On the Fourier coefficients of powers
of a finite Blaschke product
January 31, 2024
Given a finite Blaschke product \(B\) we prove asymptotically sharp estimates on the \(\ell^{\infty}\)-norm of the sequence of the Fourier coefficients of \(B^{n}\) as \(n\) tends to \(\infty\). We provide constructive examples which show that our estimates are sharp. As an application we construct a sequence of \(n\times n\) invertible matrices \(T\) with arbitrary spectrum in the unit disk and such that the quantity \(|\det{T}|\cdot\|T^{-1}\|\cdot\|T\|^{1-n}\) grows as a power of \(n\). This is motivated by Schäffer’s question on norms of inverses.
Let \(\mathbb{D}\) be the unit disk of the complex plane, \(m\ge1\), let \(\sigma=\left(\lambda_{1},\dots,\lambda_{m}\right)\in\mathbb{D}^{m}\) and let
\(B\) be the finite Blaschke product of degree \(m\ge1\) associated to \(\sigma\): \[B(z)=\prod_{j=1}^{m}b_{\lambda_j}(z),\] where \(b_{\lambda}(z)=\frac{z-\lambda}{1-\bar{\lambda}z}\) is the Blaschke factor corresponding to \(\lambda\in\mathbb{D}\). Here and later on we omit the standard unimodular factor \(\bar\lambda/|\lambda|\) which is of no importance for the questions we study here. Furthermore, we assume that
\(0\not\in\sigma\). For \(k\ge0\) we consider the \(k^{{\rm th}}\)-Fourier coefficient of \(B^{n}\) defined by: \[\widehat{B^{n}}(k)=\frac{1}{2\pi}\int_{0}^{2\pi}B^n(e^{{\rm i}\theta})e^{-\rm ik\theta}\,d\theta.\] We study the asymptotic behavior of the \(\ell^{\infty}\)-norm of the sequence \(\bigl(\widehat{B^{n}}(k)\bigr)_{k\ge0}\): \[\|\widehat{B^{n}}\|_{\ell^{\infty}}:=\sup_{k\ge0}|\widehat{B^{n}}(k)|,\] as \(n\) grows large. We use some standard
notation from asymptotic analysis. For two positive functions \(f,g\) we say that \(f\) is dominated by \(g\), denoted by \(f\lesssim g\), if there is a constant \(c>0\) such that \(f\le cg\). We say that \(f\) and \(g\) are comparable, denoted by \(f\asymp g\), if both \(f\lesssim g\) and \(g\lesssim f\).
The case \(B=b_{\lambda}\), \(\lambda\) being arbitrary in \(\mathbb{D}\setminus\{0\}\), is well-studied [1]–[4] and it is known that \[\|\widehat{b_{\lambda}^{n}}\|_{\ell^{\infty}}\asymp n^{-1/3}.\label{eq:basic}\tag{1}\] We mention in passing that Y. Meyer recently rediscovered the upper bound in 1 in view of applying it to
some sparse crystalline measure constructions [4]. Precise asymptotic formulas for the \(k^{{\rm th}}\) Fourier coefficients of
\(b_{\lambda}^{n}\), \(k\in[0,\infty)\) as \(n\rightarrow\infty\) have been recently obtained in [3] where the authors distinguish several regions of different asymptotic behavior of \(\widehat{b_{\lambda}^{n}}(k)\) in terms of \(k\) and \(n.\) These asymptotic formulas are applied to the construction of strongly annular functions with Taylor coefficients satisfying sharp summation properties, which improves and generalizes the results in [1].
