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On uniqueness for series in the general Franklin system G. G. Gevorkyan Yerevan State University, Yerevan, Republic of Armenia
Russian version:
Matematicheskii Sbornik, 2024, Volume 215, Number 3, DOI: https://doi.org/10.4213/sm9933 
Bibliographic databases:
   § 1. IntroductionUniqueness problems for series in some classical orthogonal systems play an important role in investigations of series in these systems. A central place in the study of uniqueness for trigonometric series is taken by Cantor’s theorem (see [1], and also [2], Ch. 1, § 70), which asserts that if a trigonometric series converges to zero everywhere, then each coefficient of this series is zero. The uniqueness of trigonometric series has been studied extensively. The studies of uniqueness for series in the Haar system began with [3]–[5], where a Cantor-type theorem for Haar series was established. Analogues of de la Vallée Poussin’s theorem for series in the Haar and Walsh systems whose coefficients satisfy certain necessary conditions were derived in [6]. Investigations of uniqueness problems for series in the trigonometric system, as also in the Haar and Walsh systems and their generalizations, are still going on (see, for example, [7]–[14]). The studies of uniqueness problems for series in the Franklin system began recently. The definition of the orthonormal Franklin system $\{f_n(x)\}_{n=1}^{\infty}$ (and also of the general Franklin system) is given below. The following theorem (see [16]) was announced in [15]. Theorem 1. If a series in the Franklin system converges to zero everywhere, then all coefficients of this series are zero. Further, the following theorem was proved in [17]. Theorem 2. If a series in the Franklin system converges everywhere to a finite integrable function $f$, then it is the Fourier–Franklin series of this function. In [18] it was shown that no singleton is a uniqueness set for Franklin series, that is, for each $x_0\in[0,1]$ there exists a nontrivial series in the Franklin system that converges to zero everywhere except at $x_0$. The coefficients of such a series satisfy In this regard, we mention the following theorem (see [17], Theorem 2.5). Theorem 3. Let the Franklin series $\sum_{n=0}^{\infty}a_nf_n(x)$ with coefficients converge in measure to an integrable function $f$, and let everywhere, except possibly at the points in some countable set. Then it is the Fourier–Franklin series of $f$. Wronicz (see [19]) constructed a nontrivial Franklin series satisfying
$$
\begin{equation}
\lim_{k\to\infty}\sum_{n=0}^{2^k}a_nf_n(x)=0 \quad\text{for each}\ x\in[0,1].
\end{equation}
\tag{1.2}
$$
The following problem was stated in [17]: does a Franklin series have all coefficients $a_n$ equal to zero under conditions (1.1) and (1.2)? A positive answer to this problem was given by Wronicz [20]. The following theorem was proved in [21] by methods different from those used by Wronicz [20]. Theorem 5. Let and let the partial sums $S_{n_i}(x):=\sum_{k=0}^{n_i}a_kf_k(x)$ of a series $\sum_{k=0}^{\infty}a_kf_k(x)$ with coefficients (1.1) converge in measure to a bounded function $f$, and where $B$ is a countable set. Then this series is a Fourier–Franklin series of $f$. In [22] Theorem 5 was extended to some class of general Franklin systems. The statement of this theorem, as well as the main results of the present paper, is presented after the definition of a general Franklin system. We adopt the following notation:
