On systems of non-overlapping Haar polynomials

TL;DR

This paper proves that the logarithm of n acts as an almost everywhere convergence Weyl multiplier for orthonormal systems of non-overlapping Haar polynomials and general martingale difference polynomials.

Contribution

It establishes the convergence properties of Haar polynomial systems and extends results to general martingale difference polynomials.

Findings

Log n is an almost everywhere convergence Weyl multiplier.

Convergence results hold for general systems of martingale difference polynomials.

Abstract

We prove that $\log n$ is an almost everywhere convergence Weyl multiplier for the orthonormal systems of non-overlapping Haar polynomials. Moreover, it is done for the general systems of martingale difference polynomials.

Equations

$${\Vert \underset{1\le m\le n}{\max }\left|\sum _{k=1}^m{\varphi }_k\right|\Vert }_2\le c\cdot \log n{\Vert \sum _{k=1}^n{\varphi }_k\Vert }_2,$$

$${\Vert \underset{1\le m\le n}{\max }\left|\sum _{k=1}^m{\varphi }_k\right|\Vert }_2\ge c\cdot \log n{\Vert \sum _{k=1}^n{\varphi }_k\Vert }_2,$$

$$p_k(x)=\sum _{j\in G_k}{c}_j{\varphi }_j(x),k=1,2,\dots ,n,$$

$${\mathcal{K}}_n(\Phi )=\underset{\{p_k\}\in {\mathcal{P}}_n(\Phi )}{\sup }\frac{{\Vert {\max }_{1\le m\le n}\left|p_m\right|\Vert }_2}{{\Vert p_n\Vert }_2}.$$

$${\mathcal{K}}_n(\mathcal{H})\sim \sqrt{\log n}.$$

$$\sum _{n=1}^{\infty }{a}_n{\varphi }_n(x),$$

$$p_n(x)=\sum _{j\in G_n}{c}_j{f}_j(x),n=1,2,\dots ,$$

$$\sum _{n=1}^{\infty }\frac{1}{n\omega (n)}<\infty ,$$

$${\int }_0^1{\varphi }_{n_1}{\varphi }_{n_2}{\varphi }_{n_3}{\varphi }_{n_4}=0$$

$${\xi }_a(x)=\frac{|[0,x)\cap a|}{|a|}$$

$$u_{a,n}(x)=\left\{\begin{array}[]{lcl}((\xi_{a})^{-1}\circ\eta_{n}\circ\xi_{a})(x)&\hbox{ if }&x\in a,\\ x&\hbox{ if }&x\in[0,1)\setminus a,\end{array}\right.$$

$$\underset{n\to \infty }{\lim }{\int }_{a}f(u_{a,n}(x))g(x)dx={\int }_{a}f(x)dx\cdot {\int }_{a}g(x)dx$$

$$u_{\mathcal{A},n}(x)=\sum _j{u}_{a_j,n}(x)\cdot 1_{a_j}(x)$$

$$\underset{n\to \infty }{\lim }{\int }_0^{1}f(u_{\mathcal{A},n}(x))g(x)dx=\sum _j{\int }_{a_j}f(x)dx\cdot {\int }_{a_j}g(x)dx$$

$$|\{f_{m_k}(x)>{\lambda }_k,k=1,2,\dots ,n\}|=|\{{\tilde{f}}_{m_k}(x)>{\lambda }_k,k=1,2,\dots ,n\}|.$$

$$\Vert {\tilde{f}}_k-p_k{\Vert }_2<{\epsilon }_k,k=1,2,\dots .$$

$${\int }_{{\alpha }_i}{f}_{l+1}({\tau }_l(x))dx=0,i=1,2,\dots .$$

$$\underset{n\to \infty }{\lim }{\int }_0^1{f}_{l+1}({\tau }_{l+1}(x)){\varphi }_i(x)dx$$

$$=\sum _i{\int }_{{\alpha }_i}{f}_{l+1}({\tau }_l(x))dx{\int }_{{\alpha }_i}{\varphi }_j(x)dxdx=0$$

$$p_{l+1}(x)=\sum _{i=m+1}^r{c}_i{\varphi }_i(x).$$

$$\mathcal{M}f(x)=\underset{n\ge 1}{\sup }\left|\sum _{k=1}^n{a}_k{h}_k(x)\right|,Sf(x)={\left(\sum _{k=1}^{\infty }{a}_k^2{h}_k^2(x)\right)}^{1/2}.$$

