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Information theoretic measures of perinatal cardiotocography synchronization

  • We examined the use of bivariate mutual information (MI) and its conditional variant transfer entropy (TE) to address synchronization of perinatal uterine pressure (UP) and fetal heart rate (FHR). We used a nearest-neighbour based Kraskov entropy estimator, suitable to the non-Gaussian distributions of the UP and FHR signals. Moreover, the estimates were robust to noise by use of surrogate data testing. Estimating degree of synchronicity and UP-FHR delay length is useful since they are physiological correlates to fetal hypoxia. Mutual information of the UP-FHR discriminated normal and pathological fetuses early (160 min before delivery) and discriminated normal and metabolic acidotic fetuses slightly later (110 min before delivery), with higher mutual information for progressively pathological classes. The delay in mutual information transfer was also discriminating in the last 50 min of labour. Transfer entropy discriminated normal and pathological cases 110 min before delivery with lower TE values and longer information transfer delays in pathological cases, to our knowledge, the first report of this phenomena in the literature.

    Citation: Philip A. Warrick, Emily F. Hamilton. Information theoretic measures of perinatal cardiotocography synchronization[J]. Mathematical Biosciences and Engineering, 2020, 17(3): 2179-2192. doi: 10.3934/mbe.2020116

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  • We examined the use of bivariate mutual information (MI) and its conditional variant transfer entropy (TE) to address synchronization of perinatal uterine pressure (UP) and fetal heart rate (FHR). We used a nearest-neighbour based Kraskov entropy estimator, suitable to the non-Gaussian distributions of the UP and FHR signals. Moreover, the estimates were robust to noise by use of surrogate data testing. Estimating degree of synchronicity and UP-FHR delay length is useful since they are physiological correlates to fetal hypoxia. Mutual information of the UP-FHR discriminated normal and pathological fetuses early (160 min before delivery) and discriminated normal and metabolic acidotic fetuses slightly later (110 min before delivery), with higher mutual information for progressively pathological classes. The delay in mutual information transfer was also discriminating in the last 50 min of labour. Transfer entropy discriminated normal and pathological cases 110 min before delivery with lower TE values and longer information transfer delays in pathological cases, to our knowledge, the first report of this phenomena in the literature.


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

    {ut+uux=(12x)1x(ρu)ρt+(ρu)x=0(u,ρ)(0,x)=(u0,ρ0)(x) (1.1)

    where xT=R/2πZ, tR+. Here, u=u(x,t) is the horizontal velocity of the fluid, and ρ=ρ(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 Hs(R)×Hs1(R) for s>32 in [11,12].

    Local well-posedness in Besov spaces Bsp,r(R)×Bs1p,r(R) of (1.1) was established in [4] for s>max{2+1p,52}. Besov spaces Bsp,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 Bsp,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 Bsp,r(R)×Bs1p,r(R) for s>max{2+1p,52}.

    In this paper, our objective is to prove non-uniform dependence on periodic initial data for the two-component FW system (1.1) in Bsp,r(T)×Bs1p,r(T) for s>max{2+1p,52}. We work with periodic initial data, as that simplifies our choice of approximate solutions and the resulting estimates. Setting Λ=12x, we rewrite (1.1) as

    {ut+uux=Λ1x(ρu)ρt+uρx+ρux=0(u,ρ)(0,x)=(u0,ρ0)(x) (1.2)

    where xT=R/2πZ and tR+.

    The paper is organized as follows: In Section 2, we recall the standard definitions and properties of Besov spaces, linear transport equations, the operator Λ, 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 Bsp,r(T)×Bs1p,r(T) for s>max{2+1p,52}. 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 Bsp,r(T)×Bs1p,r(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 Hs norm with s>3/2 for any Hölder exponent.

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

    For any x,yR,

    xy denotes xαy for some constant α.

    xy denotes x=βy for some constant β.

    xy denotes xγy for some constant γ.