§ 2. Definition of a general Franklin system, main results and some preliminary lemmasTo formulate the results of this paper we define a Franklin system (following [23]). Definition 1. A sequence of points $\mathcal{T}=\{t_n\colon n\geqslant 0\}$ is said to be admissible if $t_0=0$, $t_1=1$, $t_n\in(0,1)$ for each $n\geqslant2$, $\mathcal{T}$ is everywhere dense in $[0, 1]$, and each point $t\in(0,1)$ occurs in $\mathcal{T}$ at most twice. Let $\mathcal{T}=\{t_n\colon n\geqslant 0\}$ be an admissible sequence. For $n\geqslant 1$ we set $\mathcal{T}_n=\{t_i\colon 0\leqslant i\leqslant n\}$. Let $\pi_n$ be obtained from $\mathcal{T}_n$ by the nondecreasing rearrangement: $\pi_n=\{\tau_{n,i}\colon \tau_{n,i}\leqslant\tau_{n,i+1},\, 0\leqslant i\leqslant n\}$, $\pi_n=\mathcal{T}_n$ as sets. We denote by $\mathbf{S}_n$ the space of functions on $[0, 1]$ which are left-continuous, linear on $(\tau_{n,i}, \tau_{n,i+1})$ and continuous at $\tau_{n,i}$ if $\tau_{n,i-1}<\tau_{n,i}<\tau_{n,i+1}$ for each $i=0, 1, \dots, n$. It is clear that $\dim \mathbf{S}_n=n+1$ and $\mathbf{S}_{n-1}\subset \mathbf{S}_n$. Therefore, there exists a unique (up to sign) function $f_n\in \mathbf{S}_n$, $\| f_n\|_2=1$, that is orthogonal to $\mathbf{S}_{n-1}$. This function is called the $n$th Franklin function corresponding to the partition (sequence) $\mathcal{T}$. A partition $\mathcal{T}$ is called simple if each point $t\in(0,1)$ occurs at most once in $\mathcal{T}$. In what follows we consider only simple partitions. Definition 2. A general Franklin system $\{f_n \colon n\geqslant 0\}$ corresponding to a partition $\mathcal{T}$ is defined by and for $n\geqslant 2$, $f_n$ is the $n$th Franklin function corresponding to $\mathcal{T}$. Note that for
$$
\begin{equation*}
t_n=\frac{2m-1}{2^{k+1}},\quad \text{where } n=2^k+m,\quad k=0,1,2,\dots\quad\text{and}\quad m=1,2,\dots, 2^k,
\end{equation*}
\notag
$$
we obtain the classical Franklin system (for an equivalent definition, see [24]). In [25] it was shown that for each sequence of knots $\mathcal{T}$ the corresponding general Franklin system is a basis of $L^p[0, 1]$ for $1\leqslant p<\infty$; in addition if all knots in $\mathcal{T}$ are simple, then the corresponding general Franklin system is a basis of $C[0, 1]$. In [26] Gevorkyan and Kamont proved that any general Franklin system is an unconditional basis of $L^p[0,1]$ of $1<p<\infty$. We introduce some definitions following [23]. Definition 3. An admissible sequence $\mathcal{T}$ is called a quasidyadic partition of $ [0, 1]$ if $\tau_{j+1,2k}=\tau_{j,k}$ for all $j$ and $ k$ such that $0\leqslant k\leqslant 2^j$, that is, $\pi_{2^{j+1}}$ is obtained from $\pi_{2^{j}}$ by augmenting it by one point in each interval $(\tau_{j,k}, \tau_{j,k+1})$ for all $1\leqslant k\leqslant 2^j$. Definition 4. An admissible sequence $\mathcal{T}$ is called a strongly regular sequence with parameter $\gamma$ if $\gamma^{-1}\leqslant\lambda_{n,i+1}/\lambda_{n,i}\leqslant\gamma$ for all $n\geqslant2$, $i=1, 2, \dots, n$. Here and in what follows $\lambda_{n,i}=\tau_{n,i}-\tau_{n,i-1}$. In the present paper we consider only strongly regular sequences $\mathcal T$ with fixed parameter $\gamma$. It is also assumed that $\mathcal T$ satisfies the following condition. Given a fixed $x \in (0, 1)$ and a natural integer $n$, we set $\Delta_{n,x}{\kern1pt}{:=}\,[\tau_{n,i}, \tau_{n,i+1}]$ if ${x\in\Delta_{n,x}}$. For fixed $x$ let $\{m_{j}\}$ be such that $[0,1]=\Delta_{m_1,x}\supset\Delta_{m_2,x}\supset \dots\supset\Delta_{m_{j},x}\supset \dotsb$ and $\Delta_{m_j,x}\neq\Delta_{m_{j+1},x}$ for $j=1,2,\dots $ . Next, we assume that the partition $\mathcal T$ satisfies
$$
\begin{equation}
\kappa_1:=\sup_{x\in[0,1]}\sup_j\operatorname{card}\{p\colon \ [\tau_{m_{j+1},p}, \tau_{{m_{j+1}},p+1}]\subset\Delta_{m_j,x}\}<\infty.
\end{equation}
\tag{2.1}
$$
For a sequence $n_i$ we assume that
$$
\begin{equation}
\kappa_2:=\sup_i\sup_{j,p}\biggl\{\frac{\lambda_{n_i,j}}{\lambda_{n_{i+1},p}}\colon [\tau_{n_{i+1},p-1},\tau_{n_{i+1},p}]\subset[\tau_{n_i,j-1},\tau_{n_i,j}]\biggr\}<\infty.
