Indicator Functions with Uniformly Bounded Fourier Sums and Large Gaps in the Spectrum

Abstract

Indicator functions mentioned in the title are constructed on an arbitrary nondiscrete locally compact Abelian group of finite dimension. Moreover, they can be obtained by small perturbation from any indicator function fixed beforehand. In the case of a noncompact group, the term “Fourier sums” should be understood as “partial Fourier integrals”. A certain weighted version of the result is also provided. This version leads to a new Men$'$shov-type correction theorem.

1 Introduction

When stated in precise terms, any specific form of the vague claim that a function and its Fourier transform cannot be too small simultaneously (the celebrated “uncertainty principle in harmonic analysis”) often turns into the question about a frontier beyond which this claim becomes false. In the range of problems where smallness is understood as the vanishing on a large set, one such frontier was marked recently by Nazarov and Olevskii (see [11]), who constructed a set E of finite positive measure on the real line such that the Fourier transform of the indicator function $\chi _E$ has support that is fairly thin at infinity. More specifically, given arbitrary mutually nonintersecting intervals $I_k$ in $\mathbb {R}_+$ whose lengths tend to infinity, the support of $\widehat{\chi _E}$ can be placed in $K\cup \left( \bigcup _k(I_k\cup (-I_k))\right)$ for some compact set K. We refer the reader to the same paper [11] for a concise survey of known facts about the two “countries” separated by the borderline indicated.

Shortly after, the first author of the present paper observed (see [6]) that a slight modification of the construction in [11] makes it possible to turn an arbitrary set $A\subset \mathbb {R}$ of finite positive measure into a set E as above by a small perturbation. It was also shown in [6] that, basically with the same proof, a similar result holds for any nondiscrete locally compact Abelian group.¹ In fact, the invocation of the idea of “correcting” a given indicator function was motivated by the results of [1, 2] and [7]. For example, in the last paper, an analog of Men$'$shov’s classical correction theorem was proved for an arbitrary locally compact Abelian group of finite dimension, moreover, the spectrum of the corrected function was placed in a “thin” set like the above union $\bigcup _k(I_k\cup (-I_k))$ in the case of $\mathbb {R}$.

We remind the reader that, on the circle, Men$'$shov’s correction theorem says that any measurable (equivalently, any measurable and bounded) function can be modified on a set of an arbitrarily small measure so as to acquire a uniformly convergent Fourier series. Surely, the analog of this statement for general groups also involves a certain type of uniform convergence for Fourier expansions. For the indicator function that emerges after correction, one might only hope for the uniform boundedness of partial Fourier sums or integrals instead of uniform convergence, but even this was not ensured in [6], moreover, it was hinted there that the method would unlikely be suitable for that.

However, later, a more careful look at the situation showed that, even within the class of indicator functions, we can still combine “thin” spectrum, uniform boundedness of partial Fourier integrals, and the idea of correction à la Men$'$shov. Again, all this can be done on every (nondiscrete) locally compact Abelian group of finite dimension. The present paper is devoted to the exposition of this and related results. The clever nonlinear construction by Nazarov and Olevskii will again be in the core of the arguments, but here this construction will require a more substantial modification than in [6]. Also, some techniques of the paper [7] will be invoked (which, however, are rather standard in similar issues).

The paper is organized as follows. In Sect. 2, after necessary preliminaries, we state the results and comment on them. The final Sect. 3 is devoted to the proofs.

2 Preliminaries and Precise Statements

Throughout, G will be a nondiscrete locally compact Abelian group of finite dimension, and $\Gamma$ will stand for its group of characters, with Haar measures dx and $d\gamma$; it is assumed that these Haar measures are normalized so that the Fourier transform ${\mathcal {F}}$, ${\mathcal {F}} f(\gamma ) = \int {f(x)\overline{\gamma (x)}dx}, \gamma \in \Gamma , f \in L^1(G)$, is a unitary operator from $L^2(G)$ onto $L^2(\Gamma )$. We will often write |e| for the Haar measure of a measurable subset e of G or $\Gamma$.

We reproduce a definition from [7] (see also [4]).

Definition

(Sufficient pairs) A pair (R, S) of closed subsets of $\Gamma$ is said to be sufficient if for every compact set $E \subset \Gamma$ there exists a character $\gamma \in \Gamma$ with $-\gamma + E \subset R$ and $\gamma + E \subset S$.

We shall put the spectrum of corrected (indicator) functions in the union $K\cup R\cup S$, where (R, S) is a sufficient pair in $\Gamma$ and K is a certain compact set depending on the function we are going to modify. The pair $(-\bigcup _k I_k, \bigcup _k I_k)$, which occurred in the Introduction, is sufficient for the (dual) group (of) $\mathbb {R}$. Clearly, a similar construction with intervals replaced by mutually nonintersecting balls of radii tending to infinity provides a sufficient pair in the case of (the dual group of) $\mathbb {R}^n$, and examples for the dual group $\mathbb {Z}^n$ of the torus $\mathbb {T}^n$ are provided in the same way. Moreover, in the case of $\mathbb {R}^n$ or $\mathbb {T}^n$, any sufficient pair includes another one of the above form.

Next, to discuss uniformly bounded Fourier sums (or partial Fourier integrals), we need the notion of a summation basis.

Definition

(Summation bases and the norm $\Vert \cdot \Vert _u$) For a measurable subset E of $\Gamma$ we define the operator $P_E$ (at least on $L^2(G)$) by the formula

$$\begin{aligned} P_Ef = {\mathcal {F}}^{-1}(\chi _E{\mathcal {F}} f). \end{aligned}$$

A subset of $\Gamma$ is said to be bounded if it has compact closure. Let ${\mathcal {B}}$ be a family of bounded measurable subsets of $\Gamma$ such that for every compact set $K \subset \Gamma$ there exists $E \in {\mathcal {B}}$ with $K\subset E$. Such a system ${\mathcal {B}}$ will be called a summation basis. For functions $f\in L^2(G)$ we introduce the following not necessarily finite quantity:

$$\begin{aligned} \Vert f\Vert _u = \sup _{B \in {\mathcal {B}}}{(\Vert P_B f\Vert _\infty )}. \end{aligned}$$

When we talk about uniformly bounded partial Fourier integrals (or sums), we shall mean the finiteness of the norm $\Vert \cdot \Vert _u$ for a certain fixed summation basis. Also, it should be noted that we might consider the quantity $\Vert f\Vert _u$ beyond the class $L^2(G)$ (for instance, it is well defined also for $f\in L^1(G)$), but the present $L^2(G)$-version will suffice in what follows.

Surely, not all summation bases are expected to admit a Men$'$shov-type correction theorem. So, we impose a restriction on them taken from [4] and [8].