$$\begin{gathered} |\{x\in [0,1):\mathcal{M}f(x)>\lambda ,Sf(x)<\epsilon \lambda \}| \\ \lesssim \exp \left(-\frac{c}{{\epsilon }^2}\right)|\{\mathcal{M}f(x)>\lambda /2\}|,\lambda >0,0<\epsilon <1. \end{gathered}$$

$$|\{|p_k(x)|>$$

$$\lesssim \exp \left(-\frac{c}{{\epsilon }_n^2}\right)|\{\mathcal{M}{p}_k(x)>\lambda /2\}|.$$

$$\{p^{\ast }(x)>\lambda \}$$

$$\cup \{S{p}_n(x)>{\epsilon }_n\lambda \}=A(\lambda )\cup B(\lambda ),$$

$$\Vert p^{\ast }{\Vert }_2^2\le 2{\int }_0^{\infty }\lambda |A(\lambda )|d\lambda +2{\int }_0^{\infty }\lambda |B(\lambda )|d\lambda .$$

$${\int }_0^{\infty }\lambda |A(\lambda )|d\lambda$$

$$\le \exp \left(-\frac{c}{{\epsilon }_n^2}\right)\sum _{m=1}^n{\int }_0^{\infty }\lambda |\{\mathcal{M}{p}_m>\lambda /2\}|d\lambda$$

$$\lesssim \frac{1}{n}\sum _{m=1}^n\Vert \mathcal{M}{p}_m{\Vert }_2^2$$

$$\lesssim \frac{1}{n}\sum _{m=1}^n\Vert p_m{\Vert }_2^2$$

$$\le \Vert p_n{\Vert }_2^2.$$

$$2{\int }_0^{\infty }\lambda |B(\lambda )|d\lambda$$

$$\Vert p^{\ast }{\Vert }_2={\Vert \underset{1\le m\le n}{\max }|p_m(x)|\Vert }_2\lesssim \sqrt{\log n}\cdot \Vert p_n{\Vert }_2$$

Full text

Research was supported by the Science Committee of Armenia, grant 18T-1A081

1. Introduction

The following two theorems are well-known in Fourier Analysis.

Theorem A** (Menshov-Rademacher, [Men], [Rad], see also [KaSa]).**

If $\{\phi_{k}:\,\,k=1,2,\ldots,n\}\subset L^{2}(0,1)$ is an orthogonal system, then

[TABLE]

where $c>0$ is an absolute constant.

Theorem B** (Menshov, [Men]).**

For any $n\in\mathbb{N}$ there exists an orthogonal system $\phi_{k}$, $k=1,2,\ldots,n$, such that

[TABLE]

for an absolute constant $c>0$.

Let $\Phi=\{\phi_{k}(x),\,k=1,2,\ldots\}\subset L^{2}(0,1)$ be an infinite orthogonal system of functions. Denote by $\mathcal{P}_{n}(\Phi)$ the family of all monotonic sequences of $\Phi$-polynomials

[TABLE]

where $G_{1}\subset G_{2}\subset\ldots\subset G_{n}\subset\mathbb{N}$ and $\sum_{j\in G_{n}}c_{j}^{2}\neq 0$. Define

[TABLE]

From Theorem A it follows that $\mathcal{K}_{n}(\Phi)\leq c\cdot\log n$ for every orthogonal system $\Phi$, where $c$ is an absolute constant. On the other hand, applying Theorem B, one can also construct an infinite orthogonal system with the lower bound $\mathcal{K}_{n}(\Phi)\geq c\cdot\log n$, $n=1,2,\ldots$. Thus we conclude, in general, the logarithmic bound of $\mathcal{K}_{n}(\Phi)$ is optimal. We will see below that from results of Nikishin-Ulyanov [NiUl] and Olevskii [Ole] it follows that $\mathcal{K}_{n}(\Phi)\gtrsim\sqrt{\log n}$ for any complete orthonormal system $\Phi$.

In this paper we found the sharp rate of the growth of $\mathcal{K}_{n}\sim\sqrt{\log n}$ for the generalized Haar systems. The classical Haar system case of the result is also new and interesting. The upper bound $\mathcal{K}_{n}\lesssim\sqrt{\log n}$ is proved for the general systems of martingale type.