\end{equation}
\tag{2.2}
$$
A strongly regular sequence $\mathcal T$ and a sequence $n_i$ satisfy conditions (2.1) and (2.2), for example, if $\mathcal T$ is quasidyadic and (1.3) is met. Note that for the classical Franklin system $\kappa_1=2$, and if (1.3) is met, then (2.2) holds. In what follows we consider only strongly regular partitions $\mathcal T$ with parameter $\gamma$. We also consider a general Franklin system $\{f_n(x)\}_{n=0}^{\infty}$ corresponding to a fixed partition $\mathcal T$. In addition, we assume that this general Franklin system satisfies (2.1) and the sequence $n_i$ obeys (2.2). We also set
$$
\begin{equation*}
\begin{gathered} \, t_n^-:=\max\{\tau_{n,i}\colon \tau_{n,i}<t_n\}, \qquad t_n^+:=\min\{\tau_{n,i}\colon t_n<\tau_{n,i}\}, \\ \Delta_n=[t_n^-, t_n^+]\quad\text{and} \quad \delta_n= \operatorname{mes}(\Delta_{n}). \end{gathered}
\end{equation*}
\notag
$$
Below we prove the following result. Theorem 6. Let the coefficients of the series satisfy and the partial sums converge in measure to an integrable function $f$ and satisfy where $B$ is some countable set. Then this series is the Fourier–Franklin series of $f$. The next result is a direct consequence of this theorem. Theorem 7. If the partial sums (2.5) of a series $\sum_{n=0}^{\infty}a_nf_n(x)$ with coefficients (2.4) converge to a finite integrable function $f$ everywhere, except possibly on some countable set, then this series is the Fourier–Franklin series of $f$. Remark 1. For any simple partition $\mathcal T$ and a general Franklin system corresponding to the partition $\mathcal T$, Remark 2. No analogue of Theorem 1 (nor a counterexample to this result) is known for any general Franklin system. For the classical Franklin system these theorems read as follows (in Theorems 8 and 9, $\{f_n\}$ is the classical Franklin system). Theorem 8. Let Theorem 9. Let (2.7) hold, and let where $B$ is some countable set and $f$ is a finite integrable function. Then the $a_n$ are the Fourier–Franklin coefficients of $f$. Let $\delta_{j k}$ be the Kronecker delta, that is, $\delta_{j k}=1$ for $j=k$, and $\delta_{j k}=0$ for $j \neq k$. For $n \geqslant 2$ the functions $\{N_{n, j}(t)\}_{j=0}^{n}$ are defined by
$$
\begin{equation*}
N_{n, j}(x)= \begin{cases} \delta_{j k} \quad\text{for } x=\tau_{n, k}, \quad k=0, \dots, n, \\ \text {is linear on }[\tau_{n, k-1},\tau_{n, k}], \quad k=1, \dots, n . \end{cases}
\end{equation*}
\notag
$$
Each function in $\{N_{n, j}(t)\}_{j=0}^{n}$ has the unit norm in $C[0,1]$, and $N_{n, j}(s_{n, k})=\delta_{j k}$ implies that the system $\{N_{n, j}(t)\}_{i=0}^{n}$ forms a basis for $\mathbf{S}_{n}$. We set $\tau_{n,-1}=0$, $\tau_{n,n+1}=1$, and define
$$
\begin{equation*}
M_{n, j}(t):=\frac{2}{\tau_{n, j+1}-\tau_{n, j-1}} N_{n, j}(t).
\end{equation*}
\notag
$$
This gives us a different basis for $\mathbf{S}_{n}$; this basis is normalized in $L[0, 1]$. Let $\Delta_{n, j}:=[\tau_{n, j-1}, \tau_{n, j+1}]$. It is clear that $\Delta_{n, j}=\operatorname{supp} N_{n, j}=\operatorname{supp} M_{n, j}$. Since $n_{i}$ is fixed and satisfies (2.2), we simplify the notation as follows:
$$
\begin{equation*}
\begin{gathered} \, \Delta_{j}^{i}:=\Delta_{n_{i}, j}, \qquad \tau_{j}^{i}:=\tau_{n_{i}, j}, \\ N_{j}^{i}(x):=N_{n_{i}, j}(x)\quad\text{and} \quad M_{j}^{i}(x):=M_{n_{i}, j}(x). \end{gathered}
\end{equation*}
\notag
$$
The inner product of a series (2.3) and a function $g\in \mathbf S_n$, which was introduced in [16], proved to be useful in investigations of uniqueness for series in the Franklin system. By the definition of a general Franklin system, for $m>n$ we have $f_m\,{\bot}\, \mathbf S_{m-1}$ and $\mathbf S_n\subset \mathbf S_{m-1}$. Therefore,
$$
\begin{equation*}
\int_0^1f_m(t)g(t)\,dt=0 \quad\text{for}\ m>n, \quad g\in \mathbf S_m.
\end{equation*}
\notag
$$
Hence, for each series (2.3) and any function $g\in \mathbf S_n$ their inner product can be defined by
$$
\begin{equation*}
(\mathcal S, g):=\sum_{m=0}^{\infty}a_m\int_0^1f_m(t)g(t)\,dt= \sum_{m=0}^{n}a_m\int_0^1f_m(t)g(t)\,dt=\int_0^1S_n(t)g(t)\,dt;
\end{equation*}
\notag
$$
here and in what follows $\mathcal S$ denotes the formal series (2.3). It is easily seen that for all $\alpha, \beta \in \mathbb{R}$ and $g_1, g_2\in \mathbf S_n$,
$$
\begin{equation*}
(\mathcal S, \alpha g_1+\beta g_2)= \alpha(\mathcal S, g_1)+\beta(\mathcal S, g_2).