Let E be a bounded measurable subset of $\Gamma$. We say that a set $B\in {\mathcal {B}}$ splits E if the sets $E \cap B$ and $E {\setminus } B$ have positive Haar measure. (If G is compact, this simply means that the two sets are nonempty, because $\Gamma$ is discrete in this case and its Haar measure is the counting measure.) Next, we denote by $E_{{\mathcal {B}}}$ an arbitrary representative of the lowest upper bound (in the complete lattice of measurable sets $\mod 0$) of the collection $\{B \in {\mathcal {B}}:B \; \text{ splits } \; E\}$ (in symbols, with a slight abuse of notation: $E_{{\mathcal {B}}} = \cup \{B \in {\mathcal {B}}:B \, \text{ splits } \, E\}$; if G is compact, then the union in the last formula can be understood literally).

The restriction we are going to impose on a summation basis will depend on a sufficient pair in question. Here it is.

Definition

(Coordination of a summation basis and a sufficient pair) A summation basis ${\mathcal {B}}$ and a sufficient pair (R, S) are said to be coordinated if for every bounded measurable set $E \subset \Gamma$ the pair $(R {\setminus } E_{{\mathcal {B}}}, S {\setminus } E_{{\mathcal {B}}})$ is also sufficient.

We give here two simple but important examples of summation bases coordinated with any sufficient pair. See [4, 8] for more information.

Example 2.1

Let G be compact, infinite, and metrizable. Let ${\mathcal {B}}=\{B_n\}_{n\in \mathbb {N}}$ be an arbitrary strictly monotone increasing sequence of finite subsets of $\Gamma$ whose union is equal to $\Gamma$. Then this collection is a summation basis coordinated with an arbitrary sufficient pair in $\Gamma$.

Indeed, under the above assumptions all compact sets in $\Gamma$ are finite, and it is easily seen that $E_{{\mathcal {B}}}$ is finite for every finite E. So it suffices to show that, whenever (S, R) is a sufficient pair in $\Gamma$ and $A\subset \Gamma$ is finite, the pair $(S{\setminus } A, R\setminus A)$ is also sufficient. For this, taking a finite set $D\subset \Gamma$, we fix $\mu \in \Gamma$ (to be specified later), put $D_1=(-\mu +D)\cup (\mu +D)\cup D$ and find $\lambda \in \Gamma$ with $\lambda + D_1\subset S$ and $-\lambda + D_1\subset R$. Then for some choice of $\mu$ either $(\lambda + D)\cup (-\lambda + D)$ or $(\lambda +\mu + D)\cup (-\lambda -\mu + D)$ does not intersect A. For, otherwise both $\lambda$ and $\lambda +\mu$ belong to $H=(D-A)\cup (A-D)$, whence $\mu \in H-H$. Since H is finite and $\Gamma$ is infinite, there is $\mu$ for which the last condition is violated.

Example 2.2

The second example pertains to the case where G is $\mathbb {R}^n$ or $\mathbb {T}^n$ (accordingly, $\Gamma$ is either $\mathbb {R}^n$ or $\mathbb {Z}^n$). A nonempty subset B of $\mathbb {R}^n$ or $\mathbb {Z}^n$ is said to be solid if $y=(y_1,\ldots ,y_n)\in B$ whenever $|y_j|\le |x_j|$ for $j=1,\ldots ,n$ and $x=(x_1,\ldots ,x_n)\in B$. We claim that the collection ${\mathcal {B}}$ of all solid sets constitutes a summation basis coordinated with an arbitrary sufficient pair (S, R).

To explain this, suppose for definiteness that we work with $\Gamma =\mathbb {Z}^n$ . (The case of $\mathbb {R}^n$ is similar.) Taking a finite set $K\subset \mathbb {Z}^n$, for each $j=1,\ldots ,n$ consider the smallest strip $D_j$ of the form $\{x\in \mathbb {Z}^n:|x_j|\le d\}$ that includes K, and let D be the union of these strips. It is quite easy to realize that $K_{{\mathcal {B}}}\subset D$. Now, a simple direct inspection shows that $(S{\setminus } D, R\setminus D)$ is a sufficient pair. The figure illustrates the case of $n=2$.

We pass to precise statements of the results. In fact, we prove a “weighted” version of what was discussed in the Introduction. By a weight (maybe the term “a rail” would be more appropriate) we mean a uniformly continuous positive function w on $\Gamma$ that is bounded and bounded away from zero. We will modify functions of the form $\chi _a w$ (instead of $\chi _a$) up to functions of the same form. For convenience, we assume that $w\le 1$ (this is merely a normalization condition). Suppose we are given a sufficient pair (R, S) in G and a summation basis ${\mathcal {B}}$ in $\Gamma$ coordinated with this sufficient pair; the norm $\Vert \cdot \Vert _u$ will be related to this summation basis.

Theorem 2.3

For every $\varepsilon > 0$ and an arbitrary measurable subset a of G with $0<|a|<\infty$, there is a measurable subset b of G such that

• (1) $\int _{a \triangle b} w^2 <\varepsilon$,

• (2) the spectrum of $\chi _b w$ is included in $K\cup R\cup S$ for some compact set $K\subset \Gamma$ depending on a,

• (3) the norm $\Vert w\chi _b\Vert _u$ is finite.

Remark 2.4

The facts discussed in the Introduction follow if w is identically equal to 1. In this case it can also be ensured that $|b|=|a|$.

Remark 2.5

The claim that $\Vert w \chi _b\Vert _u$ is finite can be supplemented with the inequality $\Vert P_B(w \chi _b)\Vert _{\infty }\le C$ with C depending only on $\dim G$ whenever $B\in {\mathcal {B}}$ and $B\supset K$ (this will be verified in the course of the proof).

Neither the compact set K nor the norms of the $P_B(\chi _b w)$ where B splits K are under control in general, we only know a uniform bound for these norms depending on a. However, in some specific cases the estimate can be refined. For example, this is true for the groups $\mathbb {R}$ and $\mathbb {T}$, the standard summation bases $\{[-N, N], n \in \mathbb {N}\}$ for the circle and $\{[-M, M], M \in \mathbb {R}\}$ for the real line (by the way, these are precisely the bases of solid sets mentioned in Example 1.2) and an arbitrary sufficient pair.

Theorem 2.6

Under the assumptions listed in the preceding paragraph, the set b as in Theorem 2.3 can be chosen in such a way that $\Vert w \chi _b\Vert _u \le C\log (2+\varepsilon ^{-1}\int _aw)$. Here C is a universal constant.