Information about those coefficients has also recently been exploited in operator theory [5] to identify decreasing weights for which composition operators are
bounded on weighted Hardy spaces of Hilbert type. Observe also that a connection was earlier established in [6] between the asymptotics of the Fourier coefficients of
\(B^{n}\) and the boundedness of the composition operator \(\mathcal{C}_B\), \(\mathcal{C}_B(f)=f\circ B\) on the analytic Beurling–Sobolev space of analytic
functions in \(\mathbb{D}\) whose sequence of Taylor coefficients belongs to \(l^{p},\) \(p\in[1,\infty]\). Generally speaking, to verify whether \(\mathcal{C}_B\) is a bounded linear operator from one Banach space \(X\) of analytic functions into another, say \(Y\), it is often enough to know the asymptotic
behavior of \(\|B^{n}\|_Y\). It is shown in particular in [6] that for any finite Blaschke product \(B\) \[\|\widehat{B^{n}}\|_{\ell^{p}}\asymp n^{\frac{2-p}{2p}}\;\textrm{for}\;p\in[1,2]\label{eq:BS}\tag{2}\] and that
\[\|\widehat{B^{n}}\|_{\ell^{p}}\gtrsim n^{\frac{2-p}{2p}}\;\textrm{for}\;p\in[2,\infty].\label{eq:BS95Weak}\tag{3}\] The study of the case \(p=1\) in 2 with a single Blaschke factor \(B=b_{\lambda}\) was probably initiated by J.-P. Kahane [7]. Applying van der Corput type estimates on \(\widehat{b_{\lambda}^{n}}(k)\) [7], he proved that
\(\|\widehat{b_{\lambda}^{n}}\|_{\ell^{1}}\asymp n^{-1/2}\) which is in line with 2 . Notice that his motivation [7] was different from the above: he generalized a theorem by Z. K. Leibenson [8], which is a special case of a theorem [9] about homomorphisms of group algebras due to P. T. Cohen. The exact asymptotic \(n\)-dependency of \(\|\widehat{b_{\lambda}^{n}}\|_{\ell^{p}}\) for \(p\in[1,\infty]\) was described in [2] and a change of
asymptotic behavior at \(p=4\) was in particular discovered [2], [3]. In this paper we
focus on the case \(p=\infty\): in this case the estimate 3 reads \[\|\widehat{B^{n}}\|_{\ell^{\infty}}\gtrsim n^{-1/2}\] for any finite Blaschke product \(B\). We will prove that \(\|\widehat{B^{n}}\|_{\ell^{\infty}}\) is never of order \(n^{-1/2}\). More precisely, for any finite Blaschke product \(B\), there exists an integer \(N\ge3\) such that \[\|\widehat{B^{n}}\|_{\ell^{\infty}}\asymp
n^{-1/N}.\label{eq:main95estimate}\tag{4}\] For every integer \(N\ge3\) we will exhibit an example of a Blaschke product \(B\) of degree \(N\)
satisfying estimate 4 . It is known [2] that for \(N=3\) every Blaschke product \(B(z)\not=z\) of degree 1 satisfies 4 . For \(N=5\) – respectively \(N=7\) – we produce explicit examples of Blaschke products
of degree 2 – respectively degree 4 – such that 4 holds.
Finally we apply 4 to construct a new class of \(n\times n\) invertible matrices \(T\) with arbitrary spectrum in \(\mathbb{D}\) such that \[\frac{|\det{T}|\cdot\|T^{-1}\|}{\|T\|^{n-1}}\gtrsim n^{1/N}\label{eq:schaffer95arb95spec}\tag{5}\] for some
integer \(N\ge3.\) This is motivated by Schäffer’s question on the norms of \(n\times n\) invertible matrices, see Section 3.
The paper is organized as follows. In Section 2 we establish sharp estimates on the \(\ell^{\infty}\)-norm of the sequence \((\widehat{B^{n}}(k))_{k\ge0}\) for large \(n\), where \(B\) is an arbitrary Blaschke product, see Theorem 1. Concrete examples of finite Blaschke products achieving estimate 4 are provided in Theorem 2. In Section 3 we state Theorem 3, which extends the work [2], [10], [11] already undertaken to answer a question raised by Schäffer on norms of inverse matrices. In Theorem 3 we exhibit a sequence of \(n\times n\) matrices \(T\) satisfying 5 , which we obtain as an application of the techniques used to prove Theorem 1. Section 4 is devoted to the proof of Theorem 1 which combines the use of two van der Corput lemmata with a stationary phase type argument. In Section 5 we prove Theorem 2 and in particular we construct finite Blaschke products \(B\) of degree 2 – respectively 4 – such that \(\|\widehat{B^{n}}\|_{\ell^\infty}\) grows as \(n^{-1/5}\) – respectively \(n^{-1/7}\). Finally in Section 6 we prove Theorem 3 combining a duality method with the techniques that lead to the proof of Theorem 1. An alternative proof of Theorem 3 using Theorem 1 is provided in Remark 9.