    We recall the construction of a dyadic partition of unity from [8]. Consider a smooth bump function χ such that suppχ=[43,43] and χ=1 on [34,34]. For ξ>0, set φ1(ξ)=χ, φ0(ξ)=χ(ξ2)χ(ξ) and φq(ξ)=φ0(2qξ). Then, suppφq=[342q,832q] and q1φq(ξ)=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 uD(T) with the Fourier series u(x)=jZˆujeijx where ˆuj=12π2π0eijyu(y)dy, its Littlewood-Paley decomposition is given by u=q1Δqu, where Δqu are periodic dyadic blocks defined for all qZ as

    Δqu=jZφq(j)ˆujeijx.

    Using this Littlewood-Paley decomposition, Besov spaces on T are defined in [3] as follows:

    Definition 2.2 (Besov spaces). Let sR and p, r[1,]. Then the Besov spaces of functions are defined as

    Bsp,rBsp,r(T)={uD(T)|

    where

    \|u\|_{B^s_{p,r} } = \begin{cases} \left(\sum\limits_{q\geq -1}(2^{sq}\|\Delta_q u\|_{{\bf L}^p})^r\right)^{1/r}\; \; \text{if}\; 1\leq r < \infty\\ \sup\limits_{q\geq -1} 2^{sq}\|\Delta_q u\|_{{\bf L}^p}\; \; \; \; \; \; \; \; \; \; \; \; \; \text{if}\; r = \infty \end{cases} \; .

    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

    \|f\|_{B^{\theta s_1 + (1-\theta) s_2}_{p,r}} \leq \|f\|^{\theta}_{B^{s_1}_{p,r}} \|f\|^{1-\theta}_{B^{s_2}_{p,r}} \; .

    (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.

    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

    \begin{equation} \begin{cases} \partial_t f + v \partial_x f = F \\ f(x,0) = f_0(x) \end{cases} \end{equation} (2.1)

    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

    \begin{equation*} \label{transestimate} \|f(t)\|_{B^s_{p,r}}\; \leq\; e^{CV(t)}\left(\|f_0\|_{B^s_{p,r}} + C\int_0^t e^{-CV(\tau)}\|F(\tau)\|_{B^s_{p,r}}\; d\tau\right) \end{equation*}

    where

    V(t) = \int_0^t \|\partial_x v(\tau)\|_{B^{1/p}_{p,r}\cap {\bf L}^{\infty}}\; d\tau\; \; \; \; \text{if}\; s < 1+\frac{1}{p}

    and

    V(t) = \int_0^t \|\partial_x v(\tau)\|_{B^{s-1}_{p,r}}\; d\tau\; \; \; \; {otherwise}\; .

    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 .

    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

    \begin{equation} \|\Lambda^{-1}\partial_xh\|_{B^{s}_{p,r}}\; \leq\; \kappa\|h\|_{B^{s-1}_{p,r}}\; . \end{equation} (2.2)

    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

    T\; = \; \frac{C}{\left(\|u_0\|_{B^s_{p,r}} + \|\rho_0\|_{B^{s-1}_{p,r}}\right)^2}\; ,

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

    \begin{equation*} \label{solnsize} \|(u, \rho)\|_{B^s_{p,r} \times B^{s-1}_{p,r}} \; \leq\; 2\left(\|u_0\|_{B^s_{p,r}} + \|\rho_0\|_{B^{s-1}_{p,r}}\right)\; . \end{equation*}

    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].

    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

    \begin{equation*} \begin{cases} u^{\omega,n} = \frac{-\omega}{n} + \frac{1}{n^s} \sin(nx+\omega t)\\ \rho^{\omega,n} = \frac{1}{n}+\frac{1}{n^{s}} \sin(nx+\omega t) \end{cases}\; . \end{equation*}

    We take initial data

    \begin{equation*} \label{initialdata} \begin{cases} u^0_{\omega,n} = u^{\omega,n}(0) = \frac{-\omega}{n} + \frac{1}{n^s} \sin nx\\ \rho^0_{\omega,n} = \rho^{\omega,n}(0) = \frac{1}{n} + \frac{1}{n^{s}} \sin nx \end{cases}\; . \end{equation*}