\end{equation*}
\notag
$$
Lemma 1. For any $M^{i_0}_{j_0}$ and $i>i_0$, where $0\leqslant \alpha^{(i)}_j\leqslant {\lambda_{n_i,j}}/{\lambda_{n_{i_0},j_0}}$ and $\sum_j\alpha^{(i)}_j=1$. Indeed, to prove this result it suffices to note that
$$
\begin{equation*}
M^{i_0}_{j_0}(t)=\sum_{j\colon \operatorname{supp}M^{i}_j\subset \operatorname{supp}M^{i_0}_{j_0}}\alpha^{(i)}_jM^{i}_j(t), \quad\text{where} \ \alpha^{(i)}_j= \frac{M^{i_0}_{j_0}(s^{i}_j)}{M^{i}_j(s^{i}_j)},
\end{equation*}
\notag
$$
and to use the fact that the integral of each of the functions $M^{i_0}_{j_0}$ and $M^{i}_j$ is 1. This is Lemma 3 in [27]. The next three results are Lemmas 3.2, 3.1 and 3.5 in [22], respectively. Lemma 3. If the coefficients $a_{n}$ of a series in a general Franklin system satisfy (2.4), then for each $x\in(0, 1)$ where $\Delta_{n,x}=[\tau_{n,i}, \tau_{n,i+1}]$ for some $i$ such that $x\in\Delta_{n,x}$. Lemma 3 was proved in [22] under condition (2.1). Lemma 4. Let $f$ be an integrable function and $a_n$ be its Fourier–Franklin coefficients. Then $a_n=o(\delta_n^{-1/2})$. Lemma 5. For some $M^{i_0}_{j_0}$ let the series (2.3) with coefficients (2.4) satisfy the condition Then for each $x_0\in[0,1]$ there exist $i$ and $j$ such that § 3. The main lemmaFor an integrable function $f$ from Theorem 6 we set
$$
\begin{equation}
b_k:=\int_0^1f(t)f_n(t)\,dt\quad\text{and} \quad S^{(i)}_f(x):=\sum_{i=0}^{n_i}b_kf_k(x).
\end{equation}
\tag{3.1}
$$
It is known that for any general Franklin system, if
$$
\begin{equation*}
S^{\star}_f(x):=\sup_n\biggl|\sum_{k=0}^nb_kf_k(x)\biggr|,
\end{equation*}
\notag
$$
then (see [25]) where $\mathcal M(f,x)$ is the Hardy–Littlewood maximal function of the function $f$. This inequality implies (see, for example, Ch. 2, § 1 in [29]) that
$$
\begin{equation}
\lim_{\lambda\to\infty}\bigl(\lambda\operatorname{mes}\{x\in[0,1]\colon S^{\star}_f(x)>\lambda\}\bigr)=0.
\end{equation}
\tag{3.2}
$$
Let $\widetilde{\mathcal S}$ be the difference between the series (2.3) and the Fourier–Franklin series of the function $f$, so that
$$
\begin{equation}
\widetilde{\mathcal S}=\sum_{k=0}^{\infty}c_kf_k(x), \quad\text{where}\ c_k=a_k-b_k.
\end{equation}
\tag{3.3}
$$
To prove Theorem 6 we need to show that $c_k=0$ for all $k$. The main ingredient of this proof is the following lemma. Main Lemma. Let the series (2.3) with coefficients (2.4) converge in measure to $f$ on $\Delta^{i_0}_{j_0}$, and let Then, for each $K>0$ there exist $i$ and $j$ such that and Proof. We set It is clear that $A_m$ is an open set and $A_m\subset A_{m+1}$. Consider the open set which, as an open subset of the real line, is a union of disjoint open intervals, that is,
The case $A=\varnothing$ is also possible. Assume that $A\neq\varnothing$ and there exist $i$ and $j$ such that $\Delta^i_j\subset A$ and $(\widetilde{\mathcal S}, M^i_j)\neq 0$. Since $A_m$ are open sets and $A_m\subset A_{m+1}$, we find that $\Delta^i_j\subset A_m$ for some $m$. This means that if $x\in\Delta_j^i$, then $x\in A^p$ for some $p$, $i_0\leqslant p\leqslant m$, that is,
$$
\begin{equation*}
|\widetilde{\mathcal S}^{(p)}(x)|>K+|S^{(p)}_f(x)| \quad\text{for some}\ p\in[i_0, m].
\end{equation*}
\notag
$$
As a result (see (3.1) and (3.3)), if $x\in\Delta^i_j$, then $|S^{(p)}(x)|>K$ for some $p\in[i_0, m].$ It is clear that these $i$ and $j$ are the required ones.
Recall that the case $A=\varnothing$ is not excluded. (This case does not require a separate consideration.) Now assume that
$$
\begin{equation}
(\widetilde{\mathcal S}, M^i_j)=0 \quad\text{if}\ \Delta^i_j\subset A.
\end{equation}
\tag{3.5}
$$
By the definition of the sets $A^p$, if $x\notin A$, then $|\widetilde{\mathcal S}^{(p)}(x)|\leqslant K+|S_f^{(p)}(x)|$ for ${p\geqslant i_0}$. Therefore,
$$
\begin{equation}
\sup_{p\geqslant i_0}|\widetilde{\mathcal S}^{(p)}(x)|\leqslant K+S^{\star}_f(x), \quad \ x\notin A.
\end{equation}
\tag{3.6}
$$
Using (3.6) and (3.2) we can find $\lambda$ such that
$$
\begin{equation}
\lambda>\kappa_2^2\max(1,d)\quad\text{and} \quad \lambda\operatorname{mes}(E_{\lambda})<\varepsilon_1:=\frac{\min(1,d)\operatorname{mes}(\Delta^{i_0}_{j_0})}{25000\kappa_2^3\gamma^2},
\end{equation}
\tag{3.7}
$$
where
$$
\begin{equation}
E_{\lambda}:=\Bigl\{x\in\Delta^{i_0}_{j_0}\colon \sup_i|\widetilde{\mathcal S}^{(i)}(x)|>\lambda\Bigr\}.
