MRChexNet: Multi-modal bridge and relational learning for thoracic disease recognition in chest X-rays

1.   Introduction

In this paper, we consider the following two-component Fornberg-Whitham (FW) system for a fluid

where $x\in\mathbb{T} = \mathbb{R}/2\pi\mathbb{Z}$, $t\in\mathbb{R}^{+}$. Here, $u = u(x, t)$ is the horizontal velocity of the fluid, and $\rho = \rho(x, t)$ is the height of the fluid surface above a horizontal bottom. This system was first proposed in ^[5], and local well-posedness and non-uniform dependence on the initial data were established in Sobolev spaces $H^s(\mathbb{R}) \times H^{s-1}(\mathbb{R})$ for $s > \frac{3}{2}$ in ^[11,12].

Local well-posedness in Besov spaces $B^s_{p, r}(\mathbb{R}) \times B^{s-1}_{p, r}(\mathbb{R})$ of (1.1) was established in ^[4] for $s > \max\{2+\frac{1}{p}, \frac{5}{2}\}$. Besov spaces $B^s_{p, r}$ are a class of functions of interest in the study of nonlinear partial differential equations, as they are based on Sobolev spaces and introduce a measure of generalized Hölder regularity through the index $r$, along with the Sobolev index of differentiability $s$ and the Lebesgue integrability index $p$. If $s$ and $p$ are fixed, the spaces $B^s_{p, r}$ grow larger with increasing $r$. In ^[4], the FW system was shown to be well-posed in the sense of Hadamard by establishing the existence and uniqueness of the solution to the system (1.1) and then proving continuity of the data-to-solution map when the initial data belong to $B^s_{p, r}(\mathbb{R}) \times B^{s-1}_{p, r}(\mathbb{R})$ for $s > \max\{2+\frac{1}{p}, \frac{5}{2}\}$.

In this paper, our objective is to prove non-uniform dependence on periodic initial data for the two-component FW system (1.1) in $B^s_{p, r}(\mathbb{T}) \times B^{s-1}_{p, r}(\mathbb{T})$ for $s > \max\{2+\frac{1}{p}, \frac{5}{2}\}$. We work with periodic initial data, as that simplifies our choice of approximate solutions and the resulting estimates. Setting $\Lambda = 1-\partial_x^2$, we rewrite (1.1) as

where $x\in\mathbb{T} = \mathbb{R}/2\pi\mathbb{Z}$ and $t\in\mathbb{R}^{+}$.

The paper is organized as follows: In Section 2, we recall the standard definitions and properties of Besov spaces, linear transport equations, the operator $\Lambda$, and the two-component FW system. In Section 3, we prove non-uniform dependence on initial data for the FW system (1.2) when the initial data belong to $B^s_{p, r}(\mathbb{T}) \times B^{s-1}_{p, r}(\mathbb{T})$ for $s > \max\{2+\frac{1}{p}, \frac{5}{2}\}$. For this proof, we use a technique previously seen in the study of non-uniform continuity of data-to-solution maps for other nonlinear PDEs, for instance in ^{[6,7,8,10,12]}. We construct two sequences of approximate solutions such that the initial data for these sequences converge to each other in $B^s_{p, r}(\mathbb{T}) \times B^{s-1}_{p, r}(\mathbb{T})$. Non-uniform dependence is then established by proving that the approximate and hence the exact solutions remain bounded away from each other for any positive time $t > 0$. This idea was first explored by Kato in ^[9] to show that the data-to-solution map for Burgers' equation is not Hölder continuous in the $H^s$ norm with $s > 3/2$ for any Hölder exponent.

2.   Notation and preliminaries

This section is a review of relevant definitions and results on Besov spaces, linear transport equations, the operator $\Lambda$, and the two-component FW system (1.2). We begin by listing some useful notation to be used throughout Section 3.

2.1.   Notation

For any $x, y \in \mathbb{R}$,

● $x\lesssim y$ denotes $x\leq \alpha y$ for some constant $\alpha$.

● $x\approx y$ denotes $x = \beta y$ for some constant $\beta$.

● $x\gtrsim y$ denotes $x\geq \gamma y$ for some constant $\gamma$.