    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

    \begin{equation*} \label{zeroinitial} \lim\limits_{n\to\infty} \left(\|u^0_{1,n} - u^0_{-1,n}\|_{B^s_{p,r}} +\|\rho^0_{1,n} - \rho^0_{-1,n}\|_{B^{s-1}_{p,r}}\right) = \lim\limits_{n\to\infty} 2 \|n^{-1}\|_{B^s_{p,r}} = 0\; , \end{equation*}

    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

    \begin{equation} \|(u^0_{\omega,n}, \rho^0_{\omega,n})\|_{B^{\gamma}_{p,r} \times B^{\gamma-1}_{p,r}}\; \leq\; 2\|n^{-1}\|_{B^{\gamma}_{p,r}} + n^{-s}\|\sin nx\|_{B^{\gamma}_{p,r}} + n^{1-s}\|\sin nx\|_{B^{\gamma-1}_{p,r}}. \end{equation} (3.1)

    By Definition 2.2,

    \begin{equation} \|\sin nx\|_{B^{\gamma}_{p,r}} = \left(\sum\limits_{q\geq -1} 2^{\gamma q r}\|\Delta_q \sin nx\|^r_{{\bf L}^p}\right)^{\frac{1}{r}}\; . \end{equation} (3.2)

    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 ,

    \begin{equation*} \|\sin (nx)\|_{B^{\gamma}_{p,r}}\; \leq\; \left(\sum\limits_{q = \frac{1}{\ln (2)}\ln \left(\frac{3}{8}n\right)}^{ \frac{1}{\ln (2)}\ln \left(\frac{4}{3}n\right)} \left(2^q\right)^{\gamma r}\right)^{\frac{1}{r}}\; . \end{equation*}

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

    \begin{align} \|\sin (nx)\|_{B^{\gamma}_{p,r}}\; &\leq\; \left(\sum\limits_{q = \frac{1}{\ln (2)}\ln \left(\frac{3}{8}n\right)}^{ \frac{1}{\ln (2)}\ln \left(\frac{4}{3}n\right)} \left(\frac{4}{3}n\right)^{\gamma r}\right)^{\frac{1}{r}} \\ & = \; \left(\frac{1}{\ln (2)}\left[\ln \left(\frac{4}{3}n\right) - \ln \left(\frac{3}{8}n\right)\right]\right)^{\frac{1}{r}}\left(\frac{4}{3}n\right)^{\gamma} \\ & = \; \left(\frac{1}{\ln (2)} \ln \left(\frac{32}{9}\right)\right)^{\frac{1}{r}}\left(\frac{4}{3}\right)^{\gamma}n^{\gamma}\; = \; C_{\gamma} n^{\gamma}\; . \end{align} (3.3)

    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 ,

    \begin{equation} \|\cos (nx)\|_{B^{\gamma}_{p,r}}\; \leq\; C_{\gamma} n^{\gamma}. \end{equation} (3.4)

    By Definition 2.1,

    \Delta_q n^{-1} = \varphi_q(0)\; n^{-1} = \begin{cases} n^{-1}\; \; \; \; \text{if}\; q = -1 \\ 0\; \; \; \; \; \; \; \text{otherwise} \end{cases}\; .

    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

    \begin{align} \|(u^0_{\omega,n}, \rho^0_{\omega,n})\|_{B^{\gamma}_{p,r} \times B^{\gamma-1}_{p,r}}\; &\leq\; 2^{1-\gamma}n^{-1} + C_{\gamma} n^{\gamma}n^{-s}+C_{\gamma} n^{\gamma-1}n^{1-s} \\ &\leq\; C_{\gamma}\max\{n^{-1},n^{\gamma-s}\}\; . \end{align} (3.5)

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

    \begin{equation} \|(u^{\omega,n},\rho^{\omega,n})\|_{B^{\gamma}_{p,r}\times B^{\gamma-1}_{p,r}}\; \leq\; C_{\gamma}\max\{n^{-1},n^{\gamma-s}\}\; . \end{equation} (3.6)