To state the main results recall few standard notations. The relation $a\lesssim b$ ($a\gtrsim b$) will stand for the inequality $a\leq c\cdot b$ ($a\geq c\cdot b$), where $c>0$ is an absolute constant. Given two sequences of positive numbers $a_{n},b_{n}>0$, we write $a_{n}\sim b_{n}$ if we have $c_{1}\cdot a_{n}\leq b_{n}\leq c_{2}\cdot a_{n}$, $n=1,2,\ldots$ for some constants $c_{1},c_{2}>0$. Throughout the paper, the base of $\log$ is equal $2$.

Theorem 1.1**.**

If $\Phi$ is a martingale difference, then $\mathcal{K}_{n}(\Phi)\lesssim\sqrt{\log n}$.

Theorem 1.2**.**

For any generalized Haar system $\mathcal{H}$ we have the relation

[TABLE]

In the class of all martingale differences the upper bound in Theorem 1.1 is optimal that readily follows from Theorem 1.2. One can easily see that for the Rademacher system we have $\mathcal{K}_{n}\sim 1$. So relation (1.4) can not be extended for general martingale differences. Such estimates of $\mathcal{K}_{n}(\Phi)$ characterize Weyl multipliers of a given orthonormal system $\Phi$. Recall some well-known definitions in the theory of orthogonal series (see [KaSa]).

Definition 1.5**.**

Let $\Phi=\{\phi_{n}:\,n=1,2,\ldots\}$ be an orthonormal system ($\|\phi_{n}\|_{2}=1$). A sequence of positive numbers $\omega(n)\nearrow\infty$ is said to be an a.e. convergence Weyl multiplier (shortly C-multiplier) if every series

[TABLE]

with coefficients satisfying the condition $\sum_{n=1}^{\infty}a_{n}^{2}\omega(n)<\infty$ is a.e. convergent. If such series converge unconditionally a.e., then we say $\omega(n)$ is an a.e unconditional convergence Weyl multiplier (UC-multiplier) for $\Phi$.

Note that Menshov [Men] and Rademacher [Rad] used estimate (1.1) to prove that the sequence $\log^{2}n$ is a C-multiplier for any orthonormal system. Likewise, from Theorem 1.1, we will deduce the following.

Corollary 1.1**.**

If $\mathcal{F}=\{f_{n}\}$ is a martingale difference, then $\log n$ is a C-multiplier for any system of $L^{2}$-normalized non-overlapping $\mathcal{F}$-polynomials

[TABLE]

where $G_{n}\subset\mathbb{N}$ are finite and pairwise disjoint.

The following result is interesting and it immediately follows from Corollary 1.1.

Corollary 1.2**.**

The sequence $\log n$ is C-multiplier for any rearrangement of a generalized Haar system.

Corollary 1.3**.**

Let $\{p_{n}\}$ be a sequence of $L^{2}$-normalized non-overlapping polynomials with respect to a martingale difference. If $\omega(n)/\log n$ is increasing and

[TABLE]

then $\omega(n)$ is UC-multiplier for $\{p_{n}\}$.

The optimality of $\log n$ in Corollary 1.2 as well as condition (1.8) in Corollary 1.3 both follows from some results of Ulyanov for classical Haar system (see [Uly1],[Uly2] or [KaSa] ch. 2 Theorem 17). In particular, the paper [Uly1] proves that (1.8) is a necessary and sufficient condition for a sequence $\omega(n)\nearrow\infty$ to be an UC-multiplier for the classical Haar system.

We prove Theorem 1.1 using a good-$\lambda$ inequality due to Chang-Wilson-Wolff [CWW]. See also [GHS], where the same method has been first applied in the study of maximal functions of Mikhlin-Hörmander multipliers.

Remark**.**

Recall that an orthonormal system $\Phi$ is said to be a convergence system if $\omega(n)\equiv 1$ is a C-multiplier for $\Phi$. It was proved by Komlós-Révész [KoRe] that if an orthonormal system $\Phi=\{\phi_{n}\}\subset L^{2}(0,1)$ satisfies $\|\phi_{n}\|_{4}\leq M,\,n=1,2,\ldots,$ and we have

[TABLE]

for any choice of different indexes $n_{1},n_{2},n_{3},n_{4}$, then $\Phi$ is a convergence system. One can check that systems of non-overlapping martingale difference polynomials satisfy (1.9). Thus, with the extra condition $\|p_{n}\|_{4}\leq M$ in Corollary 1.1 we can claim that $\{p_{n}\}$ is a convergence system.