Remark 2.7

The same is true for the dyadic group $D=\{-1,1\}^{\mathbb {N}}$ if we mean a uniform bound for the partial sums of Walsh-Fourier series under the standard enumeration of the Walsh system. Again, a sufficient pair can be taken arbitrarily (note that $1=-1$ in the dual group of D, so the notion itself of a sufficient pair simplifies in this case). Moreover, the Walsh system here can be replaced with bounded Vilenkin systems. See Sect. 3.6 for some more information.

Theorem 2.6 will allow us to deduce our final result, which is not confined to characteristic functions, and is apparently new in the range of correction theorems. It holds for the groups $\mathbb {R}$ and $\mathbb {T}$ with the standard summation bases and arbitrary sufficient pairs, and also for certain zero-dimensional compact groups; see Sect. 3.6 for the discussion. To a certain extent, this correction theorem absorbs all developments known previously: the modified function has both thin spectrum and bounded Fourier integrals, and obeys a sharp estimate like in Theorem 2.6. We give the statement for the group $\mathbb {R}$ for definiteness (and with a slightly weaker inequality for the norm $\Vert \cdot \Vert _u$ than in Sect. 3.6).

Theorem 2.8

Let $\varepsilon > 0$. Given a function $h\in L^{\infty }(\mathbb {R})$ supported on a set of finite measure and with $\Vert h\Vert _{\infty }\le 1$, there is a function f such that $\Vert f\Vert _\infty \le 20$, $|\{h\ne f\}|\le \varepsilon$, and $\Vert w \chi _b\Vert _u \le C\log (2+\varepsilon ^{-1}|{{\,\mathrm{supp}\,}}h|)$. Furthermore, the spectrum of f is included in $K\cup R\cup S$, where K is a compact set depending on h.

3 Proofs

3.1 Approximate Identities

We need a certain analog of the Fejér kernels for our group G. Let U be a compact symmetric neighborhood of zero in the dual group $\Gamma$. We put $\psi _U = (|U|^{-1/2}\chi _{U}) * (|U|^{-1/2}\chi _{U})$. This is a continuous function on $\Gamma$ with values in [0, 1], supported on the compact set $K = U + U$, and satisfying $\psi _{U}(0) = 1$. Define $\Phi _{U} = {\mathcal {F}}^{-1}\psi _{U}$; then $\Vert \Phi _{U}\Vert _1 = 1$ and $\Phi _{U}\ge 0$. In this subsection, the assumption that $\dim G<\infty$ is not required.

Lemma 3.1

For a certain family of neighborhoods U of zero in $\Gamma$, the corresponding functions $\Phi _{U}$ form an approximate identity² for G.

The claim is standard but not quite straightforward because the invocation of the structure theorem seems to be obligatory for the proof. Next, surprisingly, we have not found precisely this statement in standard handbooks. So, for completeness, we sketch the arguments. By the structure theorem (see, e.g., [3]), G splits in the direct product of $\mathbb {R}^k$ and a group containing an open compact subgroup $G_1$. It is quite easy to see that it suffices to prove the claim separately for $\mathbb {R}^k$ and $G_1$. The group $\mathbb {R}^k$ presents no problems (when U runs through the family of cubes centered at zero, we obtain the family of genuine multiple Fejér kernels on $\mathbb {R}^k$). We will see that the case of the compact group $G_1$ reduces to considering similar cubes, this time in $\mathbb {Z}^s$ for some s. Denote by $\Gamma _1$ the (discrete) dual of $G_1$. Since the operators of convolution with $\Phi _{U}$ have norm at most one on $L^1(G_1)$, it suffices, given a finite set C of characters on $G_1$, to find U such that this convolution operator is as close to the identity on C as we wish. This means that we must find a symmetric finite subset U of $\Gamma _1$ containing zero such that the function $\psi _U$ is very close to 1 on C.

Now, let $\Delta$ be the subgroup of $\Gamma _1$ generated by C. Since C is finite, $\Delta$ is a direct sum of finitely many cyclic groups, whence $\Delta =\mathbb {Z}^s\oplus \Omega$ for some finite Abelian group $\Omega$. If $s=0$, take $\Omega$ for U. Otherwise, take a large cube Q centered at zero in $\mathbb {Z}^s$ and put $U=Q\oplus \Omega$. A short reflection shows that the required property of $\psi _U$ is ensured as soon as the diameter of Q is much greater then the diameter of the projection of C to $\mathbb {Z}^s$, and we are done.

Lemma 3.2

Given a compact neighborhood V of zero in G and $\varepsilon >0$, there exists a compact symmetric neighborhood U of zero in $\Gamma$ such that $\int _{G{\setminus } V}\Phi _U<\varepsilon$.

Proof

This is standard for approximate identities. Indeed, given f in $L^1(G)$, we can find U with $\Vert f-f*\Phi _U\Vert _{L^1(G)}<\varepsilon$. Now, take a symmetric neighborhood W of zero in G such that $W-W\subset V$ and find such a U for $f=|W|^{-1}\chi _W$. Since now f vanishes outside W, we have $\int _{G{\setminus } W} f*\Phi _U (x) dx <\varepsilon$. After plugging the integral formula for convolution in the expression on the left and changing the order of integration, this becomes

$$\begin{aligned} \int _G\Phi _U(t)\frac{|(W+t){\setminus } W|}{|W|}dt<\varepsilon . \end{aligned}$$

Now, the fraction under the integral sign is equal to 1 if $t\notin W-W\subset V$. So, it suffices to restrict integration in the last formula to the complement of V. $\square$

The last lemma allows us to prove the following statement, which will be useful in the main construction below. Let w be a weight as in Theorem 2.3, i.e., a uniformly continuous positive function on $\Gamma$ that is bounded above by 1 and bounded away from zero.

Lemma 3.3

For every $\eta >0$, we have eventually $w*\Phi _U\le (1+\eta ) w$ for the above approximate identity.

Proof

Take $\varepsilon >0$ and find a compact symmetric neighborhood V of zero in G such that $|w(x)-w(y)|\le \varepsilon$ whenever $x-y\in V$. Then take $\Phi _U$ as in Lemma 3.2 for this V and $\varepsilon$ and write (denoting by d some positive lower bound for w):

$$\begin{aligned} \Phi _U*w(x)\le & {} \int \limits _V w(x-y)\Phi _U(y) dy+\int \limits _{G{\setminus } V}\Phi _U (y)dy \\\le & {} w(x)+2\varepsilon \le w(x)\left( 1+\frac{2\varepsilon }{d}\right) \le w(x)(1+\eta ) \end{aligned}$$

if $\varepsilon$ is sufficiently small. $\square$

3.2 Covering Neighborhoods

This is a technical ingredient used in various proofs of Men$'$shov-type correction theorems.