    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

    \begin{equation} \|u_{1,n} - u_{-1,n}\|_{B^{s}_{p,r}}\; \geq\; \|u^{1,n} - u^{-1,n}\|_{B^{s}_{p,r}} - \|u^{1,n} - u_{1,n}\|_{B^{s}_{p,r}} - \|u^{-1,n} - u_{-1,n}\|_{B^{s}_{p,r}} \end{equation} (3.7)

    and

    \begin{equation} \|\rho_{1,n} - \rho_{-1,n}\|_{B^{s-1}_{p,r}}\; \geq\; \|\rho^{1,n} - \rho^{-1,n}\|_{B^{s-1}_{p,r}} - \|\rho^{1,n} - \rho_{1,n}\|_{B^{s-1}_{p,r}} - \|\rho^{-1,n} - \rho_{-1,n}\|_{B^{s-1}_{p,r}}\; , \end{equation} (3.8)

    adding (3.7) and (3.8) we obtain

    \begin{eqnarray} \|u_{1,n} - u_{-1,n}\|_{B^{s}_{p,r}} + \|\rho_{1,n} - \rho_{-1,n}\|_{B^{s-1}_{p,r}} \geq\; \|u^{1,n} - u^{-1,n}\|_{B^{s}_{p,r}} + \|\rho^{1,n} - \rho^{-1,n}\|_{B^{s-1}_{p,r}} - \|\mathcal{E}^{1,n}(t)\|_{B^s_{p,r}\times B^{s-1}_{p,r}} - \|\mathcal{E}^{-1,n}(t)\|_{B^s_{p,r}\times B^{s-1}_{p,r}} \\ \geq n^{-s}\left(\|\sin(nx+t) - \sin(nx-t)\|_{B^{s}_{p,r}} + \|\sin(nx+t) - \sin(nx-t)\|_{B^{s-1}_{p,r}}\right) \\- 2\|n^{-1}\|_{B^s_{p,r}} - \|\mathcal{E}^{1,n}(t)\|_{B^s_{p,r}\times B^{s-1}_{p,r}} - \|\mathcal{E}^{-1,n}(t)\|_{B^s_{p,r}\times B^{s-1}_{p,r}}\\ = 2n^{-s}\left(\|\cos (nx) \|_{B^{s}_{p,r}}|\sin (t)| + \|\cos (nx) \|_{B^{s-1}_{p,r}}|\sin (t)|\right) - 2^{1-\gamma}n^{-1} - \|\mathcal{E}^{1,n}(t)\|_{B^s_{p,r}\times B^{s-1}_{p,r}} - \|\mathcal{E}^{-1,n}(t)\|_{B^s_{p,r}\times B^{s-1}_{p,r}}\; . \end{eqnarray} (3.9)

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

    \begin{equation} \|\cos (nx)\|_{B^{s}_{p,r}} \geq C_s n^s\; , \end{equation} (3.10)

    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

    \begin{eqnarray} \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)\; \\\geq\; 2 C_s\left(\liminf\limits_{n\to\infty} |\sin (t)| + \liminf\limits_{n\to\infty} n^{-1} |\sin (t)|\right)\; \approx\; |\sin(t)|\; > \; 0\; . \end{eqnarray}

    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

    \begin{equation} \begin{cases} \partial_t \mathcal{E}^{\omega,n}_1 + u^{\omega,n}\partial_x \mathcal{E}^{\omega,n}_1 = -\mathcal{E}^{\omega,n}_1\partial_x u_{\omega, n} + \Lambda^{-1}\partial_x (\mathcal{E}^{\omega,n}_2 - \mathcal{E}^{\omega,n}_1) - R_1\\ \partial_t \mathcal{E}^{\omega,n}_2 + u_{\omega,n}\partial_x \mathcal{E}^{\omega,n}_2 = -\mathcal{E}^{\omega,n}_2\partial_x u_{\omega, n} -\rho^{\omega,n}\partial_x\mathcal{E}^{\omega,n}_1 - \mathcal{E}^{\omega,n}_1\partial_x\rho^{\omega,n} - R_2 \end{cases}\; . \end{equation} (3.11)

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

    \begin{equation*} \label{r1r2defn} \begin{cases} R_1 = \partial_t u^{\omega, n} + u^{\omega, n}\partial_x u^{\omega, n} - \Lambda^{-1}\partial_x(\rho^{\omega, n} - u^{\omega, n}) \\ R_2 = \partial_t \rho^{\omega, n} + \partial_x (\rho^{\omega,n}u^{\omega, n}) \end{cases}\; . \end{equation*}