2.2.   Besov spaces

We recall the construction of a dyadic partition of unity from ^[8]. Consider a smooth bump function $\chi$ such that $\mathrm{supp}\; \chi = [-\frac{4}{3}, \frac{4}{3}]$ and $\chi = 1$ on $[-\frac{3}{4}, \frac{3}{4}]$. For $\xi > 0$, set $\varphi_{-1}(\xi) = \chi$, $\varphi_0(\xi) = \chi\left(\frac{\xi}{2}\right) - \chi(\xi)$ and $\varphi_q(\xi) = \varphi_0(2^{-q}\xi)$. Then, $\mathrm{supp}\; \varphi_q = [\frac{3}{4}\cdot 2^q, \frac{8}{3}\cdot 2^q]$ and $\sum\limits_{q\geq -1}\varphi_q(\xi) = 1$. Using this partition, a Littlewood-Paley decomposition of any periodic distribution $u$ is defined in ^[3] as follows:

Definition 2.1 (Littlewood-Paley decomposition). For any $u\in\mathcal{D}'(\mathbb{T})$ with the Fourier series $u(x) = \sum\limits_{j\in\mathbb{Z}} \hat{u}_j e^{ijx}$ where $\hat{u}_j = \frac{1}{2\pi}\int\limits_0^{2\pi}e^{-ijy}u(y)\; dy$, its Littlewood-Paley decomposition is given by $u = \sum\limits_{q\geq -1} \Delta_q u$, where $\Delta_q u$ are periodic dyadic blocks defined for all $q\in\mathbb{Z}$ as

Using this Littlewood-Paley decomposition, Besov spaces on $\mathbb{T}$ are defined in ^[3] as follows:

Definition 2.2 (Besov spaces). Let $s\in \mathbb{R}$ and $p$, $r \; \in [1, \infty]$. Then the Besov spaces of functions are defined as

where

Following are some properties proved in [1, Section 2.8] and [3, Section 1.3] that facilitate the study of nonlinear partial differential equations in Besov spaces.

Lemma 2.3. Let $s, s_j \in \mathbb{R}$ for $j = 1, 2$ and $1\leq p, r \leq \infty$. Then the following properties hold:

(1) Topological property: $B^s_{p, r}$ is a Banach space continuously embedded in $\mathcal{D}'(\mathbb{T})$.

(2) Algebraic property: For all $s > 0$, $B^s_{p, r} \cap {\bf L}^{\infty}$ is a Banach algebra.

(3) Interpolation: If $f \in B^{s_1}_{p, r} \cap B^{s_2}_{p, r}$ and $\theta \in [0, 1]$, then $f \in B^{\theta s_1 + (1-\theta) s_2}_{p, r}$ and

(4) Embedding: $B^{s_1}_{p, r} \hookrightarrow B^{s_2}_{p, r}$ whenever $s_1 \geq s_2$. In particular, $B^{s}_{p, r} \hookrightarrow B^{s-1}_{p, r}$ for all $s \in \mathbb{R}$.

Remark on (2) in Lemma 2.3: When $s > \frac{1}{p}$ (or $s\geq \frac{1}{p}$ and $r = 1$), $B^s_{p, r} \hookrightarrow {\bf L}^{\infty}$. We will use the fact that for $0 < s < \frac{1}{p}$, the result is still true as long as the functions are bounded.

2.3.   Linear transport equation

Given a linear transport equation, Proposition A.1 in ^[2] proves the following estimate for its solution size in Besov spaces:

Proposition 2.4. Consider the linear transport equation

where $f_0 \in B^s_{p, r}(\mathbb{T})$, $F \in {\bf L}^1((0, T); B^{s}_{p, r}(\mathbb{T}))$ and $v$ is such that $\partial_x v \in {\bf L}^1((0, T); B^{s-1}_{p, r}(\mathbb{T}))$. Suppose $f \in {\bf L}^{\infty}((0, T); B^{s}_{p, r}(\mathbb{T})) \cap C([0, T]; \mathcal{D}'(\mathbb{T}))$ is a solution to (2.1). Let $1\leq p, r \leq \infty$. If either $s\neq 1+\frac{1}{p}$, or $s = 1+\frac{1}{p}$ and $r = 1$, then for a positive constant $C$ that depends on $s$, $p$, and $r$, we have

where

and

For $r < \infty$, $f\in C([0, T], B^{s}_{p, r}(\mathbb{T}))$, and if $r = \infty$, then $f\in C([0, T], B^{s'}_{p, 1}(\mathbb{T}))$ for all $s' < s$.