\end{equation}
\tag{3.8}
$$
Let $q_0$ be such that
$$
\begin{equation}
\operatorname{mes}\biggl(\bigcup_{q\geqslant q_0}I_q\biggr) < \frac{\varepsilon_1}{\lambda}
\end{equation}
\tag{3.9}
$$
(if $A=\varnothing$, then no $q_0$ is required). Then we set
$$
\begin{equation}
P:=\bigcup_{q<q_0}I_q\quad\text{and} \quad Q:=\bigcup_{q\geqslant q_0}I_q,
\end{equation}
\tag{3.10}
$$
and define
$$
\begin{equation}
G:=E_{\lambda}\cup Q\quad\text{and} \quad D:=\biggl\{x\in\Delta^{i_0}_{j_0}\colon \mathcal M_\mathcal T(\chi_G,x)>\frac{1}{6\kappa_2\gamma}\biggr\},
\end{equation}
\tag{3.11}
$$
where $\mathcal M_\mathcal T(\chi_G,x)$ is the maximal function of the characteristic function of the set $G$ over the intervals $[s^i_j,s^i_{j+1}]$, that is,
$$
\begin{equation*}
\mathcal M_\mathcal T(\chi_{G},x):=\sup_{x\in[s^i_j,s^i_{j+1}]} \frac{\operatorname{mes}(G\cap[s^i_j,s^i_{j+1}])}{s^i_{j+1}-s^i_{j}}.
\end{equation*}
\notag
$$
It is clear that if $A=\varnothing$, then $P=Q=\varnothing$. In general, from (3.8)–(3.11) we obtain
$$
\begin{equation}
\operatorname{mes}(G)<\frac{2\varepsilon_1}{\lambda}
\end{equation}
\tag{3.12}
$$
and
$$
\begin{equation}
\operatorname{mes}(D)<30\cdot\kappa_2\gamma\cdot\operatorname{mes}(G) <\frac{60\min(1,d)\operatorname{mes}(\Delta^{i_0}_{j_0})}{25000\kappa_2^2\gamma\lambda}.
\end{equation}
\tag{3.13}
$$
We prove by induction the representations
$$
\begin{equation}
\begin{aligned} \, \notag M^{i_0}_{j_0}(x) &=\sum_{\nu=i_0}^i\sum_{\Delta^{\nu}_j\subset P}\alpha^{(\nu)}_jM^{\nu}_j(x)+ \sum_{\nu=i_0+1}^i\sum_{\Delta^{\nu}_j\subset D}\alpha^{(\nu)}_jM^{\nu}_j(x) +\sum_{\substack{\Delta^i_j\not\subset P\\ \Delta^i_j\not\subset D}}\alpha^{(i)}_jM^i_j(x) \\ &=:\Sigma^i_1(x)+\Sigma^i_2(x)+\Sigma^i_3(x), \qquad i>i_0, \end{aligned}
\end{equation}
\tag{3.14}
$$
with coefficients satisfying
$$
\begin{equation}
\alpha^{(\nu)}_j\geqslant 0\quad\text{and} \quad \sum_{\nu=i_0}^i\sum_j\alpha^{(\nu)}_j=1.
\end{equation}
\tag{3.15}
$$
For $i=i_0+1$, from Lemma 1 we obtain
$$
\begin{equation}
M^{i_0}_{j_0}(x)=\sum_{j\colon \Delta^{i_0+1}_j\subset\Delta^{i_0}_{j_0}}\alpha^{(i_0+1)}_jM^{i_0+1}_j(x).
\end{equation}
\tag{3.16}
$$
Note that $\Delta^{i_0+1}_j\!\not\subset\! D$ for each $j$. Indeed, suppose for a contradiction that ${\Delta^{i_0+1}_j\!\subset\! D}$. Then (see (2.2))
$$
\begin{equation*}
\operatorname{mes}(G)\geqslant \operatorname{mes}(\Delta^{i_0+1}_j\cap G)\geqslant \frac{1}{6\kappa_2\gamma}\operatorname{mes}(\Delta^{i_0+1}_{j_0})> \frac{\operatorname{mes}(\Delta^{i_0}_{j_0})}{6\kappa^2_2\gamma},
\end{equation*}
\notag
$$
which contradicts (3.12) and (3.7).