Definition 3.4

A compact neighborhood V of 0 in G is said to be covering if there exists a family $\{x_i\}_{i \in I}$ of points in G such that $G = \cup _{i \in I}{(x_i + V)}$ and $|(x_i + V) \cap (x_j + V)| = 0,\,\,\, i \ne j$.

With a covering neighborhood V, we associate the family $\{\alpha _i\}_{i \in I}$,

$$\begin{aligned} \alpha _i(t) =\frac{\chi _V * \chi _V (t - x_i)}{|V|}, \; t \in G, \end{aligned}$$

(3.1)

of functions on G, where $\{x_i\}$ is the family of points mentioned in Definition 3.4. Observe that $\Vert \alpha _i\Vert _\infty = 1$ and $\Vert {\mathcal {F}}\alpha _i\Vert _1 = 1$. The last identity will enable us to use combinations of these functions to provide Fourier expansions with uniformly bounded partial integrals (or sums).

Lemma 3.5

• (1) Let D be a compact subset of G, and J a finite subset of I such that $D - V \subset \cup _{i \in J}{(x_i + V)}$. Then $\sum \limits _{i \in J}{\alpha _i} = 1$ on D, i.e., $\{\alpha _i\}_{i\in J}$ is a partition of unity on D.

• (2) There exists a base ${\mathcal {V}}$ of neighborhoods of zero in G such that every $V \in {\mathcal {V}}$ is a covering neighborhood and $m(V + V) \le 2^{dim G}m(V)$.

Proof

• (1) This is clear (however, see [7] or [5] for details).

• (2) This fact is obvious for the groups $\mathbb {R}^n$ and $\mathbb {T}^n$: the role of ${\mathcal {V}}$ can be played by a certain family of cubes centered at zero. For an arbitrary group G, the claim is deduced form these elementary cases with the help of the structure theorem. See again the above references.

$\square$

Remark 3.6

The construction of a covering neighborhood (see the above hint) shows that the supports of the associated functions $\alpha _i$ form a covering of G whose multiplicity is at most $2^{\dim G}$. Again, in the cases of $\mathbb {R}^n$ and the tori, this is straightforward.

3.3 Inductive Construction

Here we present a principal ingredient of the proofs of the main results. As it has already been said, we use the ideas of [11]. However, some complications arise. Besides the fact that now we must ensure also the boundedness of Fourier sums (or partial integrals), a technical difference is that presently we shall need a certain double sequence $\{f_k^{(n)}\}_{n \in \mathbb {Z}_+,\,\, 0\le k \le n}$ of functions on G (instead of a single sequence in [11]). It will turn out eventually that the functions $f_n^{(n)}$ converge as $n\rightarrow \infty$, and this limit yields the desired function $\chi _b w$ after multiplication by a constant. We shall proceed by induction on the upper index n (if we view the required functions as the entries of a triangular matrix, this means that at each step we add an entire new row to this matrix).

So, let w and a be as in Theorem 2.3. Fix a small $\varepsilon >0$ and a strictly monotone increasing sequence $\{t_n\}_{n\ge 0}$, $t_n>1$, whose limit t does not exceed $1+\varepsilon$. Also, fix a sequence of positive numbers $\{\rho _n\}_{n\ge 0}$ with $\sum _{n\ge 0}\sqrt{\rho _n}<\varepsilon$. Below we gather certain properties of the functions $f_k^{(n)}$ that will be ensured by induction. The construction will imply some important supplements to these properties, which we do not indicate now.

• (i) For every $n\ge 0$, we have

  $$\begin{aligned} 0\le f_k^{(n)}\le t_n w,\quad k=0,\ldots ,n. \end{aligned}$$

• (ii) The spectra of all $f_k^{(n)}$ are compact and all these functions belong to $L^1(G)\cap C_0(G)$. By $C_0(G)$ we mean the set of all continuous functions on G tending to 0 at infinity; $C_0(G)=C(G)$ if G is compact. Consequently, all $f_k^{(n)}$ are square integrable.

• (iii) There exists a compact subset K of G such that all functions $f_k^{(n)}$ have compact spectra included in $K\cup R\cup S$, where (R, S) is the sufficient pair mentioned in Theorem 2.3.

• (iv) We have

  $$\begin{aligned} \Vert f_0^{(0)}-\chi _a w\Vert _1<\rho _0. \end{aligned}$$

  (3.2)

  Next,

  $$\begin{aligned} \Vert f_k^{(n)}-f_k^{(n-1)}\Vert _1<\rho _{n} \end{aligned}$$

  (3.3)

  for $n\ge 1$ and $k=0,\ldots ,n-1$ (this relates all functions in the $(n-1)$st row of the matrix mentioned above with the first n functions in the nth row).

Now, we start the construction with $n=0$. To ensure (3.2), we put $f_0^{(0)}=(\chi _a w)*\Phi _{U_0}$, where $U_0\subset \Gamma$ is chosen in such a way that $\Vert (\chi _a w)*\Phi _{U_0} - \chi _a w\Vert _1\le \rho _0$ (see Lemma 3.1) and $w*\Phi _{U_0}\le t_0 w$ (see Lemma 3.3). Since $\chi _a w\le w$, the inequalities in (i) for $n=0$ follow. Next, clearly, $f_0^{(0)}\in L^1(G)\cap C_0 (G)$. Moreover, ${\mathcal {F}}(f_0^{(0)})$ is supported on the compact set $K=U_0-U_0$; this will be the “K” mentioned in Theorem 2.3 and in (iii) above. Next, since ${\mathcal {F}}(f_0^{(0)})$ is integrable, the norm $\Vert f_0^{(0)}\Vert _u$ is finite, though no reasonable control of it is available.

Next, suppose that for some $n> 0$ the functions $f_0^{(n - 1)},\ldots ,f_{n-1}^{(n-1)}$ have already been constructed. We are going to construct the required collection with the upper index n. If $k\le n-1$, we take $f_k^{(n)}=\Phi _{U_n}*f_k^{(n-1)}$, where $U_n$ is chosen so as to ensure (3.3) and also the inequality $\Phi _{U_n}*w\le \frac{t_n}{t_{n-1}}w$ (see Lemmas 3.1 and 3.3). In the sequel, we will impose more restrictions on $U_{n}$, compatible with the above. Surely, the present choice of $U_n$ ensures the estimates in (i) for all k except for $k=n$.

The construction of $f_{n}^{(n)}$ is more tricky. First, we introduce the auxiliary function

$$\begin{aligned} g_n = f_{n-1}^{(n-1)}\left( 1 - \frac{f_{n-1}^{(n-1)}}{t_{n-1}w}\right) . \end{aligned}$$

(3.4)

Observe that the spectrum of $g_n$ is a compact subset of $\Gamma$ and $g_n$ is nonnegative by (i).