In this section we give sharp asymptotics for the \(\ell^{\infty}\)-norm of the sequence \(\bigl(\widehat{B^{n}}(k)\bigr)_{k\ge0}\), see Theorem 1 below, and constructive examples that achieve these asymptotics, see Theorem 2 below. We denote by \(\psi_{B}(\theta)\) the continuous argument (determined modulo \(2\pi\)) of \(B(e^{{\rm i}\theta})\): \[B(e^{{\rm i}\theta})=\exp\left({\rm i}\psi_{B}(\theta)\right)\] and by \((\xi_\ell)_{\ell=1}^s\) the sequence of (consecutive) zeros of \(\psi_{B}''\) on \([0,2\pi)\) with respective multiplicities \((N_\ell-2)_{\ell=1}^s\), \(N_\ell\ge 3\). This means that \[0\le \xi_{1}<\xi_{2}<\ldots<\xi_{s}<2\pi\] and that for any \(\ell=1,\ldots, s\) \[\psi''_{B}(\xi_\ell)=\ldots=\psi_{B}^{(N_\ell-1)}(\xi_\ell)=0,\qquad\psi_{B}^{(N_\ell)}(\xi_\ell)\not=0.\] Without loss of generality (using a simple rotation if necessary), we can assume that \(\xi_1>0\).
1. Let \(B\) be a finite Blaschke product, and let \(\psi_{B}\), \((\xi_\ell)_{\ell=1}^{s}\), \((N_\ell)_{\ell=1}^{s}\) be defined as above. Then we have \[\|\widehat{B^n}\|_{\ell^\infty}\asymp n^{-1/N},\] where \(N=\max_{1\le \ell\le s}N_\ell\).
It is known [3] that if \(B=b_{\lambda}\) is the Blaschke factor associated to any fixed \(\lambda\in\mathbb{D}\setminus\{0\},\) then \(N=3\) and \[\|\widehat{B^{n}}\|_{\ell^{\infty}}\asymp n^{-1/3},\] which proves the sharpness of Theorem 1 for \(N=3\). Here we establish the following result.
2. For every \(N\ge3\), there exists a Blaschke product \(B\) of degree \(N\) such that \[\|\widehat{B^{n}}\|_{\ell^\infty}\asymp n^{-1/N}.\label{B}\qquad{(1)}\] Moreover:
(1) For \(N=3\) every Blaschke product of degree 1, different from identity, satisfies ?? .
(2) For \(N=5\) there exists a Blaschke product of degree 2 such that ?? holds.
(3) For \(N=7\) there exists a Blaschke product of degree 4 such that ?? holds.
1. The following two questions remain open:
(1) Every nontrivial Blaschke product \(B\) of degree \(1\) satisfies ?? with \(N=3\). In the case \(N=5\), a Blaschke
product of degree \(2\) satisfies ?? , and in the case \(N=7\), a Blaschke product of degree \(4\) satisfies ?? . What is the minimal degree of a Blaschke
product \(B_{N}\) such that ?? holds for a fixed \(N\ge 6\)?
(2) Is there an infinite Blaschke product \(B\) such that for \(n\ge1\) we have \[\|\widehat{B^{n}}\|_{\ell^\infty}\asymp\frac{1}{\log(n+1)}\,?\]
A well-established circle of questions in Operator Theory concerns estimating the norm of the inverse of an invertible \(n\times n\) matrix \(T\) on an \(n\)-dimensional Banach space \(X\). In the early 1970’s, B. L. Van der Waerden, W. A. Coppel and J. J. Schäffer [12] studied the smallest quantity \(C\), denoted \(C_{\mathcal{B}}(n)\), such that \[|\det{T}|\cdot\|T^{-1}\|\le C
\|T\|^{n-1}\] for any invertible \(T\) and for any \(X\). In 1970, Schäffer [12] proved the
estimate \(C_{\mathcal{B}}(n)\le\sqrt{en}\), and showed that the inequality \[|\det T|\cdot\|T^{-1}\|\le
2\|T\|^{n-1}\label{eq:l1linfty-1}\tag{6}\] holds for any invertible \(T\) acting on \(\mathbb{C}^{n}\) endowed with the \(\ell^{1}\)-norm or
with the \(\ell^{\infty}\)-norm (and that 6 is optimal in both cases). This led him to make the conjecture – nowadays known as Schäffer’s conjecture – that \(C_{\mathcal{B}}(n)=2\), \(n\in\mathbb{N}\). The latter was disproved first by E. Gluskin, M. Meyer and A. Pajor [13], J. Bourgain [13] 2 and later by H. Queffélec
[14] who also proved that \(C_{\mathcal{B}}(n)\gtrsim\sqrt{n}\). The above mentioned results make use of the following analytic
expression for \(C_{\mathcal{B}}(n)\), given in [13], in terms of a “max-min-type” optimization problem : \[C_{\mathcal{B}}(n)=\sup_{(\lambda_1,\ldots,\lambda_n)\in\mathbb{D}^{n}}\Phi(\lambda_1,\ldots,\lambda_n),\] where \[\begin{gather}
\Phi(\lambda_1,\ldots,\lambda_n)\\:=\inf\biggl\{ \sum_{k=1}^{\infty}|a_k|: f(z)=\prod_{j=1}^{n}\lambda_j+\sum_{k=1}^{\infty}a_kz^k,\,f(\lambda_j)=0,\,j=1,\ldots, n\biggr\} .
