar{Y}_\rho ^i(\theta _1,\ldots ,\theta _{d-1},\phi )d\mu (\theta _1,\ldots ,\theta _{d-1},\phi ),\nonumber \\ \end{aligned}$$

(88)

$$\begin{aligned}&d\mu (\theta _1,\ldots ,\theta _{d-1},\phi )=\frac{\Gamma (\frac{d+1}{2})}{2\pi ^{\frac{d+1}{2}}} \sin ^{d-1}(\theta _1)d\theta _1\ldots \sin (\theta _{d-1})d\theta _{d-1}d\phi . \end{aligned}$$

(89)

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 - $\theta _j=0$, $j=1,\ldots ,d-1$, and $\theta _j=\pi$, $j=1,\ldots ,d-1$, 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

$$\begin{aligned} \varphi :\mathrm {SO}(2)\times \mathrm {SO}(2)\backslash \mathrm {SO}(d+1)/\mathrm {SO}(d)\rightarrow {\mathbb {S}}^d \end{aligned}$$

(90)

given by $\varphi (\phi ,(\theta _1,\ldots ,\theta _{d-1}))=(\theta _1,\ldots ,\theta _{d-1},\phi )$, 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

$$\begin{aligned}{}[0,1]\times \mathrm {SO}(2)\backslash \mathrm {SO}(d+1)/\mathrm {SO}(d)\rightarrow {\mathbb {S}}^d. \end{aligned}$$

(91)

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 ${\hat{N}}_0(1)=1$, ${\hat{N}}_0(k)={\hat{N}}_{\nu _0(k)}(k-1)$ for $k>1$, and ${\hat{Q}}_l^k=\sum _{n={\hat{N}}_{l-1}(k)}^{{\hat{N}}_l(k)-1}Y_n({\hat{g}}_l^k)$ ( 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

  $$\begin{aligned} {\tilde{Q}}_k=\sum _{l=1}^{\nu _0(k)}{\hat{Q}}_l^k=\sum _{n=N_{k-1}}^{N_k-1}{\tilde{Y}}_n. \end{aligned}$$

• 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 $\{\nu _s\}_{s=1}^\infty$, $\nu _s>\nu _{s-1}$ for $s>1$, and measurable functions $g_s\in L^\infty ({\mathcal {M}})$ by choosing $\nu _1$ so that $N_{\nu _1-1}>\max \sigma (R_{k_0})$ and

  $$\begin{aligned} \left\| R_{\nu _s}-R_{k_s}+\sum _{j=1}^{s-1}\left[ \tilde{Q}_{\nu _j}-g_j\right] \right\| _1 \end{aligned}$$

  is sufficiently small, and setting $g_s=R_{ks}+\tilde{g}_{\nu _s}-R_{\nu _s}$.

• Finally, set

  $$\begin{aligned} g=R_{k_0}+\sum _{s=1}^\infty g_s. \end{aligned}$$