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

    \begin{eqnarray} R_1 = \partial_t u^{\omega, n} + u^{\omega, n}\partial_x u^{\omega, n} - \Lambda^{-1}\partial_x(\rho^{\omega,n} - u^{\omega,n}) = \frac{1}{2n^{2s-1}}\sin\left(2(nx+\omega t)\right)\; . \end{eqnarray}

    Then by (3.3),

    \begin{equation*} \label{r1est} \|R_1\|_{B^{\gamma}_{p,r}} \leq C_{\gamma}n^{\gamma-2s+1}\; . \end{equation*}

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

    \begin{eqnarray} R_2 = \partial_t \rho^{\omega, n} + \partial_x (\rho^{\omega,n}u^{\omega, n}) = \frac{1}{n^{s}}\cos(nx+\omega t) + \frac{1}{n^{2s-1}}\sin\left(2(nx+\omega t)\right)\; . \end{eqnarray}

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

    \begin{equation*} \label{r2est} \|R_2\|_{B^{\gamma-1}_{p,r}} \leq C_{\gamma} \left(n^{-s}n^{\gamma-1} + n^{1-2s}n^{\gamma-1}\right) \leq C_{\gamma} n^{\gamma-s-1}\; . \end{equation*}

    Therefore,

    \begin{equation} \|R_1\|_{B^{\gamma}_{p,r}} + \|R_2\|_{B^{\gamma-1}_{p,r}}\; \lesssim\; n^{\gamma-s-1}\; . \end{equation} (3.12)

    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

    \begin{equation} \|\mathcal{E}^{\omega,n}_1(t)\|_{B^{\gamma}_{p,r}} \leq K_1e^{K_1V_1(t)}\int_0^t e^{- K_1V_1(\tau)}\|F_1({\tau})\|_{B^{\gamma}_{p,r}} \; d\tau \end{equation} (3.13)

    and

    \begin{equation} \|\mathcal{E}^{\omega,n}_2(t)\|_{B^{\gamma-1}_{p,r}} \leq K_2e^{K_2V_2(t)}\int_0^t e^{- K_2V_2(\tau)}\|F_2({\tau})\|_{B^{\gamma-1}_{p,r}} \; d\tau \end{equation} (3.14)

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

    \begin{equation} F_1(t) = -\mathcal{E}^{\omega,n}_1\partial_x u_{\omega, n} + \Lambda^{-1}\partial_x (\mathcal{E}^{\omega,n}_2 - \mathcal{E}^{\omega,n}_1) - R_1\; , \end{equation} (3.15)
    \begin{equation} \; \; \; \; \; F_2(t) = -\mathcal{E}^{\omega,n}_2\partial_x u_{\omega, n} -\rho^{\omega,n}\partial_x\mathcal{E}^{\omega,n}_1 - \mathcal{E}^{\omega,n}_1\partial_x\rho^{\omega,n} - R_2\; . \end{equation} (3.16)
    \begin{equation*} V_1(t) = \int_0^t \|\partial_x u^{\omega,n}(\tau)\|_{B^{\gamma-1}_{p,r}} \; d\tau\; , \end{equation*}
    \begin{equation*} V_2(t) = \begin{cases} \int_0^t \|\partial_x u_{\omega,n}(\tau)\|_{B^{1/p}_{p,r}\cap{\bf L}^{\infty}} \; d\tau\; \; \; \; \text{if}\; \gamma < 2+\frac{1}{p}\\ \int_0^t \|\partial_x u_{\omega,n}(\tau)\|_{B^{\gamma-2}_{p,r}} \; d\tau\; \; \; \; \; \; \; \; \text{otherwise} \end{cases}\; . \end{equation*}

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

    \begin{equation} V_1(t) \lesssim \; n^{\gamma - s}t \leq\; n^{-1}t \; \; \text{ and} \; \; V_2(t) \leq C\int_0^t \|u_{\omega,n}(\tau)\|_{B^{\gamma}_{p,r}}\; d\tau \end{equation} (3.17)