\end{gather}\] The proof by Queffélec makes use of Bourgain’s lower bound on \(\Phi\) [11], which he combines with a number
theoretic argument to prove the existence of a family \((\lambda_1,\ldots,\lambda_n)\) on the circle of radius \(1-1/n\) such that \(\Phi(\lambda_1,\ldots,\lambda_n)\gtrsim\sqrt{n}\).
The question of finding a concrete exemple of a sequence \((\lambda_1,\ldots,\lambda_n)\) in \(\mathbb{D}^n\) such that \(\Phi(\lambda_1,\ldots,\lambda_n)\gtrsim\sqrt{n}\) has been recently addressed by O. Szehr–R. Zarouf [11] who proved that for every \(\lambda\in\mathbb{D}\setminus\{0\}\) we have \(\Phi(\lambda,\ldots,\lambda)\gtrsim\sqrt{n}\), where \(\lambda\) is repeated \(n\) times. As a consequence, they construct in [11] an explicit class of counterexamples to Schäffer’s conjecture: a sequence of
invertible lower triangular Toeplitz matrices \(T_{\lambda}\in\mathcal{M}_{n}\) with singleton spectrum \(\{\lambda\}\subset\mathbb{D}\backslash\{0\}\) such that \[|\lambda|^{n}\|T_\lambda^{-1}\|\ge c(\lambda)\sqrt{n}\|T_\lambda\|^{n-1},\] where \(c(\lambda)>0\) depends only on \(\lambda\).
Furthermore, Gluskin–Meyer–Pajor [13] mention that it is of interest to find concrete examples \((\lambda_1,\ldots,\lambda_n)\) for which \(\Phi(\lambda_1,\ldots,\lambda_n)\) grows. We will combine the approach in [11] with Theorem 1 to prove that given a fixed \(m\ge1\) and an arbitrary sequence \((\lambda_1,\lambda_2,\ldots,\lambda_m)\in\mathbb{D}^{m}\), the sequence \[(\lambda_1,\ldots,\lambda_1,\lambda_2,\ldots,\lambda_2,\ldots,\lambda_m,\ldots,\lambda_m)\in\mathbb{D}^{n\times m}\] (i.e. each \(\lambda_{i}\) is repeated according to its multiplicity \(n\)) satisfies the estimate \[\Phi(\lambda_1,\ldots,\lambda_1,\lambda_2,\ldots,\lambda_2,\ldots,\lambda_m,\ldots,\lambda_m)\gtrsim n^{1/N}\] for some integer \(N\ge3\), as \(n\) tends to infinity. As a consequence, we obtain a family of invertible lower triangular matrices \(T\in\mathcal{M}_{nm}\) with arbitrary spectrum \((\lambda_1,\lambda_2,\ldots,\lambda_m)\) such that \[|\det T|\cdot\|T^{-1}\|\gtrsim n^{1/N}\|T\|^{n-1}\] for some integer \(N\ge3\), as \(n\) tends to infinity. Our construction of \(T\) is given in Section 3.2.
We denote by \(H(\mathbb{D})\) the space of the functions analytic in the open unit disk \(\mathbb{D}\), and by \(H^{\infty}\) the Banach algebra of bounded analytic functions in \(\mathbb{D}\), endowed with the supremum norm. We recall that the Hardy space \(H^{2}\) is defined as the subspace of \(H(\mathbb{D})\) consisting of the functions \(f\) such that \[\Vert f\Vert_{H^{2}}^{2}:=\sup_{0\le r<1}\int_{\mathbb{T}}\left|f(rz)\right|^{2}\,d\nu(z)<\infty,\] where \(\nu\) is the normalized Lebesgue measure on the unit circle \(\mathbb{T}:=\left\{ z\in\mathbb{C}:\,|z|=1\right\}\). Let \(\sigma\) be a finite sequence of points in \(\mathbb{D}.\) The finite Blaschke product \(B=B_{\sigma}\) corresponding to \(\sigma\) is defined by \[B=B_{\sigma}=\prod_{\lambda\in\sigma}b_{\lambda},\] where \(b_{\lambda}=\frac{z-\lambda}{1-\overline{\lambda}z}\) is the Blaschke factor corresponding to \(\lambda\in\mathbb{D}\). Then one defines the model space \(K_{B}\) as the finite dimensional subspace of \(H^{2}\) given by \[K_B:=\left(BH^{2}\right)^{\perp}=H^{2}\ominus BH^{2}.\] Let \(\sigma=(\lambda_{1},\dots,\lambda_{q})\in\mathbb{D}^q\). We set \(f_k=\dfrac{1}{1-\overline{\lambda_{k}}z}\), \(k=1,\ldots,q\). Observe that \(\|f_k\|_{H^{2}}=\left(1-\vert\lambda_k\vert^{2}\right)^{-1/2}\). The family \((f_k)_{1\le k\le n}\) is a basis of \(K_B\). Furthermore, the family \((e_k)_{1\le k\le n}\) given by \[e_1=\frac{f_1}{\|f_1\|_{H^2}}\,\quadand\quad e_{k}=\frac{f_{k}}{\|f_k\|_{H^2}}{\displaystyle \prod_{j=1}^{k-1}}b_{\lambda_{j}},\quad k=2,\ldots,q,\] is an orthonormal basis of \(K_B\) (known as the Malmquist–Walsh basis, see [15]).