The subsequent arguments will involve certain objects (sets, functions, coefficients, parameters) depending in fact on n. But since now n is fixed, this dependence will not always be reflected in the notation. We shall approximate $g_n$ from below by a function suitable for further constructions.

For this, observe that $g_n$ is square-integrable, so the quantity $\int _G \min (g_n(x),\delta )^2dx$ tends to zero as $\delta \rightarrow +0$. Hence, we can find a (small) $\delta >0$ such that for the compact set $C=\{x\in G:g_n(x)\ge \delta \}$ we have $\Vert (g_n-\delta )\chi _C\Vert _2 > (9/10)\Vert g_n\Vert _2$. Denote $g= (g_n-\delta )\chi _C$, then g is continuous and compactly supported, hence uniformly continuous. Next, clearly, $g(x)+\delta /2< g_n(x)$ in a neighborhood W of C with compact closure. By using Lemma 3.5 (with C in the role of “D”), it is easy to realize that there is a (small) covering neighborhood V in G (among other things, we need that $C-V\subset W$) such that g is approximated uniformly and in $L^2(G)$ within any precision prescribed beforehand by a function of the form

$$\begin{aligned} h_n=\sum \limits _{i\in J} c_i\alpha _i,\quad c_i=g(x_i)\ge 0, \end{aligned}$$

(the $\alpha _j$ are given by (3.1); we also use the notation from Lemma 3.5). Clearly, $h_n\le g_n$ if V is sufficiently small, and all this can be arranged so as to ensure the inequality

$$\begin{aligned} \Vert h_n\Vert _2^2\ge \frac{1}{2} \Vert g_n\Vert _2^2. \end{aligned}$$

(3.5)

Next, by Remark 3.6, we have

$$\begin{aligned} h_n(x)^2\le 2^{\dim G}\sum \limits _{i\in J}(c_i\alpha _i (x))^2,\quad x\in G. \end{aligned}$$

Integrating, we arrive at

$$\begin{aligned} \Vert g_n\Vert _2^2\le 2^{\dim G+1}\sum \limits _{i\in J}(c_i)^2\Vert \alpha _i\Vert _2^2. \end{aligned}$$

(3.6)

Recall that the set J in the last sum is finite. For a detail in what follows, it is convenient to assume that J is a segment of positive integers. Now, we want to replace the functions $\alpha _i$, $i\in J$, by the functions $\beta _i=\Phi _{U_{n}}*\alpha _i$, $i\in J$, with compact spectrum. In addition to the restrictions on $U_{n}$ imposed above, we demand that

$$\begin{aligned} \Vert \beta _i\Vert _2^2=\Vert \Phi _{U_{n}}*\alpha _i\Vert _2^2\ge \frac{1}{2}\Vert \alpha _i\Vert _2^2. \end{aligned}$$

(3.7)

(See Lemma 3.1.)

Finally, we can define the function $f_{n}^{(n)}$: put

$$\begin{aligned} f_{n}^{(n)}=f_{n-1}^{(n)}+\widetilde{h_n},\,\text { where }\, \widetilde{h_n}=\mathop {\mathrm {Re}}\nolimits \sum \limits _{i\in J}{c_i\beta _i\gamma _i}. \end{aligned}$$

(3.8)

Here the $\gamma _i$ are certain characters of the group G (of course, they depend also on n, but we do not reflect this in the notation for short). These characters are introduced to eliminate the interference between the summands in (3.8) (and elsewhere), which will provide the desired estimate for $\Vert \cdot \Vert _u$. The choice of the $\gamma _i$ is described in the following lemma, whose proof is much similar to the proof of Lemma 1 in [6],³ and is based entirely on the definition of a sufficient pair and on the fact that the summation basis and the sufficient pair in question are coordinated. We do not reproduce the arguments here. Note that the lemma is quite transparent for the groups $\mathbb {R}^n$ and $\mathbb {T}^n$ in the role of G. Surely, in these cases $\Gamma$ has no elements of order two, so item (2) below can be shortened accordingly. In any case, the $\gamma _i$ are chosen one after another as $i\in J$ grows (we remind the reader that we have assumed that J is a segment of integers). Recall also that the sufficient pair in question is denoted by (R, S).

Lemma 3.7

The characters $\gamma _i$ can be chosen in such a way that

• (1) the support of the Fourier transform of   $\mathop {\mathrm {Re}}\nolimits (\beta _i\gamma _i)=\beta _i(\gamma _i+\overline{\gamma _i})/2$ lies in $S \cup R$ and intersects neither the spectra of all functions $f_k^{(j)}$ constructed previously (i.e., with $j\le n$ and $k\le n-1$), nor the spectra of all functions $\mathop {\mathrm {Re}}\nolimits \beta _s\gamma _s$ for $1 \le s < i$, nor the “union⁴” of all sets in the summation basis ${\mathcal {B}}$ that split any of these spectra;

• (2) either $2\gamma _i = 0$ (i.e., $\gamma _i(\cdot )^2 = 1$) or the $\pm 2\gamma _i$ do not lie in the spectrum of $\beta _i^2$.

Now, we verify the inequality in (i) for the function $f_{n}^{(n)}$. By the definition of the $\beta _i$, we have

$$\begin{aligned} \left| \mathop {\mathrm {Re}}\nolimits \sum \limits _{i\in J}{c_i\beta _i\gamma _i}\right| \le \sum \limits _{i\in J}c_i\alpha _i*\Phi _{U_{n}}=h_n*\Phi _{U_{n}}\le g_n*\Phi _{U_{n}}. \end{aligned}$$

So, by (3.8) and the definition of $f_{n-1}^{(n)}$, we have

$$\begin{aligned} \Phi _{U_{n}}*(f_{n-1}^{(n-1)}-g_n)\le f_{n}^{(n)}\le \Phi _{U_{n}}*(f_{n-1}^{(n-1)}+g_n). \end{aligned}$$

Finally, we use the inductive hypothesis in (i) and the definition (3.4) to conclude that $f_{n-1}^{(n-1)}-g_n\ge f_{n-1}^{(n-1)}-f_{n-1}^{(n-1)}\ge 0$ and

$$\begin{aligned} f_{n-1}^{(n-1)}+g_n\le f_{n-1}^{(n-1)}+ t_{n-1} w\left( 1 - \frac{f_{n-1}^{(n-1)}}{t_{n-1}w}\right) =t_{n-1} w. \end{aligned}$$

The desired result follows because $\Phi _{U_{n}}*(t_{n-1} w)\le t_{n} w$ by the choice of $U_{n}$.