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

    \begin{equation} V_2(t) \leq 2C \int_0^t \|\left(u^0_{\omega,n}, \rho^0_{\omega,n}\right)\|_{B^{\gamma}_{p,r}\times B^{\gamma-1}_{p,r}}\; d\tau\; \lesssim\; n^{-1}t\; . \end{equation} (3.18)

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

    \begin{equation} \|\mathcal{E}^{\omega,n}_1(t)\|_{B^{\gamma}_{p,r}} + \|\mathcal{E}^{\omega,n}_2(t)\|_{B^{\gamma-1}_{p,r}} \lesssim \int_0^t e^{K(t-\tau)/n}\left(\|F_1({\tau})\|_{B^{\gamma}_{p,r}} + \|F_2(\tau)\|_{B^{\gamma-1}_{p,r}}\right) \; d\tau\; . \end{equation} (3.19)

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

    \begin{align} \|F_1\|_{B^{\gamma}_{p,r}} & \leq \|\mathcal{E}^{\omega,n}_1\|_{B^{\gamma}_{p,r}}\|\partial_x u_{\omega, n}\|_{B^{\gamma}_{p,r}} + \|\Lambda^{-1}\partial_x (\mathcal{E}^{\omega,n}_2 - \mathcal{E}^{\omega,n}_1)\|_{B^{\gamma}_{p,r}}+ \|R_1\|_{B^{\gamma}_{p,r}}\; \\ & \leq \|\mathcal{E}^{\omega,n}_1\|_{B^{\gamma}_{p,r}}\|u_{\omega, n}\|_{B^{\gamma+1}_{p,r}} + \|\Lambda^{-1}\partial_x (\mathcal{E}^{\omega,n}_2 - \mathcal{E}^{\omega,n}_1)\|_{B^{\gamma}_{p,r}}+ \|R_1\|_{B^{\gamma}_{p,r}}\; . \end{align} (3.20)

    From (2.2),

    \begin{equation} \|\Lambda^{-1}\partial_x (\mathcal{E}^{\omega,n}_2 - \mathcal{E}^{\omega,n}_1)\|_{B^{\gamma}_{p,r}} \leq \kappa \|\mathcal{E}^{\omega,n}_2 - \mathcal{E}^{\omega,n}_1\|_{B^{\gamma-1}_{p,r}} \leq M\left(\|\mathcal{E}^{\omega,n}_1\|_{B^{\gamma}_{p,r}} + \|\mathcal{E}^{\omega,n}_2\|_{B^{\gamma-1}_{p,r}}\right) \end{equation} (3.21)

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

    \begin{equation*} \|u_{\omega, n}\|_{B^{\gamma+1}_{p,r}} \leq 2\|\left(u^0_{\omega,n}, \rho^0_{\omega,n}\right)\|_{B^{\gamma+1}_{p,r}\times B^{\gamma}_{p,r}}\; , \end{equation*}

    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}\} ,

    \begin{equation} \|u_{\omega, n}\|_{B^{\gamma+1}_{p,r}} \lesssim n^{\gamma+1-s}\; . \end{equation} (3.22)

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

    \begin{equation} \|F_1(\tau)\|_{B^{\gamma}_{p,r}} \lesssim \left(M + n^{\gamma+1-s}\right)\|\mathcal{E}^{\omega,n}_1(\tau)\|_{B^{\gamma}_{p,r}} + M\|\mathcal{E}^{\omega,n}_2(\tau)\|_{B^{\gamma-1}_{p,r}} + \|R_1(\tau)\|_{B^{\gamma}_{p,r}}\; . \end{equation} (3.23)