The backward shift operator \(S: f\mapsto (f-f(0))/z\) acts on \(K_B\).
We put \[\sigma=(\lambda_{1},\dots,\lambda_{1},\lambda_{2},\dots,\lambda_{2},\dots,\lambda_{m},\dots,\lambda_{m})\in\mathbb{D}^{n m},\] where distinct \(\lambda_i\) are arbitrary in \(\mathbb{D}\backslash\{0\}\) and are repeated \(n\) times. We consider the Malmquist–Walsh basis of \(K_B\) given by \[e_{sn+t}=\biggl(\prod_{j=1}^sb^n_{\lambda_j} \biggr)\frac{f_{s+1}}{\|f_{s+1}\|_{H^2}}b_{\lambda_{s+1}}^{t-1},\qquad 0\le s<m,\,1\le t\le n.\]The backward shift operator \(S\) has a lower triangular matrix with respect to this basis, with diagonal \((\lambda_{1},\dots,\lambda_{1},\dots,\lambda_{m},\dots,\lambda_{m})\), see [16] for the entry-wise description of this matrix. Hence, \[\label{det1} \det S= \prod_{j=1}^m\lambda^n_j.\tag{7}\] We will use the duality method from [11] and combine it with our upper estimates on the Fourier coefficients of the \(n^{\text{th}}\) power of a finite Blaschke product \(B\) (see Theorem 1) to show that there exists a Banach space norm on \(K_B\) such that the operator \(S\) with spectrum \((\lambda_1,\lambda_2,\ldots,\lambda_m)\) acting there, satisfies the asymptotic relation \(|\det S|\cdot \|S^{-1} \|\cdot \| S \|^{1-n} \gtrsim n^{1/N}\).
Following [13], we choose the norm \(\|\cdot\|_{\ell^\infty_A}\) on \(K_B\): \[\|f\|_{\ell^\infty_A}=\|\widehat{f}\|_{\ell^\infty}.\] Then \(\|S\|_{\ell^\infty_A\to \ell^\infty_A}\le 1\). Furthermore, if \(\dim K_B>1\), then \(\|S\|_{\ell^\infty_A\to \ell^\infty_A}=1\). We are now ready to state the corresponding result.
3. Let \[\sigma=(\lambda_1,\ldots,\lambda_1,\lambda_2,\ldots,\lambda_2,\ldots,\lambda_m,\ldots,\lambda_m)\in\mathbb{D}^{n m},\] where distinct \(\lambda_i\) are arbitrary in \(\mathbb{D}\backslash\{0\}\) and are repeated \(n\) times. There exists an integer \(N \ge 3\) such that
\(|\det S|\cdot \|S^{-1}\|_{\ell^\infty_A\to \ell^\infty_A}
\gtrsim {n^{1/N}}\)
and
\(\Phi(\lambda_1, \ldots, \lambda_1, \ldots, \lambda_m,\ldots, \lambda_m) \gtrsim n^{1/N}\).
Here \(N\) is the integer associated with the Blaschke product \(B= \prod_{j=1}^m b_{\lambda_j}\) in Theorem 1.
The proof of Theorem 1 makes use of the following van der Corput lemmata, see, for example, [17].