Let $\Sigma^{i_0+1}_1(x)$ be the sum of the terms in (3.16) for which $\Delta^{i_0+1}_j\subset P$, and let $\Sigma^{i_0+1}_3(x)$ be the sum of the remaining terms. This completes the first induction step. Assume that representation (3.14) is true for $i$. Let us verify (3.14) for $i+1$. Applying Lemma 1 to the terms $M^i_j$ involved in $\Sigma^i_3$ and collecting similar terms, we obtain the sum Let $\Sigma^{i+1}_1(x)$ be the sum of the terms in (3.17) for which $\Delta_j^{i+1}\subset P$ and let $\Sigma^{i+1}_2(x)$ involve the terms in (3.17) for which $\Delta_j^{i+1}\subset D$. Let $\Sigma^{i+1}_3(x)$ be the sum of the remaining terms. Thus, representation (3.14) holds for each $i$. That condition (3.15) is satisfied is clear. Now, for each $i$, using representation (3.14) we obtain
$$
\begin{equation}
\begin{aligned} \, \notag (\widetilde{\mathcal S}, M^{i_0}_{j_0})&=\sum_{\nu=i_0}^i\sum_{\Delta^{\nu}_j\subset P}\alpha^{(\nu)}_j(\widetilde{\mathcal S}, M^{\nu}_j)+ \sum_{\nu=i_0+1}^i\sum_{\Delta^{\nu}_j\subset D}\alpha^{(\nu)}_j(\widetilde{\mathcal S}, M^{\nu}_j) \\ &\qquad +\sum_{\substack{\Delta^i_j\not\subset P \\ \Delta^i_j\not\subset D}}\alpha^{(i)}_j(\widetilde{\mathcal S},M^i_j)=:\sigma^i_1+\sigma^i_2+\sigma^i_3. \end{aligned}
\end{equation}
\tag{3.18}
$$
The sum $\sigma_1^i$ is zero by assumption (see (3.5)).
Let us show that (see (3.14))
$$
\begin{equation}
|(\widetilde{\mathcal S}, M^i_j)|\leqslant 2\lambda \quad\text{if}\ M^i_j(x)\text{ is involved in the sum }\sigma^i_2.
\end{equation}
\tag{3.19}
$$
Suppose that $|(\widetilde{\mathcal S}, M^i_j)|> 2\lambda $. In this case Lemma 4 implies that
$$
\begin{equation}
\operatorname{mes}\{x\in\Delta^i_j\colon |\widetilde{\mathcal S}^{(i)}(x)|>\lambda\}\geqslant\frac{\operatorname{mes}(\Delta^i_j)}{3(\gamma+1)}.
\end{equation}
\tag{3.20}
$$
Let $\Delta^{i-1}_{\nu}$ be an interval such that $\Delta^i_j\subset\Delta^{i-1}_{\nu}$. Then from (3.20) we have
$$
\begin{equation*}
\operatorname{mes}\{x\in\Delta^{i-1}_{\nu}\colon |\widetilde{\mathcal S}^{(i)}(x)|\geqslant\lambda\}\geqslant\frac{\operatorname{mes}(\Delta^{i-1}_{\nu})}{3(\gamma+1)\cdot \kappa_2}>\frac{\operatorname{mes}(\Delta^{i-1}_{\nu})}{6\gamma\cdot \kappa_2}
\end{equation*}
\notag
$$
(also see (2.2)), which means that $\Delta^{i-1}_{\nu}\subset D$. But this is impossible by the construction of (3.14). This proves (3.19). As a result (see (3.11)–(3.14) and (3.7)),
$$
\begin{equation}
\begin{aligned} \, \notag |\sigma_2^i| &\leqslant 2\lambda\sum_{\Delta^i_j\subset D}\alpha^{(i)}_j=2\lambda\int\sum_{\Delta^i_j\subset D}\alpha^{(i)}_jM^i_j(x)\,dx\leqslant 2\lambda \int_DM^{i_0}_{j_0}(x)\,dx \\ &\leqslant 2\lambda\cdot\operatorname{mes}(D)\cdot\|M^{i_0}_{j_0}\|_{\infty} <2\lambda\frac{d\operatorname{mes}(\Delta^{i_0}_{j_0})}{400\lambda} \,\frac{2}{\operatorname{mes}(\Delta^{i_0}_{j_0})}<\frac{d}{100}. \end{aligned}
\end{equation}
\tag{3.21}
$$
It is known that $S_f^{(i)}(x)$ converges to $f(x)$ almost everywhere (see [25]). Therefore, the series $\widetilde{\mathcal S}(x)$ converges to zero in measure. Hence for sufficiently large $i$
$$
\begin{equation}
\operatorname{mes}\biggl\{x\in\Delta^{i_0}_{j_0}\colon |\widetilde{\mathcal S}^{(i)}(x)|>\frac{d}{4}\biggr\} <\varepsilon_2:=\frac{d\operatorname{mes}(\Delta^{i_0}_{j_0})}{100\gamma\lambda}.
\end{equation}
\tag{3.22}
$$
Setting
$$
\begin{equation*}
\begin{gathered} \, H^i:=\{j\colon \Delta^i_j\subset \Delta^{i_0}_{j_0},\, \Delta^i_j\not\subset P,\, \Delta^i_j\not\subset D\}, \\ H^i_1:=\{j\in H^i\colon \Delta^i_j\cap P\neq \varnothing\}\quad\text{and} \quad H^i_2:=\{j\in J^i\colon \Delta^i_j\cap P=\varnothing\}, \end{gathered}
\end{equation*}
\notag
$$
we obtain
$$
\begin{equation*}
\sigma^i_3=\sum_{j\in H^i_1}\alpha^{(i)}_j(\widetilde{\mathcal S}, M^i_j)+\sum_{j\in H^i_2}\alpha^{(i)}_j( \widetilde{\mathcal S}, M^i_j)=:h^i_1+h^i_2.