2.4.   The Operator $\Lambda$

Let $\Lambda = 1-\partial_x^2$; then, for any test function $g$, the Fourier transform of $\Lambda^{-1}g$ is given by $\mathcal{F}\left(\Lambda^{-1}g\right) = \frac{1}{1+\xi^2}\hat{g}(\xi)$. Moreover, for any $s\in\mathbb{R}$, $\Lambda^{-1}\partial_x$ is continuous from $B^{s-1}_{p, r}$ to $B^{s}_{p, r}$; that is, for all $h\in B^{s-1}_{p, r}$, there exists a constant $\kappa > 0$ depending on $s$, $p$, and $r$ such that

2.5.   The Fornberg-Whitham system

The well-posedness of the two-component FW system (1.2) in Besov spaces was established on the real line in ^[4] with the following result:

Theorem 2.5. Let $s > \max\{2+\frac{1}{p}, \frac{5}{2}\}$, $p \in [1, \infty]$, $r \in [1, \infty]$ and $(u_0, \rho_0) \in B^s_{p, r}(\mathbb{R}) \times B^{s-1}_{p, r}(\mathbb{R})$. Then the system (1.2) has a unique solution $(u, \rho) \in C\left([0, T];B^s_{p, r}(\mathbb{R}) \times B^{s-1}_{p, r}(\mathbb{R}) \right)$, where the doubling time $T$ is given by

with $C$ being a constant that depends on $s$, $p$, and $r$, and the solution size is estimated as

Moreover, the data-to-solution map is continuous.

Since we work with $B^s_{p, r}(\mathbb{T}) \times B^{s-1}_{p, r}(\mathbb{T})$ in this paper, we state the following:

Corollary 2.6. Theorem 2.5 holds when $\mathbb{R}$ is replaced by $\mathbb{T}$.

Proof. The existence of a solution to (1.2) is proved by altering the mollifier used to prove Theorem 2.5. This adaptation of the mollifier was done for the single Fornberg-Whitham equation in [7, Section 3.1]. Uniqueness and continuous dependence on periodic initial data for the system (1.2) are established by approximation arguments similar to those in [4, Sections 3.2 and 3.3]. □

3.   Non-uniform dependence on initial data

In this section, we establish non-uniform dependence on initial data in the periodic case for the two-component FW system (1.2) in Besov spaces.

Theorem 3.1. Let $s > \max\{2+\frac{1}{p}, \frac{5}{2}\}$ and $r \in [1, \infty]$. The data-to-solution map $(u_0, \rho_0) \mapsto (u(t), \rho(t))$ of the Cauchy problem (1.2) is not uniformly continuous from any bounded subset of $B^s_{p, r}(\mathbb{T}) \times B^{s-1}_{p, r}(\mathbb{T})$ into $\mathcal{C}([0, T];B^s_{p, r}(\mathbb{T}))\times\mathcal{C}([0, T];B^{s-1}_{p, r}(\mathbb{T}))$ where $T$ is given by Theorem 2.5. In particular, there exist two sequences of solutions $\{(u_{\omega, n}, \rho_{\omega, n})\}$ with $\omega = \pm 1$ such that the following hold:

(i) $\lim\limits_{n\to\infty} \left(\|u_{1, n}(0) - u_{-1, n}(0)\|_{B^s_{p, r}} +\|\rho_{1, n}(0) - \rho_{-1, n}(0)\|_{B^{s-1}_{p, r}}\right) = 0$.

(ii) $\liminf\limits_{n\to\infty}\left(\|u_{1, n} - u_{-1, n}\|_{B^s_{p, r}} + \|\rho_{1, n} - \rho_{-1, n}\|_{B^{s-1}_{p, r}}\right)\gtrsim |\sin t|$.

Proof. For $n\in\mathbb{N}$, we consider two sequences of functions $\{(u^{\omega, n}, \rho^{\omega, n})\}$ with $\omega = \pm 1$, defined by

We take initial data

Let the solutions to the FW system (1.2) with these initial data be denoted by $(u_{\omega, n}, \rho_{\omega, n})$. At $t = 0$, we have

which proves part (i) of Theorem 3.1.