4. Let \(g\) be a real function continuously differentiable on the interval \([a,b]\subset\mathbb{R}\) such that \(g\) and \(g'\) are monotone and \(g'\) does not vanish on \([a,b].\) Then \[\left|\int_{a}^{b}e^{{\rm i}g(t)}\,dt\right|\le\frac{2}{|g'(a)|}+\frac{2}{|g'(b)|}.\]
5. Let \(F(x)\) be a real, twice differentiable function in \([a,b]\) such that \(F''(x)\ge\mu>0\) or \(F''(x)\le-\mu<0\). Let \(G(x)\) be a positive monotonic function in \([a,b]\) such that \(G(x)\le M\). Then \[\left|\int_{a}^{b}G(x)e^{{\rm i}F(x)}\,dx\right|\le\frac{8M}{\sqrt{\mu}}.\]
Proof of Theorem 1.. For \(j=1,\ldots,m\) we write \(\lambda_j=\rho_{j}\exp({\rm i}\theta_j)\), where \(\rho_{j}\in(0,1)\) and \(\theta_{j}\in(-\pi,\pi]\). A direct computation shows that \[\psi'_{B}(\theta)=|B'(e^{{\rm i}\theta})|=\Bigl|\frac{B'(e^{{\rm i}\theta})}{B(e^{{\rm i}\theta})}\Bigr|=\biggl|\sum_{1\le j\le m}\frac{b'_{\lambda_j} (e^{{\rm i}\theta})}{b_{\lambda_j}(e^{{\rm i}\theta})}\biggr|.\] Since \[\frac{zb'_{\lambda}(z)}{b_{\lambda}(z)}=\Re\frac{1+\overline{\lambda}z}{1-\overline{\lambda}z}\ge0,\qquad\lambda\in\mathbb{D},\,z\in\mathbb{T},\] we have \[\psi'_{B}(\theta)=\Re\sum_{1\le j\le m}\frac{1+\overline{\lambda_{j}}e^{{\rm i}\theta}}{1-\overline{\lambda_j}e^{{\rm i}\theta}}.\label{eq1}\tag{8}\]
Therefore, \[\psi_{B}'(\theta)=\sum_{j=1}^{m}\frac{1-\rho_{j}^{2}}{1+\rho_{j}^{2}-2\rho_{j}\cos\left(\theta-\theta_{j}\right)}.\] The \(k^{th}\)-Fourier coefficient of \(B^{n}\) is given by \[\widehat{B^{n}}(k)=\frac{1}{2\pi}\int_{0}^{2\pi}\exp\left({\rm i}\left(n\psi_{B}(\theta)-k\theta\right)\right)\,d\theta.\] We put \[f(\theta)=\psi_{B}(\theta)-\frac{k}{n}\theta,\] and given \(\varepsilon>0\) we write (recall that \(\xi_{1}>0\)) \[\begin{gather} \int_{0}^{2\pi}\exp\left({\rm i}n\, f(\theta)\right){\rm d}\theta =\sum_{\ell=1}^{s}\int_{\xi_{\ell}-\varepsilon}^{\xi_{\ell}+\varepsilon}\exp\left({\rm i}n\,f(\theta)\right)\,d\theta \\+\sum_{\ell=1}^{s-1}\int_{\xi_{\ell}+\varepsilon}^{\xi_{\ell+1}-\varepsilon}\exp\left({\rm i}n\,f(\theta)\right)\,d\theta +\int_{0}^{\xi_{1}-\varepsilon}\exp\left({\rm i}n\,f(\theta)\right)\,d\theta\\ +\int_{\xi_{s}+\varepsilon}^{2\pi}\exp\left({\rm i}n\,f(\theta)\right)\,d\theta. \end{gather}\] We put \[\varepsilon=n^{-1/N}\rightarrow0,\qquad n\rightarrow\infty.\] Clearly \[\biggl|\int_{\xi_{\ell}-\varepsilon}^{\xi_{\ell}+\varepsilon}\exp\left({\rm i}n\,f(\theta)\right)\,d\theta\biggr|\le2\varepsilon,\] and therefore the first sum is bounded from above by \[\biggl|\sum_{\ell=1}^{s}\int_{\xi_{\ell}-\varepsilon}^{\xi_{\ell}+\varepsilon}\exp\left({\rm i}n\,f(\theta)\right)\,d\theta\biggr|\lesssim n^{-1/N}.\]