    ● 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),

    \begin{align} \|F_2\|_{B^{\gamma-1}_{p,r}} &\leq \|\mathcal{E}^{\omega,n}_2\|_{B^{\gamma-1}_{p,r}}\|\partial_x u_{\omega, n}\|_{B^{\gamma-1}_{p,r}} + \|\rho^{\omega,n}\|_{B^{\gamma-1}_{p,r}}\|\partial_x \mathcal{E}^{\omega,n}_1\|_{B^{\gamma-1}_{p,r}} \\ &\; \; \; \; + \|\partial_x\rho^{\omega,n}\|_{B^{\gamma-1}_{p,r}}\|\mathcal{E}^{\omega,n}_1\|_{B^{\gamma-1}_{p,r}} + \|R_2\|_{B^{\gamma-1}_{p,r}} \\ &\lesssim n^{-1}\|\mathcal{E}^{\omega,n}_1\|_{B^{\gamma}_{p,r}} + \|\mathcal{E}^{\omega,n}_2\|_{B^{\gamma-1}_{p,r}}\|u_{\omega, n}\|_{B^{\gamma}_{p,r}} + \|R_2\|_{B^{\gamma-1}_{p,r}}\; . \end{align} (3.24)

    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

    \|u_{\omega, n}\|_{B^{\gamma}_{p,r}} \leq 2C_{\gamma}\max\{n^{-1}, n^{\gamma-s}\}

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

    \begin{equation} \|F_2(\tau)\|_{B^{\gamma-1}_{p,r}} \lesssim n^{-1}\|\mathcal{E}^{\omega,n}_1(\tau)\|_{B^{\gamma}_{p,r}} + n^{-1}\|\mathcal{E}^{\omega,n}_2(\tau)\|_{B^{\gamma-1}_{p,r}} + \|R_2(\tau)\|_{B^{\gamma-1}_{p,r}}\; . \end{equation} (3.25)

    Adding (3.23) and (3.25) gives

    \begin{align} \|F_1(\tau)\|_{B^{\gamma}_{p,r}} + \|F_2(\tau)\|_{B^{\gamma-1}_{p,r}}\; & \lesssim\; (M+n^{\gamma+1-s})\left(\|\mathcal{E}^{\omega,n}_1(\tau)\|_{B^{\gamma}_{p,r}} + \|\mathcal{E}^{\omega,n}_2(\tau)\|_{B^{\gamma-1}_{p,r}}\right) \\ &\; \; \; \; \; + \|R_1(\tau)\|_{B^{\gamma}_{p,r}} + \|R_2(\tau)\|_{B^{\gamma-1}_{p,r}}\; . \end{align} (3.26)

    Substituting (3.26) into (3.19), we obtain

    \begin{equation} \|\mathcal{E}^{\omega,n}(t)\|_{B^{\gamma}_{p,r} \times B^{\gamma-1}_{p,r}} \; \lesssim\; f(t) + \int_0^t g(\tau)\|\mathcal{E}^{\omega,n}(\tau)\|_{B^{\gamma}_{p,r} \times B^{\gamma-1}_{p,r}}\; d\tau \end{equation} (3.27)

    where

    \begin{equation} f(t) \approx \int_0^t e^{K(t-\tau)/n}\left(\|R_1(\tau)\|_{B^{\gamma}_{p,r}} + \|R_2(\tau)\|_{B^{\gamma-1}_{p,r}}\right)\; d\tau \end{equation} (3.28)

    and

    \begin{equation} g(\tau) \approx (M+n^{\gamma+1-s})e^{K(t-\tau)/n}\; \leq\; (M+1)e^{K(t-\tau)/n}\; . \end{equation} (3.29)

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

    \begin{equation} \|\mathcal{E}^{\omega,n}(t)\|_{B^{\gamma}_{p,r} \times B^{\gamma-1}_{p,r}}\; \lesssim\; f(t) + \int_0^t g(\tau)f(\tau)e^{\int_{\tau}^t g(z)\; dz}\; d\tau\; . \end{equation} (3.30)

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

    \begin{equation} \|\mathcal{E}^{\omega,n}(t)\|_{B^{\gamma}_{p,r}\times B^{\gamma-1}_{p,r}} \lesssim n^{\gamma-s-1}\; , \end{equation} (3.31)