This finishes the induction.

Remark 3.8

It can easily be arranged that $U_0\subset U_1\subset U_2\subset \ldots$. Then the spectrum of $f_k^{(n)}$ does not change when n varies with k fixed (i.e., within each column of the matrix). In the sequel, we will assume that the $U_n$ have this property.

3.4 Proof of All Claims Except the Uniform Boundedness of Partial Fourier Integrals

In this subsection we show that the sequence $\{t_n^{-1}f_n^{(n)}\}_{n \in \mathbb {Z}_+}$ converges to a function of the form $\chi _b w$, and we verify all metric and spectral conditions for the limit function, except the finiteness of the norm $\Vert \cdot \Vert _u$ for it.

3.4.1 Convergence

First, we observe that for every k the limit $F_k = \lim _{j \ge k, j \rightarrow \infty }{f_k^{(j)}}$ exists in $L^2(G)$. (These are “the limits along all columns”.) Indeed, by (3.3) and (i), we have $\Vert f_k^{(j)}-f_k^{(j - 1)}\Vert _2<c\sqrt{\rho _{j}}$, and the quantities on the right were chosen to constitute a convergent series.

Next, the functions $\{F_k\}$ form partial sums of an orthogonal series. Indeed, it can easily be seen by induction that, for each n, the spectra of the functions $f_0^{(n)}, f_1^{(n)}-f_0^{(n)},\ldots , f_n^{(n)}-f_{n-1}^{(n)}$ are mutually disjoint (see Lemma 3.7), hence, these functions are mutually orthogonal, and the claim follows by the limit passage as $n\rightarrow \infty$.

It is also easily seen by induction that

$$\begin{aligned} \int \limits _G f_k^{(n)}(x)dx = \int \limits _a w(x)dx \end{aligned}$$

(3.9)

for all $n\ge 0$ and all $k=0,\ldots ,n$. Indeed, this is clear for $k=n=0$ and then for $k=0,\,n=1$, because these two functions are obtained from $\chi _a w$ by convolution with positive functions of unit $L^1$-norm. Hence, the spectrum of $f_0^{(1)}$ includes a (neighborhood of) zero. So, $f_1^{(1)}$ is obtained from $f_0^{(1)}$ by adding a function with zero integral (see again Lemma 3.7). This proves (3.9) for $n=1$. Then we pass to $n=2$ in the same way, etc.

Now, we see that

$$\begin{aligned} \int \limits _G (F_k)^2 dx=\lim _{n\rightarrow \infty }\int \limits _G (f_k^{(n)})^2 dx\le c\int \limits _a w dx, \end{aligned}$$

hence the functions $F_k$ converge to some function F in $L^2(G)$ as $k\rightarrow \infty$. Since

$$\begin{aligned} \Vert F_k - f_k^{(k)}\Vert _2 \le c \sum \limits _{i> k}\sqrt{\rho _i}\,\text { and }\,\Vert F_{k-1} - f_{k-1}^{(k)}\Vert _2 \le c \sum \limits _{i > k}\sqrt{\rho _i}, \end{aligned}$$

we see that the sequences $\{f_k^{(k)}\}$ and $\{f_{k-1}^{(k)}\}$ also tend to F in $L^2(G)$ as $k\rightarrow \infty$. Hence, $\Vert \widetilde{h_k}\Vert _2=\Vert f_k^{(k)}-f_{k-1}^{(k)}\Vert _2\rightarrow 0$ as $k\rightarrow \infty$.

But the terms $\mathop {\mathrm {Re}}\nolimits ( c_i\beta _i\gamma _i)$ in the formula for $\widetilde{h_k}$ (see (3.8)) are mutually orthogonal by construction (see Lemma 3.7), hence

$$\begin{aligned} \Vert \widetilde{h_k}\Vert _2^2= \sum \limits _{i\in J}\Vert \mathop {\mathrm {Re}}\nolimits (c_i\beta _i\gamma _i)\Vert _2^2. \end{aligned}$$

Next, we observe that

$$\begin{aligned} \Vert \mathop {\mathrm {Re}}\nolimits (\beta _i\gamma _i)\Vert _2^2 = \frac{1}{4}\int \limits _G{(\overline{\gamma _i} + \gamma _i)^2 \beta _i^2} = \left\{ \begin{matrix} \int \limits _G{\beta _i^2}, \, 2\gamma _i = 0,\\ \frac{1}{2}\int \limits _G{\beta _i^2}, \, 2\gamma _i \ne 0 \end{matrix}\right. \end{aligned}$$

(in the second line, we have used the fact that the characters $\pm 2\gamma _i$ are not in the spectrum of $\beta _i^2$ if $\gamma _i$ is not of order 2, see Lemma 3.7). Combining (3.7), (3.6), and (3.5), we see that $g_k\rightarrow 0$ in $L^2(G)$. Since some subsequence of $\{f_k^{(k)}\}$ must converge to F a.e., looking at formula (3.4) for $g_k$ we realize that at every $x\in G$, either $F(x)=0$, or $F(x)=tw(x)$ (recall that $t_n\rightarrow t$). Hence, $F=t\chi _b w$ for some measurable set b. We shall show that this b is the required set.

Observe, by the way, that

$$\begin{aligned} \int \limits _b tw(x)dx=\int \limits _a w(x)dx. \end{aligned}$$

(3.10)

Indeed, inequality (3.3) implies that $f_k^{(n)}\rightarrow F_k$ also in $L^1(G)$ as $n\rightarrow \infty$, hence $\int _G F_k(x) dx=\int _a w(x) dx$ for all k by (3.9). Since also all $F_k$ are nonnegative, F is integrable and $\int _G F(x) dx\le \int _a w(x) dx$ by the Fatou lemma. In fact, equality occurs here. Indeed, the construction and the Plancherel theorem show that ${\mathcal {F}}(F)$ and ${\mathcal {F}}(F_0)$ coincide a.e. in some neighborhood of zero in $\Gamma$. Since both functions are continuous, they coincide at 0, whence the claim.

Since $t>1$, we see that $\int _b w\le \int _a w$.