Question**.**

Is the additional condition $\|p_{n}\|_{p}\leq M$ in Corollary 1.1, with a fixed $2<p<4$, is sufficient for $\{p_{n}\}$ to be a convergence system?

2. Measure-preserving transformations

A mapping $\tau:[0,1)\to[0,1)$ is said to be measure-preserving (MP) transformation if $|\tau^{-1}(A)|=|A|$ for any Lebesgue measurable set $A\subset[0,1)$. A set in $[0,1)$ is said to be simple, if it is a finite union of intervals (of the form $[\alpha,\beta)$). Let $a$ be a simple set. One can easily check, that the function

[TABLE]

defines a one to one mapping from $a$ to $[0,1)$, such that $|\xi_{a}(E)|=|E|/|a|$ for any Lebesgue measurable set $E\subset a$. Given integer $n\geq 1$ the mapping $\eta_{n}(x)=\{nx\}$ defines an MP-transformation of $[0,1)$. Observe that if $a$ is a simple set, then for any integer $n\geq 1$ the mapping

[TABLE]

determines an MP-transformation of $[0,1)$ that maps the set $a$ to itself. Moreover, for any functions $f,g\in L^{2}(0,1)$ we have

[TABLE]

that is a well-known standard argument. A partition of $[0,1)$ is a sequence of pairwise disjoint sets $\mathcal{A}=\{E_{k}\}$ such that $\cup_{k}E_{k}=[0,1)$. We say $\mathcal{A}$ is a simple partition if each $E_{k}$ is simple. Let $\mathcal{A}=\{a_{j}\}$ be a simple partition of $[0,1)$. Given integer $n\geq 1$ we consider the MP-transformation

[TABLE]

that maps every $a_{j}$ to itself. This is an MP-transformation on $[0,1)$ that maps each set $a_{j}$ to itself and from (2.3) it follows that

[TABLE]

for any functions $f,g\in L^{2}(0,1)$. An MP-transformation $\tau$ is said to be simple if $\tau^{-1}(a)$ is simple set whenever $a$ is simple. Obviously all above described MP-transformations are simple.

A sequence $\mathcal{A}_{n}$, $n=1,2,\ldots$, of partitions of $[0,1)$ is said to be a filtration if any $A\in\mathcal{A}_{n}$ is a union of some sets from $\mathcal{A}_{n+1}$ called children of $A$. A martingale difference based on a filtration $\{\mathcal{A}_{n}:\,n=1,2,\ldots\}$ is a sequence of functions $f_{n}\in L^{1}(0,1)$, satisfying the conditions

1. (1)

Every function $f_{n}$ is constant on each $A\in\mathcal{A}_{n}$. 2. (2)

We have $\int_{A}f_{n}=0$ for any $A\in\mathcal{A}_{n-1}$, $n\geq 2$.

Consider a filtration $\{\mathcal{A}_{n}\}$ for which 1) $\mathcal{A}_{1}$ consists of a single element $[0,1)$, 2) each element $A\in\mathcal{A}_{n}$ has only two children intervals in $\mathcal{A}_{n+1}$, 3) $\max_{A\in\mathcal{A}_{n}}|A|\to 0$ as $n\to\infty$. A generalized Haar system is a $L^{2}$-normalized martingale difference based on such filtration. If two children intervals of any $A\in\mathcal{A}_{n}$ are equal, then it gives a signed classical Haar system. It is well-known that any generalized Haar system is complete.

We say that a function system $\{\tilde{f}_{n}\}$ is a transformation of another system $\{f_{n}\}$ if for every choice of numbers $m_{k}\in\mathbb{N}$ and $\lambda_{k}\in\mathbb{R}$ it holds the equality

[TABLE]

For example, this relation occurs when $\tilde{f}_{k}(x)=f_{k}(\tau(x))$ for some MP-transformation $\tau$.

The following lemma is an extension of a lemma of Olevksii [Ole] (see also [KaSa], ch. 10, Lemma 1) proving the same for the classical Haar system.