Writing the Taylor expansion (of order \(N_{\ell}-2\)) of \(\psi_{B}''\) near \(\xi_{\ell}\) we obtain that \[\psi_{B}''(\theta)=\frac{\psi_{B}^{(N_{\ell})}(\xi_{\ell})}{(N_{\ell}-2)!}\left(\theta-\xi_{\ell}\right)^{N_{\ell}-2}\left(1+\mathcal{O}\left(\theta-\xi_{\ell}\right)\right),\qquad\theta\to\xi_{\ell}.\] Therefore, if \(\theta\) is close to \(\xi_{\ell}\) but satisfies \(|\theta-\xi_{\ell}|\ge\varepsilon\), then \[|\psi_{B}''(\theta)|\gtrsim n^{-(N_{l}-2)/N}\label{eq2}.\tag{9}\]
Since \(\psi_{B}''\) does not vanish on the intervals \((0,\xi_{1}-\varepsilon)\), \((\xi_{s}+\varepsilon,2\pi)\) and \((\xi_{\ell}+\varepsilon,\xi_{\ell+1}-\varepsilon)\) for \(\ell=1,\ldots,s-1\), we obtain that 9 holds everywhere outside of the intervals \([\xi_{\ell}-\varepsilon,\xi_{\ell}+\varepsilon]\). Now we can apply Lemma 5 to the remaining integrals \[\int_{\xi_{\ell}+\varepsilon}^{\xi_{\ell+1}-\varepsilon}e^{{\rm i}n\,f(\theta)}\,d\theta, \qquad \int_{0}^{\xi_{1}-\varepsilon}e^{{\rm i}n\,f(\theta)}\,d\theta,\qquad \int_{\xi_{s}+\varepsilon}^{2\pi}e^{{\rm i}n\,f(\theta)}\,d\theta.\] Since \[|nf''|\gtrsim n^{2/N}\] on these intervals, these integrals are also \(\mathcal{O}\left(n^{-1/N}\right)\). Thus, \[\|\widehat{B^n}\|_{\ell^\infty}\lesssim n^{-1/N}.\]
Let \(r\in\{1,\ldots,s\}\) be such that \(N_r=N\) and define \[\mathcal{L}=\bigl\{\ell:1\le \ell\le s,\, N_\ell=N,\, \psi_{B}'(\xi_\ell)=\psi_{B}'(\xi_r)\bigr\},\quad D=\mathop{\rm Card}\mathcal{L}.\]
We will show that the maximal value of \(|\widehat{B^{n}}(k)|\) for \(k\ge0\) is (asymptotically as \(n\) grows large) attained at \(k=k_d=k_d(n)=[n\psi_{B}'(\xi_r)]+d\), \(0\le d<D\) (where \(\left[A\right]\) means the integer part of \(A\)), that is, \[\max_{0\le d<D}|\widehat{B^{n}}(k_d(n))|\asymp n^{-1/N}.\] Recall that the \(k_d^{th}\)-Fourier coefficient of \(B^{n}\) is given by \[\widehat{B^n}(k_d) =\frac{1}{2\pi}\int_{0}^{2\pi}\exp\left({\rm i}n\,f_d(\theta)\right)\,d\theta,\] where \[f_d(\theta)=\psi_{B}(\theta)-\frac{k_d}{n}\theta.\] Let \((\varepsilon_{\ell})_{\ell=1}^{s}\) be a sequence of nonnegative numbers. Each \(\varepsilon_\ell\) for \(\ell=1,\ldots, s\) will be chosen below over the proof, depending on the nature of \(\xi_{\ell}\). Let \(\tau_0=0\), \(\tau_\ell\in(\xi_\ell+\varepsilon_{\ell},\xi_{\ell+1}-\varepsilon_{\ell+1})\), \(1\le\ell\le s-1\), \(\tau_s=2\pi\), \(J_\ell=[\xi_\ell-\varepsilon_{\ell},\xi_{\ell}+\varepsilon_{\ell}]\), \(1\le\ell\le s\). Then \[[0,2\pi)\setminus\bigsqcup_{1\le\ell\le s}J_\ell=\bigsqcup_{1\le\ell\le s}[\tau_{\ell-1},\xi_\ell-\varepsilon_{\ell})\sqcup (\xi_{\ell}+\varepsilon_{\ell},\tau_\ell)=:\bigsqcup_{1\le\ell\le s}(J'_\ell\sqcup J''_\ell).\]
We have \[\int_{0}^{2\pi}e^{{\rm i}n\,f(\theta)}\,d\theta =\sum_{1\le\ell\le s}\int_{J_\ell}e^{{\rm i}n\,f(\theta)}\,d\theta+\sum_{1\le\ell\le s} \int_{J'_\ell\sqcup J''_\ell}e^{{\rm i}n\,f(\theta)}\,d\theta.\] We divide our argument into three steps in order to show that the main contribution of \(\int_{0}^{2\pi}\exp\left({\rm i}n\,f(\theta)\right)\,d\theta\) is due to \(J_\ell\), \(\ell\in\mathcal{L}\).