3.4.2 Correction

Here we prove that a and b differ only slightly, as claimed in Theorem 2.3. By the above discussion, the functions $F_0$ and $F-F_0$ are orthogonal, hence $\int _G F_0F dx=\int _G (F_0)^2 dx$. Since $f_0^{(0)}$ is at the distance of at most (${{\,\mathrm{const}\,}}\varepsilon$) from $F_0$ in $L^2(G)$, we see that

$$\begin{aligned} \int \limits _G f_0^{(0)}F =\int \limits _G (f_0^{(0)})^2 +O(\varepsilon ). \end{aligned}$$

Now,

$$\begin{aligned} \int \limits _{a\cap b}w^2=\frac{1}{t} \int \limits _G (\chi _aw)F= \frac{1}{t}\left( \int \limits _G(\chi _a w-f_0^{(0)})F+\int \limits _G (f_0^{(0)})^2 +O(\varepsilon )\right) . \end{aligned}$$

Clearly, the first integral in parentheses is $O(\varepsilon )$ by (3.2). For the second integral, we write

$$\begin{aligned} \int \limits _G (f_0^{(0)})^2\ge \int \limits _G(\chi _a w)^2-\int \limits _G| (f_0^{(0)})^2 -(\chi _a w)^2|\ge \int \limits _a w^2-A_0\int \limits _G| f_0^{(0)} -\chi _a w|. \end{aligned}$$

The subtrahend in the last expression is again $O(\varepsilon )$. Collecting the estimates, we arrive at $\int \limits _{a\cap b}w^2\ge \frac{1}{t} \int \limits _a w^2 -A_1\varepsilon \ge \int \limits _a w^2-A_2\varepsilon$ if t has been chosen sufficiently close to 1. Since $\int _b w\le \int _a w$, we arrive at $\int \limits _{a\Delta b}w^2\le A_3\varepsilon$, as required.

3.4.3 About Remark 2.4

We have $1*\Phi _U=1$ for every U. Hence, the numbers $t_n$ are not required in the case where w is identically equal to 1: the above arguments work with $t_n=1$ for all n (accordingly, $t=1$). Now the claim of the remark follows from (3.10).

3.4.4 Spectrum

Needless to say that condition (2) in Theorem 2.3 is clear from the construction.

3.5 Uniform Boundedness of Partial Fourier Integrals

We remind the reader that by Remark 3.8, the spectrum of $f_k^{(n)}$ does not change when $n\ge k$ varies with k fixed.

Now, take a set B in the summation basis ${\mathcal {B}}$ in question and find the minimal k such that B does not include the spectrum of $f_k^{(k)}$ (here and in the next several lines, all inclusions are understood up to a set of zero measure in $\Gamma$). We shall provide some uniform bound for $|P_B f_k^{(k)}|$, and this will suffice. Indeed, suppose we have ensured an upper bound D for this function. Since for every $j>k$ the function $f_k^{(j)}$ is obtained form $f_k^{(k)}$ by convolution with a nonnegative function with unit integral and since convolution commutes with $P_B$, we see that $|P_B f_k^{(j)}|\le D$. However, by construction (see Lemma 3.7), we have $P_B f_j^{(j)}=P_B f_k^{(k)}$, and we see that $|P_B f_j^{(j)}|\le D$ for all $j\ge k$, hence also $|P_B F|\le D$ in the limit.

The nature of the required uniform estimate depends heavily on whether $k=0$ or $k>0$. If $k=0$ (i.e., B splits the support of $f_0^{(0)}$), then $|P_B f_0^{(0)}|\le \Vert {\mathcal {F}}f_0^{(0)}\Vert _1$. The last quantity does not depend on B, as required, but otherwise it is out of our control, we know only that it is finite because ${\mathcal {F}}f_0^{(0)}$ is bounded and compactly supported. But if $k>0$ (which is true for sure if $B\supset K$, as in Remark 2.5), we see that B includes the support of $f_{k-1}^{(k)}$ by the minimality of k. Hence, $P_B f_k^{(k)}=f_{k-1}^{(k)}+P_B \widetilde{h_k}$, see (3.8). Since (it can be arranged that) all functions $f_j^{(n)}$ are uniformly bounded, say, by 2, it suffices to estimate the second summand on the right in the last formula.

Recall that in the expression given for $\widetilde{h_k}$ in (3.8) we assumed that J is a segment of integers. The way in which we used the order on J in the construction (see again Lemma 3.7) shows that there is a unique $l\in J$ with

$$\begin{aligned} P_B\widetilde{h_k}=\sum \limits _{i<l}\mathop {\mathrm {Re}}\nolimits (c_i\beta _i\gamma _i)+P_B(\mathop {\mathrm {Re}}\nolimits c_l\beta _l\gamma _l). \end{aligned}$$

Now, we remind the reader that $\beta _i=\Phi _{U_{n}}*\alpha _i$, $i\in J$. Hence, recalling the properties of the functions $\alpha _i$ (see (3.1) and Lemma 3.5) and the fact that $|c_i|=|g(x_i)|\le 2$, we obtain

$$\begin{aligned} \left| \sum \limits _{i<l}\mathop {\mathrm {Re}}\nolimits c_i\beta _i\gamma _i\right| \le 2\sum \limits _{i<l}\Phi _{U_{n}}*\alpha _i= 2\Phi _{U_{n}}*\left( \sum \limits _{i<l}\alpha _i\right) \le 2 \end{aligned}$$

and

$$\begin{aligned} |P_B(\mathop {\mathrm {Re}}\nolimits c_l\beta _l\gamma _l)|\le 2\Vert {\mathcal {F}}(\frac{\gamma _l+\bar{\gamma _l}}{2} \beta _l)\Vert _1\le \Vert {\mathcal {F}}\alpha _l\Vert _1\le 1. \end{aligned}$$

Collecting the estimates, we see that we have proved Theorem 2.3 together with Remark 2.5.

3.6 Sharp Inequalities

Here we prove Theorem 2.6. We saw in the preceding subsection that only $f_0^{(0)}$ presents an obstruction to what we are going to do, and in order to prove the desired claim we must ensure an appropriate control of the partial Fourier integrals (or sums) for this function. In some specific but important cases, this can be done indeed, with the help of a theorem proved in [9] and formulated below.

Let X be a Banach space whose elements are locally integrable functions on a measure space $(S,\mu )$. In a natural way, the space $L^1_{loc}(\mu )$ is locally convex. Next, we denote by $L^{\infty }_0(\mu )$ the space of all essentially bounded functions supported on a set of finite measure. If $g\in L^{\infty }_0(\mu )$, the formula $\Psi _g(u)=\int \limits _S gu d\mu$ defines a linear functional on $L^1_{loc}(\mu )$, hence on every linear subspace of this space. Suppose that the following two conditions are satisfied.

• A1. The natural embedding $X \hookrightarrow L^1_{loc}(\mu )$ is continuous and the unit ball of X is weakly compact in $L^1_{loc}(\mu )$.

• A2. For every $g \in L^\infty _0(\mu )$ we have the weak type estimate

  $$\begin{aligned} m(\{|g| > t\}) \le c\frac{ \Vert \Phi _g\Vert _{X^*}}{t}, \end{aligned}$$

  where c depends only on X.