Lemma 2.1**.**

Let $\Phi=\{\phi_{k}(x)\}$ be a complete orthonormal system and $\mathcal{F}=\{f_{n}\}$ be a martingale difference based on a filtration consistiong of intervals. Then for any sequence of numbers $\varepsilon_{k}>0$ there exists a transformation $\tilde{}\mathcal{F}=\{\tilde{f}_{n}\}$ of the system $\mathcal{F}$ and a sequence of non-overlapping $\Phi$-polynomials $p_{k}$ such that

[TABLE]

Proof.

We shall realize the constructions of sequences $\tilde{f}_{k}$ and $p_{k}$ by induction. First, we take $\tilde{f}_{1}=f_{1}$. Approximation of $f_{1}$ by a $\Phi$-polynomial $p_{1}$ gives (2.7) for $k=1$ that is the base of induction. Then suppose that we have already defined $\tilde{f}_{k},\,p_{k}$, $k=1,2,\ldots,l$, satisfying the condition (2.7) such that $\tilde{f}_{k}(x)=f_{k}(\tau_{l}(x))$, $k=1,2,\ldots,l$, where $\tau_{l}$ is a for a simple MP-transformation (maps a simple set to a simple set) . Let $\mathcal{A}=\{a_{j}\}$ be the partition of $[0,1)$ that is formed by the maximal sets, where each function $\tilde{f}_{k}$, $k=1,2,\ldots,l$ is constant. Clearly each $a_{j}$ is a simple set. Since $u_{\mathcal{A},n}$ maps each $a_{j}$ to itself, $\tau_{l+1}=\tau_{l}\circ u_{\mathcal{A},n}$ determines a simple MP-transformation so that $f_{k}(\tau_{l+1}(x))=f_{k}(\tau_{l}(x))=\tilde{f}_{k}(x)$, $k=1,2,\ldots,l$, and

[TABLE]

From (2.5) and (2.8) it follows that

[TABLE]

for any $i=1,2,\ldots$. We will chose $n$ bigger enough and define $\tilde{f}_{l+1}(x)=f_{l+1}(\tau_{l+1}(x))$. Let $c_{i}$ be the Fourier coefficients of the function $\tilde{f}_{l+1}$ in system $\Phi$. Suppose that each polynomial $p_{k}$, $k=1,2,\ldots,l$, is a linear combination of functions $\phi_{j}$ , $j=1,2,\ldots,m$. From (2.9) it follows that for a bigger enough $n$ we have $\sum_{i=1}^{m}c_{i}^{2}<\varepsilon_{l+1}^{2}/4$. Then we can chose an integer $r>m$ such that $\sum_{i=r+1}^{\infty}c_{i}^{2}<\varepsilon_{l+1}^{2}/4$. Define

[TABLE]

Since $\Phi$ is a complete system, one can easily check that (2.7) is satisfied for $k=l+1$ that finalizes the induction and so the proof of lemma. ∎

3. Proof of Theorem 1.1

We will first prove the theorem for the classical Haar system. Let $h_{n}$ be the $L^{2}$-normalized classical Haar system. For a given function $f\in L^{1}(0,1)$ let $\sum_{k=1}^{\infty}a_{k}h_{k}$ be the Fourier-Haar series of $f$. Recall the maximal and the square functions operators defined by

[TABLE]

It is well known the boundedness of both operators on $L^{p}$, $1<p<\infty$. A key point in the proof of Theorem 1.1 is the following good-$\lambda$ inequality due to Chang-Wilson-Wolff (see [CWW], Corollary 3.1):

[TABLE]

So let $p_{k}$, $k=1,2,\ldots,n$, be a monotonic sequence of Haar polynomials. We have $|g(x)|\leq\mathcal{M}g(x)$ a.e. for any function $g\in L^{1}$, as well as $Sp_{k}(x)\leq Sp_{n}(x)$, $k=1,2,\ldots,n$. Thus, applying inequality (3.1) with $\varepsilon_{n}=(c/\ln n)^{1/2}$, we obtain

[TABLE]

For $p^{*}(x)=\max_{1\leq m\leq n}|p_{m}(x)|$ we obviously have

[TABLE]

and thus

[TABLE]

From (3.2) it follows that

[TABLE]