    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)

    \begin{align} \|\mathcal{E}^{\omega,n}(t)\|_{B^{\delta}_{p,r}\times B^{\delta-1}_{p,r}}\; &\; \leq\; \|(u_{\omega,n},\rho_{\omega,n})\|_{B^{\delta}_{p,r}\times B^{\delta-1}_{p,r}}\; +\; \|(u^{\omega,n},\rho^{\omega,n})\|_{B^{\delta}_{p,r}\times B^{\delta-1}_{p,r}} \\ &\; \leq\; 2\|(u^0_{\omega,n},\rho^0_{\omega,n})\|_{B^{\delta}_{p,r}\times B^{\delta-1}_{p,r}}\; +\; \|(u^{\omega,n},\rho^{\omega,n})\|_{B^{\delta}_{p,r}\times B^{\delta-1}_{p,r}}\; , \end{align} (3.32)

    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

    \begin{equation} \|\mathcal{E}^{\omega,n}(t)\|_{B^{\delta}_{p,r}\times B^{\delta-1}_{p,r}}\; \lesssim \; n^{\delta-s}\; . \end{equation} (3.33)

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

    \begin{equation} \|\mathcal{E}^{\omega,n}(t)\|_{B^{s}_{p,r}\times B^{s-1}_{p,r}}\; \leq\; \|\mathcal{E}^{\omega,n}(t)\|^{\theta}_{B^{\gamma}_{p,r}\times B^{\gamma-1}_{p,r}}\|\mathcal{E}^{\omega,n}(t)\|^{1-\theta}_{B^{\delta}_{p,r}\times B^{\delta-1}_{p,r}}\; \; . \end{equation} (3.34)

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

    \begin{equation} \|\mathcal{E}^{\omega,n}(t)\|_{B^{s}_{p,r}\times B^{s-1}_{p,r}} \; \lesssim\; \left(n^{\gamma-s-1}\right)^{\frac{\delta - s}{\delta - \gamma}} \left(n^{\delta-s}\right)^{\frac{s-\gamma}{\delta-\gamma}}\; = \; n^{-\theta}\; . \end{equation} (3.35)

    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} .

    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} .

    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,

    \begin{equation} \|\cos (nx)\|_{B^{s}_{p,r}} = \left(\sum\limits_{q\geq -1} 2^{s q r}\|\Delta_q \cos nx\|^r_{{\bf L}^p}\right)^{\frac{1}{r}}\; . \end{equation} (4.1)

    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

    \begin{equation*} \|\cos (nx)\|_{B^{s}_{p,r}}\; = \; \left(\sum\limits_{q = \frac{1}{\ln (2)}\ln \left(\frac{3}{8}n\right)}^{ \frac{1}{\ln (2)}\ln \left(\frac{4}{3}n\right)} \left(2^q\right)^{sr}\varphi_q^r(n)\right)^{\frac{1}{r}}\; . \end{equation*}

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

    \begin{equation} \|\cos (nx)\|_{B^{s}_{p,r}}\; \geq\; \left(\frac{3}{8}\right)^{s}n^s\left(\sum\limits_{q = \frac{1}{\ln (2)}\ln \left(\frac{3}{8}n\right)}^{ \frac{1}{\ln (2)}\ln \left(\frac{4}{3}n\right)} \varphi_q^r(n)\right)^{\frac{1}{r}} \; . \end{equation} (4.2)

    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

    \begin{align*} \label{app2} \|\cos (nx)\|_{B^{s}_{p,r}}\; & \geq\; \left(\frac{3}{8}\right)^{s}n^s\left(\sum\limits_{q = \frac{1}{\ln (2)}\ln \left(\frac{2}{3}n\right)}^{ \frac{1}{\ln (2)}\ln \left(\frac{3}{4}n\right)} 1\right)^{\frac{1}{r}} \nonumber \\ & = \left(\frac{3}{8}\right)^{s}n^s\left(\frac{1}{\ln (2)}\left[\ln \left(\frac{3}{4}n\right) - \ln \left(\frac{2}{3}n\right)\right]\right)^{\frac{1}{r}} \nonumber \\ & = \left(\frac{1}{\ln (2)} \ln \left(\frac{9}{8}\right) \right)^{\frac{1}{r}}\left(\frac{3}{8}\right)^{s}n^s\; = \; C_s n^s, \end{align*}

    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.

    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.

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

    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.

    All authors declare no conflicts of interest in this paper.



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