To prove part (ii), first we estimate $\|(u^0_{\omega, n}, \rho^0_{\omega, n})\|_{B^{\gamma}_{p, r} \times B^{\gamma-1}_{p, r}}$ and $\|(u^{\omega, n}, \rho^{\omega, n})\|_{B^{\gamma}_{p, r} \times B^{\gamma-1}_{p, r}}$ for any $\gamma > 0$ and $r < \infty$. Using the triangle inequality, we have

By Definition 2.2,

From Definition 2.1, as shown in the Appendix, we have $\|\Delta_q \sin(nx)\|_{{\bf L}^p} = \varphi_q(n)$, where $0 < \varphi_q(n) \leq 1$ for all $q$ such that $\frac{1}{\ln (2)}\ln \left(\frac{3}{8}n\right)\leq q \leq \frac{1}{\ln (2)}\ln \left(\frac{4}{3}n\right)$ and $\varphi_q\left(n\right) = 0$ otherwise. Hence, (3.2) implies that for any $\gamma > 0$,

As $2^q \leq \frac{4}{3}n$ for every term in the summation, from the above, we obtain

Here and in what follows, $C_{\gamma}$ is a generic constant that depends only on $\gamma$ for fixed $p$ and $r$. Similarly, it follows that for any $\gamma > 0$,

By Definition 2.1,

So, $\|n^{-1}\|_{B^{\gamma}_{p, r}} = \left(\sum\limits_{q\geq -1} 2^{\gamma q r}\|\Delta_q n^{-1}\|^r_{{\bf L}^p}\right)^{\frac{1}{r}} = 2^{-\gamma}n^{-1}\;$. Using this and (3.3), it follows from (3.1) that

Since $(u^{\omega, n}, \rho^{\omega, n})$ is a phase shift of $(u^0_{\omega, n}, \rho^0_{\omega, n})$, we have

If $r = \infty$, (3.5) and (3.6) follow immediately from Definition 2.2.

We complete the proof of Theorem 3.1 by establishing (ii) for $\{(u^{\omega, n}, \rho^{\omega, n})\}$, taking advantage of the following lemma, whose proof follows the proof of Theorem 3.1. Lemma 3.2 establishes that for each $n$ and $\omega$, $(u^{\omega, n}, \rho^{\omega, n})$ approximates $(u_{\omega, n}, \rho_{\omega, n})$ in $B^s_{p, r}(\mathbb{T}) \times B^{s-1}_{p, r}(\mathbb{T})$ uniformly on $[0, T]$ for some $T > 0$.

Lemma 3.2. Let $\mathcal{E}^{\omega, n} = (\mathcal{E}^{\omega, n}_1, \mathcal{E}^{\omega, n}_2)$ where $\mathcal{E}^{\omega, n}_1 = u_{\omega, n} - u^{\omega, n}$ and $\mathcal{E}^{\omega, n}_2 = \rho_{\omega, n} - \rho^{\omega, n}$, with $\omega = \pm 1$. Then for all $t \in (0, T)$, where $T$ is given by Theorem 2.5, $\|\mathcal{E}^{\omega, n}(t)\|_{B^s_{p, r}\times B^{s-1}_{p, r}} = \|\mathcal{E}^{\omega, n}_1(t)\|_{B^{s}_{p, r}} + \|\mathcal{E}^{\omega, n}_2(t)\|_{B^{s-1}_{p, r}} \to 0$ as $n \to \infty$.

We show that $(u_{-1, n}, \rho_{-1, n})$ and $(u_{1, n}, \rho_{1, n})$ stay bounded away from each other for any $t > 0$. Since

and

adding (3.7) and (3.8) we obtain

By Definition 2.2, if $r = \infty$, we immediately have

where $C_s$ is a constant that depends only on $s$ for a given $p$. For $1\leq r < \infty$, there is a similar estimate, whose proof is given in the Appendix. Also, by Lemma 3.2, we have $\|\mathcal{E}^{\omega, n}(t)\|_{B^s_{p, r}\times B^{s-1}_{p, r}} \to 0$ for $\omega = \pm 1$, as $n\to\infty$. Using this and (3.10), it follows from (3.9) that

This proves part (ii) of Theorem 3.1 and completes the proof of non-uniform dependence on initial data for the two-component FW system (1.2) in $B^s_{p, r}(\mathbb{T}) \times B^{s-1}_{p, r}(\mathbb{T})$ for $s > \max\{2+\frac{1}{p}, \frac{5}{2}\}$. □

Now we prove Lemma 3.2.