Theorem 3.9

For every $f\in L^\infty (\mu )\cap L^1(\mu )$ with $\Vert f\Vert _\infty \le 1$ and every $\varepsilon >0$, there exists a measurable function $\varphi$ with $0 \le \varphi \le 1$ such that $\varphi f\in X$, $\mu (\{\varphi \ne 1\}) \le \varepsilon$, and $\Vert \varphi f\Vert _X \le {{\,\mathrm{const}\,}}\log (2+\varepsilon ^{-1}\Vert f\Vert _1)$. The constant in the last inequality depends only on c in A2.

Now, as usual, let G be a nondiscrete locally compact Abelian group. For a subset e of G with $0<|e|<\infty$, we introduce the space

$$\begin{aligned} u(G, e; {\mathcal {B}})= \{f:f\in L^2(G),\, {{\,\mathrm{supp}\,}}f\subset e, \text{ and } \Vert f\Vert _u = \sup _{B \in {\mathcal {B}}}{\Vert P_B f\Vert _\infty }<\infty \}. \end{aligned}$$

It should be noted that all functions f from this space lie in fact in $L^{\infty }(G)$ with $\Vert f\Vert _{\infty }\le \Vert f\Vert _u$. If G itself is compact, we need only the case where $e=G$ and write $u(G, {\mathcal {B}})$ in place of $u(G, G; {\mathcal {B}})$. For some discussion around the space $u(G, e; {\mathcal {B}})$ and the norm $\Vert \cdot \Vert _u$, see Sect. 1 of [7].

Now, we explain which spaces will play the role of X in Theorem 3.9. Let ${\mathcal {B}}$ be the summation basis for the unit circle or the real line consisting of all symmetric intervals centered at zero in the dual group $\mathbb {Z}$ or $\mathbb {R}$. For short, we denote the corresponding spaces $u(\mathbb {T},{\mathcal {B}})$ and $u(\mathbb {R}, e;{\mathcal {B}})$ (where $e\subset \mathbb {R}$ is of finite positive measure) by $u(\mathbb {T})$ and $u(\mathbb {R}, e)$, or even simply by u. Next, we take for $\mu$ the Lebesgue measure on the circle or on the set e. Then the two above spaces do satisfy Axiom A2 (A1 being a triviality), but this is quite involved. Indeed, eventually this is based on the Carleson almost everywhere convergence theorem for classical Fourier expansions. See [13] and [12] for the proof of A2 in these cases, and also Sects. 2.5 and 2.6 in [9] for some explanations. Hint: in the case of $u(\mathbb {R},e)$, it is convenient to consider first the situation where e is open, and then pass to its subsets of positive measure by using the arguments at the beginning of Sect. 3 in [9]. It is important to note that the constant in A2 does not depend on the set e in the case of the real line.

Thus, given a weight w (we still assume that $w\le 1$), a number $\varepsilon >0$, and a measurable set a of finite measure on the line or on the circle, we start as at the beginning of Sect. 3.3, but first we modify $f = \chi _a w$ in accordance with Theorem 3.9 (with $u(\mathbb {T})$ or $u(\mathbb {R}, a)$ in the role of X) The resulting function ${\widetilde{f}}=f\varphi$ satisfies $0\le {\widetilde{f}}\le w$ and

$$\begin{aligned} \Vert {\widetilde{f}}\Vert _u \le {{\,\mathrm{const}\,}}\log \left( 2+\frac{\int _aw}{\varepsilon }\right) , \end{aligned}$$

(3.11)

where the constant is independent on a, and also $\Vert \chi _a w-{\widetilde{f}}\Vert _1\le \varepsilon$. Next, we put $f_0^{(0)} = {\widetilde{f}} * \varphi _{U_0}$ as before, so as to ensure (3.2) but with $\varepsilon +\rho _0$ instead of $\varepsilon$ on the right. This new function $f_0^{(0)}$ will satisfy (3.11) because ${\widetilde{f}}$ does. Then we proceed as previously. All $O(\varepsilon )$‘s in Sect. 3.4.2 will remain $O(\varepsilon )$. We do not enter in further details.

Remark 3.10

By [10], the conclusion of Theorem 2.6 is also true for certain zero-dimensional compact groups, specifically, for those linked with bounded Vilenkin systems (in particular, the dyadic group with the Walsh system in the usual ordering fits). Again, the verification of Axiom A2 for the corresponding space of functions with uniformly bounded Fourier sums is based eventually on an analog of the Carleson almost everywhere convergence theorem for Vilenkin systems.

Finally, we prove the announced correction theorem about essentially bounded functions with support of finite measure, as opposed to the mere indicator functions. We restate it and sketch the proof for the group $\mathbb {R}$, but it will be clear that similar arguments apply to $\mathbb {T}$ and to the zero-dimensional groups mentioned in the last remark. As above, on $\mathbb {R}$ we consider the summation basis of symmetric intervals, and also we are given a sufficient pair (R, S) of subsets of $\mathbb {R}$.

Theorem 3.11

Let $\varepsilon > 0$. Given a function $h\in L^{\infty }(\mathbb {R})$ supported on a set of finite measure and with $\Vert h\Vert _{\infty }\le 1$, there is a function $f\in u(\mathbb {R})$ such that $\Vert f\Vert _\infty \le 20$, $|\{h\ne f\}|\le \varepsilon$, and f satisfies an estimate like (3.11). Furthermore, the spectrum of f is included in $K\cup R\cup S$, where K is a compact set depending on h.

Proof

Here h is, in general, complex-valued, but the claim can be reduced to the case of a positive h if we ensure a smaller constant (say, 3) in place of 20. So, let $0\le h\le 1$, and let A be the support of h. Find a compact set $a\subset A$ with $|A{\setminus } a|\le \varepsilon$ such that h is continuous on a, and then extend $h|_a$ up to a nonnegative uniformly continuous function v on $\mathbb {R}$ with $v\le 1$. Finally, consider the weights (“rails”) $w_1=v+1$ and $w_2=1$ on $\mathbb {R}$.⁵

We apply Theorem 2.6 to the set a consecutively with the weights $w_1$ and $w_2$, obtaining two sets $b_1$ and $b_2$. The function $\chi _{b_1}w_1-\chi _{b_2}w_2$ does the job. $\square$

Note that we cannot eliminate the uncontrollable set K in this statement because of the sharp estimate (3.11). (Should this be possible, all $L^1$-functions would have spectrum in $R\cup S$.) To the contrary, in the results of [1, 2], and [7], the spectrum of a corrected function always lies in $R\cup S$, but, naturally, the control of the size of this function and its partial Fourier integrals is much weaker.