Combining this and

[TABLE]

we get

[TABLE]

that proves the theorem for the Haar system. Clearly we will have the same bound also for any transformation of the Haar system. To proceed the general case we suppose that $\mathcal{F}=\{f_{n}\}$ is an arbitrary martingale difference and let

[TABLE]

be an arbitrary monotonic sequence of $\mathcal{F}$-polynomials. Apply Lemma 2.1, choosing $\Phi$ to be the Haar classical system and $\varepsilon_{j}=\varepsilon$ for $j\in G_{n}$. So we get (2.7) for non-overlapping Haar polynomials $p_{k}$. Denote $\tilde{F}_{k}=\sum_{j\in G_{k}}c_{j}\tilde{f}_{j}$. Obviously,

[TABLE]

forms a monotonic sequence of Haar polynomials. For a small enough $\varepsilon$ we will have

[TABLE]

Therefore, taking into account that the theorem is true for the Haar system, we get

[TABLE]

This completes the proof of theorem.

4. Proof of Theorem 1.2

The upper bound $\mathcal{K}_{n}(\mathcal{H})\lesssim\sqrt{\log n}$ follows from Theorem 1.1. The lower bound

[TABLE]

for the classical Haar system follows from the Nikishin-Ulyanov [NiUl] inequality

[TABLE]

valid for appropriate coefficients $a_{k}$ and permutation $\sigma$ of the numbers $\{1,2,\ldots,n\}$. We will have the same estimate (4.1) also for any transformation of the classical Haar system. Then we apply Olevskii lemma ([KaSa], ch. 10, Lemma 1), that is the case of Lemma 2.1 when $\mathcal{F}$ coincides with the classical Haar system. So we get a transformed Haar system $\{\tilde{h}_{n}\}$ and a sequence of non-overlapping $\Phi$-polynomials $p_{k}$ such that

[TABLE]

Since $\varepsilon_{k}$’s here can be arbitrarily small, one can conclude $\mathcal{K}_{n}(\Phi)\geq\mathcal{K}_{n}(\mathcal{H})$. Combining this and (4.1) we get the following.

Proposition 4.1**.**

If $\Phi$ is a complete orthonormal system, then $\mathcal{K}_{n}(\Phi)\gtrsim\sqrt{\log n}$.

Since any generalized Haar system is complete, the lower bound (4.1) immediately follows from Proposition 4.1.

5. Proof of corollaries

Lemma 5.1** ([KaSt], Theorem 5.3.2).**

Let $\{\phi_{n}(x)\}$ be an orthonormal system and $\omega(n)\nearrow\infty$ be a sequence of positive numbers. If an increasing sequence of indexes $n_{k}$ satisfy the bound $\omega(n_{k})\geq k$, then the condition $\sum_{k=1}^{\infty}a_{k}^{2}\omega(k)<\infty$ implies a.e. convergence of sums $\sum_{j=1}^{n_{k}}a_{j}\phi_{j}(x)$ as $k\to\infty$.

Proof of Corollary 1.1.

Consider the series

[TABLE]

with coefficients satisfying the condition $\sum_{k=1}^{\infty}a_{k}^{2}\log k<\infty$ and denote $S_{n}=\sum_{k=1}^{n}p_{k}$. Since $\omega(n)=\log n$ satisfies the condition $\omega(2^{k})\geq k$, from Lemma 5.1 we have a.e. convergence of subsequences $S_{2^{k}}(x)$. So we just need to show that

[TABLE]

We have

[TABLE]

So we get

[TABLE]

which implies (5.2). ∎

To prove the next corollary we will need another lemma.

Lemma 5.2** ([Uly3], [Pol]).**

Let $u(n)$ be a C-multiplier for any rearrangement of the orthonormal system $\Phi=\{\phi_{n}(x)\}$. If an increasing sequence of positive numbers $\delta(k)$ satisfies the condition

[TABLE]

then $\delta(n)u(n)$ turnes to be a UC-multiplier for $\Phi$.

Proof of Corollary 1.3.

According to Corollary 1.1 $u(n)=\log n$ is a C-multiplier for the systems of non-overlapping MD-polynomials and their rearrangements. By the hypothesis of Corollary 1.3 the sequence $\delta(n)=\omega(n)/\log n$ is increasing and satisfies (5.3). Thus, the combination of Corollary 1.1 and Lemma 5.2 completes the proof. ∎