\end{equation*}
\notag
$$
It is clear that $\operatorname{card}(H^i_1)\leqslant 4q_0$. By Lemmas 3 and 4, for $j\in H^i_1$,
$$
\begin{equation}
\alpha^{(i)}_j|(\mathcal S,M^i_j)|=\alpha^{(i)}_j|(S^{(i)},M^i_j)|\leqslant\alpha^{(i)}_j \max_{x\in\Delta^i_j}|\widetilde{\mathcal S}^{(i)}(x)|=\alpha^{(i)}_jo((\operatorname{mes}(\Delta^i_j))^{-1}).
\end{equation}
\tag{3.23}
$$
By (3.14) we have
$$
\begin{equation}
\alpha^{(i)}_j=\alpha^{(i)}_j\int_{\Delta^i_j}M^i_j(x)\,dx \leqslant\int_{\Delta^{i}_{j}}M^{i_0}_{j_0}(x)\,dx \leqslant\|M^{i_0}_{j_0}\|_{\infty}\cdot\operatorname{mes}(\Delta^i_j).
\end{equation}
\tag{3.24}
$$
Since $\operatorname{card}(H^i_1)\leqslant 4q_0$, from (3.23) and (3.24) we obtain
$$
\begin{equation}
|h^i_1|<\frac{d}{4} \quad\text{for sufficiently large}\ i.
\end{equation}
\tag{3.25}
$$
We set
$$
\begin{equation*}
J^i_1:=\biggl\{j\in H^i_2\colon |(\widetilde{\mathcal S}, M^i_j)|>\frac{d}{2} \text{ and }j\text{ is odd}\biggr\}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
J^i_2:=\biggl\{j\in H^i_2\colon |(\widetilde{\mathcal S}, M^i_j)|>\frac{d}{2} \text{ and }j\text{ is even}\biggr\}.
\end{equation*}
\notag
$$
By Lemma 4, if $j\in J^i_1$, then
$$
\begin{equation}
\operatorname{mes}\biggl\{x\in\Delta^i_j\colon |\widetilde{S}^{(i)}(x)|>\frac{d}{4}\biggr\} \geqslant \frac{\operatorname{mes}(\Delta^i_j)}{3(\gamma+1)}.
\end{equation}
\tag{3.26}
$$
Since $\operatorname{mes}(\Delta^i_{j_1}\cap\Delta^i_{j_2})=0$ if $j_1, j_2\in J^i_1$ and $j_1\neq j_2$, from (3.26) we obtain
$$
\begin{equation}
\operatorname{mes}\biggl\{x\in\Delta^{i_0}_{j_0}\colon |\widetilde{S}^{(i)}(x)|>\frac{d}{4}\biggr\} \geqslant\frac{\operatorname{mes}\bigl(\bigcup_{j\in J^i_1}\Delta^i_j\bigr)}{3(\gamma+1)}.
\end{equation}
\tag{3.27}
$$
A similar analysis shows that
$$
\begin{equation}
\operatorname{mes}\biggl\{x\in\Delta^{i_0}_{j_0}\colon |\widetilde{\mathcal S}^{(i)}(x)|>\frac{d}{4}\biggr\} \geqslant\frac{\operatorname{mes}\bigl(\bigcup_{j\in J^i_2}\Delta^i_j\bigr)}{3(\gamma+1)}.
\end{equation}
\tag{3.28}
$$
From (3.27), (3.28) and (3.22) it follows that
$$
\begin{equation}
\operatorname{mes}\biggl(\bigcup_{j\in J^i}\Delta^i_j\biggr)<6(\gamma+1)\varepsilon_2, \quad\text{where}\ J^i=J^i_1\cup J^i_2.
\end{equation}
\tag{3.29}
$$
We set Proceeding as in (3.19) we find that
$$
\begin{equation}
|(\widetilde{\mathcal S}, M^i_j)|\leqslant 2\lambda \quad\text{if}\ j\in H^i_2.