Proof. (Lemma 3.2) We show that $\|\mathcal{E}^{\omega, n}(t)\|_{B^{\gamma}_{p, r}\times B^{\gamma-1}_{p, r}} \to 0$ as $n \to \infty$ for any $\gamma$ with $\max\{s-\frac{3}{2}, 1+\frac{1}{p}\} < \gamma < s-1$, and then interpolate between such a $\gamma$ and a value $\delta > s$. Recall that $\mathcal{E}^{\omega, n}_1 = u_{\omega, n} - u^{\omega, n}$ and $\mathcal{E}^{\omega, n}_2 = \rho_{\omega, n} - \rho^{\omega, n}$. It can be seen that $\mathcal{E}^{\omega, n}_1$ and $\mathcal{E}^{\omega, n}_2$ vanish at $t = 0$ and that they satisfy the equations

Here, $R_1$ and $R_2$ are the approximate solutions for the FW system, that is,

● Estimate for $\|R_1\|_{B^{\gamma}_{p, r}}$: Using the definitions of $u^{\omega, n}$ and $\rho^{\omega, n}$, we have

Then by (3.3),

● Estimate for $\|R_2\|_{B^{\gamma-1}_{p, r}}$: Using the definitions of $u^{\omega, n}$ and $\rho^{\omega, n}$,

So from (3.3) and (3.4), it follows that

Therefore,

Since $\mathcal{E}^{\omega, n}_1(t)$ and $\mathcal{E}^{\omega, n}_2(t)$ satisfy the linear transport equations (3.11), to estimate the error $\|\mathcal{E}^{\omega, n}(t)\|_{B^{\gamma}_{p, r} \times B^{\gamma-1}_{p, r}}$, we apply Proposition 2.4 to obtain

and

where $K_1$, $K_2$ are positive constants depending on $\gamma$ and

Since $\max\{s-\frac{3}{2}, 1+\frac{1}{p}\} < \gamma < s-1$, we have

for some constant $C$ that depends on $\gamma$, $p$, and $r$. By Theorem 2.5 and Eq (3.5), it follows that

Let $K = \max\{K_1, K_2\}$. Using (3.17) and (3.18), we combine (3.13) and (3.14) to obtain

● Estimate for $\|F_1(\tau)\|_{B^{\gamma}_{p, r}}$: From (3.15), as $B^{\gamma}_{p, r}$ is a Banach algebra, we have

From (2.2),

where $M$ is a constant depending on $\gamma, p$, and $r$. By Theorem 2.5, we have

so by (3.5), $\|u_{\omega, n}\|_{B^{\gamma+1}_{p, r}} \leq 2C_{\gamma}\max\{n^{-1}, n^{\gamma+1-s}\}$. As $\gamma > \max\{s-\frac{3}{2}, 1+\frac{1}{p}\}$,

Using (3.21) and (3.22), from (3.20), we obtain

● Estimate for $\|F_2(\tau)\|_{B^{\gamma-1}_{p, r}}$: We may use the algebra property, item (2) of Lemma 2.3, for $B^{\gamma-1}_{p, r}$ since $\gamma -1 > \max\{s-\frac{5}{2}, \frac{1}{p}\} > 0$ and the functions we are dealing with are bounded. Then, from (3.16),

By Corollary 2.6, $\|u_{\omega, n}\|_{B^{\gamma}_{p, r}} \leq 2\|\left(u^0_{\omega, n}, \rho^0_{\omega, n}\right)\|_{B^{\gamma}_{p, r}\times B^{\gamma-1}_{p, r}}$, which implies

by (3.5). As $\gamma < s-1$, $\|u_{\omega, n}\|_{B^{\gamma}_{p, r}} \lesssim n^{-1}$. Using this in (3.24) yields

Adding (3.23) and (3.25) gives

Substituting (3.26) into (3.19), we obtain

where

and

Using Grönwall's inequality, from (3.27) we obtain

Using (3.12) along with (3.28) and (3.29), from (3.30), we obtain

which means that $\|\mathcal{E}^{\omega, n}(t)\|_{B^{\gamma}_{p, r} \times B^{\gamma-1}_{p, r}} \to 0$ as $n\to\infty$ for any $\max\{s-\frac{3}{2}, 1+\frac{1}{p}\} < \gamma < s-1$.

On the other hand, if $\delta \in (s, s+1)$, then noting that the solution with the given data is in $B^{\delta}_{p, r} \times B^{\delta-1}_{p, r}$ for any $\delta$ we have, for $0 < t < T$ (from Theorem 2.5)

where we have used the solution size estimate in Theorem 2.5. Now, for $\delta < s+1$, Eqs (3.5) and (3.6) imply that $\|(u^0_{\omega, n}, \rho^0_{\omega, n})\|_{B^{\delta}_{p, r}\times B^{\delta-1}_{p, r}} \leq C_{\delta}n^{\delta-s}$ and $\|(u^{\omega, n}, \rho^{\omega, n})\|_{B^{\delta}_{p, r}\times B^{\delta-1}_{p, r}} \leq C_{\delta}n^{\delta-s}$, where $C_{\delta}$ denotes a constant that depends only on $\delta$, for a given $p$ and $r$. So (3.32) yields