\end{equation}
\tag{3.31}
$$
Now from (3.30), (3.31) and (3.15), for sufficiently large $i$ we obtain
$$
\begin{equation}
\begin{aligned} \, \notag |h_2^i| &\leqslant \sum_{j\in J^i_3}\alpha^{(i)}_j|(\widetilde{\mathcal S}, M^i_j)|+\sum_{j\in J^i}\alpha^{(i)}_j|(\widetilde{\mathcal S}, M^i_j)|\leqslant\frac{d}{4}+2\lambda\sum_{j\in J^i}\alpha^{(i)}_j \\ &\leqslant \frac{d}{4}+2\lambda\sum_{j\in J^i}\alpha^{(i)}_j\int M^i_j(x)\,dx \leqslant\frac{d}{4}+2\lambda \int_{F} M^{i_0}_{j_0}(x)\,dx, \quad\text{where}\ F:=\bigcup_{j\in J^i}\Delta^i_j. \end{aligned}
\end{equation}
\tag{3.32}
$$
From (3.29) we find that $\operatorname{mes}(F)<6(\gamma+1)\varepsilon_2$. Now by (3.32) and (3.22) we have Next, using (3.33), (3.21), (3.18), (3.25), and (3.4), for sufficiently large $i$ we have This contradiction proves the lemma. § 4. Proof of the main result Proof of Theorem 6. Let the series (2.3) satisfy the assumptions of Theorem 6, and let $B=:\{y_p\}_{p=1}^{\infty}$. Also assume that (3.1) is the Fourier–Franklin series of a function $f$ and the series $\widetilde{\mathcal S}$ is defined by (3.3). Let us show that all the coefficients of $\widetilde{\mathcal S}$ are zero. Suppose for a contradiction that there exists $k$ such that $c_k\neq 0$. For $i_0$ such that $n_{i_0}>k$ there exists $j_0$ such that $(\widetilde{\mathcal S}, M^{i_0}_{j_0})\neq 0$ (recall that $\{M^{i}_j\}_{j=0}^{2^{n_i}}$ is a basis for $\mathbf S_{n_i})$.
Applying Lemma 5 to the series $\widetilde{\mathcal S}$ we find a closed interval $\Delta^{i'_1}_{j'_1}$ such that
$$
\begin{equation*}
y_1\notin\Delta^{i'_1}_{j'_1}\subset \Delta^{i_0}_{j_0}\quad\text{and} \quad (\widetilde{\mathcal S}, M^{i_1'}_{j_1'})\neq 0.
\end{equation*}
\notag
$$
Next, an application of the main lemma produces a closed interval $\Delta^{i_1}_{j_1}$ such that
$$
\begin{equation*}
\Delta^{i_1}_{j_1}\subset \Delta^{i'_1}_{j'_1}, \qquad |(\mathcal S, M^{i_1}_{j_1})|\neq 0,
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\max_{i_0\leqslant\nu\leqslant i_1} |S^{(\nu)}(x)|>1, \qquad x\in\Delta^{i_1}_{j_1}.
\end{equation*}
\notag
$$
Assume that we have already found $i_1,\dots, i_p$ and $j_1,\dots, j_p$ such that
$$
\begin{equation*}
y_{\nu}\notin\Delta^{i_\nu}_{j_\nu}\subset \Delta^{i_{\nu-1}}_{j_{\nu-1}}, \qquad \nu=1,\dots ,p,
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\max_{i_{\nu-1}\leqslant \mu\leqslant i_{\nu}}|S^{(\mu)}(x)|>\nu, \qquad\ x\in\Delta^{i_{\nu}}_{j_{\nu}}, \quad \nu=1,\dots ,p.
\end{equation*}
\notag
$$
Applying Lemma 5 to the series $\widetilde{\mathcal S}$ we find a closed interval $\Delta^{i'_{p+1}}_{j'_{p+1}}$ such that
$$
\begin{equation*}
y_{p+1}\notin\Delta^{i'_{p+1}}_{j'_{p+1}}\subset \Delta^{i_p}_{j_p}\quad\text{and} \quad (\widetilde{\mathcal S}, M^{i_{p+1}'}_{j_{p+1}'})\neq 0.
\end{equation*}
\notag
$$
Next, an appeal to the main lemma produces numbers $i_{p+1}$ and $j_{p+1}$ such that
$$
\begin{equation*}
\begin{gathered} \, y_{p+1}\notin\Delta^{i_{p+1}}_{j_{p+1}}\subset \Delta^{i'_{p+1}}_{j'_{p+1}}\subset \Delta^{i_p}_{j_p}, \qquad (\widetilde{\mathcal S}, M^{i_{p+1}}_{j_{p+1}})\neq 0, \\ \max_{i_p\leqslant\nu\leqslant i_{p+1}} |S^{(\nu)}(x)|>p+1, \qquad x\in\Delta^{i_{p+1}}_{j_{p+1}}. \end{gathered}
\end{equation*}
\notag
$$
Proceeding in this way we construct a nested sequence of closed intervals $\Delta^{i_p}_{j_p}$ such that
$$
\begin{equation}
y_p\notin\Delta^{i_p}_{j_p}\subset\Delta^{i_{p-1}}_{j_{p-1}}, \qquad p=1,2,\dots,
\end{equation}
\tag{4.1}
$$
and
$$
\begin{equation}
\max_{i_{p-1}\leqslant\nu\leqslant i_{p}}|S^{(\nu)}(x)|>p, \qquad x\in\Delta^{i_p}_{j_p}.
\end{equation}
\tag{4.2}
$$
From (4.1) and (4.2) it follows that there exists $z$ such that
$$
\begin{equation*}
z\notin\{y_p\}_{p=1}^{\infty}=B\quad\text{and} \quad \sup_p|S^{(i_p)}(z)|=\infty,
\end{equation*}
\notag
$$
but this contradicts (2.6).
This proves the theorem. 
Citation in format AMSBIB
\Bibitem{Gev24} |
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