We use the interpolation property, item (3) from Lemma 2.3, with $\theta = \frac{\delta - s}{\delta - \gamma}$, to obtain

From (3.34), using (3.31) and (3.33), we obtain

As $\theta \in (0, 1)$, (3.35) implies that $\|\mathcal{E}^{\omega, n}(t)\|_{B^{s}_{p, r}\times B^{s-1}_{p, r}} \to 0$ as $n \to \infty$ for any $s > \max\{2+\frac{1}{p}, \frac{5}{2}\}$. This completes the proof of Lemma 3.2. □

When $p = r = 2$, $B^s_{2, 2}$ and $H^s$ are equivalent by [2, Proposition 1.2], and so we obtain the following corollary:

Corollary 3.3. The data-to-solution map for the two-component FW system (1.2) is not uniformly continuous from any bounded subset of $H^s(\mathbb{T})\times H^{s-1}(\mathbb{T})$ into $\mathcal{C}([0, T]; H^s(\mathbb{T}))\times\mathcal{C}([0, T]; H^{s-1}(\mathbb{T}))$ for $s > \frac{5}{2}$.

4.   Conclusions

In this paper, we considered the two-component Fornberg-Whitham (FW) system (1.2) and used a sequential approach to prove that its data-to-solution map is not uniformly continuous for periodic initial data belonging to Besov spaces $B^s_{p, r}(\mathbb{T}) \times B^{s-1}_{p, r}(\mathbb{T})$ where $s > \max\{2+\frac{1}{p}, \frac{5}{2}\}$. As a corollary, this establishes non-uniform dependence on periodic initial data for the FW system (1.2) in Sobolev spaces $H^s(\mathbb{T})\times H^{s-1}(\mathbb{T})$ for $s > \frac{5}{2}$.

Appendix

In this appendix, we provide a lower bound on $\|\cos (nx)\|_{B^{s}_{p, r}}$ for any $s > 0$ and $1\leq r < \infty$. By Definition 2.2,

By Definition 2.1, $\Delta_q \cos(nx) = \varphi_q(n)e^{i n x}$. Therefore, $\|\Delta_q \cos(nx)\|_{{\bf L}^p} = \varphi_q(n)$, where $0 < \varphi_q(n) \leq 1$ for all $q$ such that $\frac{1}{\ln (2)}\ln \left(\frac{3}{8}n\right)\leq q \leq \frac{1}{\ln (2)}\ln \left(\frac{4}{3}n\right)$ and $\varphi_q\left(n\right) = 0$ otherwise, (4.1) implies that

Since $2^q \geq \frac{3}{8}n$ for all terms in the summation, from the above we have

Recall that $\varphi_0(\xi) = \chi\left(\frac{\xi}{2}\right) - \chi(\xi)$ and $\varphi_q(\xi) = \varphi_0(2^{-q}\xi)$ for any $q > -1$, where $\mathrm{supp}\; \chi = [-\frac{4}{3}, \frac{4}{3}]$ and $\chi = 1$ on $[-\frac{3}{4}, \frac{3}{4}]$. This means that $\mathrm{supp}\; \varphi_q = [\frac{3}{4}\cdot 2^q, \frac{8}{3}\cdot 2^q]$ for any $q\geq 1$ and furthermore, $\varphi_q = 1$ on the interval $[\frac{4}{3}\cdot 2^q, \frac{3}{2}\cdot 2^q]$. In other words, $\varphi_q(n) = 1$ for $\frac{1}{\ln (2)}\ln \left(\frac{2}{3}n\right)\leq q \leq \frac{1}{\ln (2)}\ln \left(\frac{3}{4}n\right)$. Therefore, from (4.2) we have

where $C_s$ is a constant that depends only on $s$, for a given $p$ and $r$. The same estimate holds for $\|\sin (nx)\|_{B^{s}_{p, r}}$ as well.

Author contributions

All authors contributed equally towards conceptualization, formal analysis, investigation and methodology in this project; Writing of the original draft was done by Prerona Dutta; thereafter all authors together completed the review and editing process. All authors have read and approved the final version of the manuscript for publication.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

We thank John Holmes, at The Ohio State University, for his valuable suggestions on this project. We would also like to thank the anonymous referees for their comments which greatly helped in improving the paper overall.

Conflict of interest

All authors declare no conflicts of interest in this paper.