The content of Step 1 is computing an asymptotic formula for \(\Sigma=\Sigma_d\), the sum of the integrals \[I_{\ell,d}=\int_{J_\ell}\exp\left({\rm i}n\,f_d(\theta)\right)\,d\theta,\qquad
\ell\in\mathcal{L},\] with \(0\le d<D\), for suitable choices of \(\varepsilon_\ell\). In Step 2 we estimate from above the integrals \[\int_{J_\ell}\exp\left({\rm i}n\,f(\theta)\right)\,d\theta,\] for \(1\le \ell\le s\), \(\ell\not\in\mathcal{L}\), and show that they are asymptotically much
smaller than \(\Omega=\max_{0\le d<D}|\Sigma_d|\), again for suitable choices of \(\varepsilon_{\ell}\). Finally the goal of Step 3 is to show – in the same spirit as in Step 2 – that the
integrals \[\int_{J'_\ell}\exp\left({\rm i}n\,f(\theta)\right)\,d\theta,\qquad \int_{J''_\ell}\exp\left({\rm i}n\,f(\theta)\right)\,d\theta,\] are also asymptotically much smaller than \(\Omega\).
Step 1. To make the notation less cluttered we set \(\xi=\xi_\ell\), \(F=F_d=nf_d\), \(\varepsilon=\varepsilon_\ell\), so that \[I=I_\ell=I_{\ell,d}=\int_{\xi-\varepsilon}^{\xi+\varepsilon}\exp\left({\rm i}F_d(\theta)\right)\,d\theta.\] Without loss of generality we may assume that \[\psi_B^{(N)}(\xi)>0.\] In the
opposite case the argument is analogous.
Furthermore, we fix \(\delta\in(0,1/(2N^2))\) and set \[\varepsilon=n^{\delta-\frac{1}{N}}.
\label{st1}\tag{10}\] This choice of \(\varepsilon\) is motivated below in the proofs of formulas 11 and 12 .
We will first establish that:
(a) If \(N\) is even, then \[\begin{gather} I_{\ell,d}=\frac{2}{N}\left(\frac{N!}{n\psi_{B}^{(N)}(\xi)}\right)^{1/N}\times \\ \times \exp\left({\rm i}n\psi_{B}(\xi)-{\rm i}k_d\xi+\frac{{\rm i}\pi}{2N}\right)\Gamma(1/N)+o\left(\frac{1}{n^{1/N}}\right).\label{eq:I95N95even} \end{gather}\tag{11}\]
(b) If \(N\) is odd, then \[\begin{gather} I_{\ell,d}=\frac{2}{N}\left(\frac{N!}{n\psi_{B}^{(N)}(\xi)}\right)^{1/N}\times \\ \times\exp\left({\rm i}n\psi_{B}(\xi)-{\rm i}k_d\xi\right)\cos\left(\frac{\pi}{2N}\right)\Gamma(1/N)+o\left(\frac{1}{n^{1/N}}\right).\label{eq:I95N95odd} \end{gather}\tag{12}\]
Here and later on, \(\Gamma\) is the Gamma function.
Writing the Taylor expansion of \(\psi_{B}\) near \(\xi\) we obtain that \[\begin{gather} \psi_{B}(\theta)=\psi_{B}(\xi)+(\theta-\xi)\psi_{B}'(\xi)\\+\frac{(\theta-\xi)^{N}}{N!}\psi_{B}^{(N)}(\xi)+\frac{(\theta-\xi)^{N+1}}{(N+1)!}\psi_{B}^{(N+1)}(\xi+t_\theta(\theta-\xi)), \end{gather}\] for some \(t_\theta\in(0,1)\). Therefore, \[\begin{gather} \psi_{B}(\theta)-\frac{k}{n}\theta =\psi_{B}(\xi)-\frac{k}{n}\xi+(\theta-\xi)\left(\psi_{B}'(\xi)-\frac{k}{n}\right)+\frac{(\theta-\xi)^{N}}{N!}\psi_{B}^{(N)}(\xi)\\ +\frac{(\theta-\xi)^{N+1}}{(N+1)!}\psi_{B}^{(N+1)}(\xi+t_\theta(\theta-\xi)), \end{gather}\] and \[\begin{gather} F(\theta)=F(\xi)+(\theta-\xi)F'(\xi)+\frac{(\theta-\xi)^{N}}{N!}F^{(N)}(\xi)\\+\frac{(\theta-\xi)^{N+1}}{(N+1)!}F^{(N+1)}(\xi+t_\theta(\theta-\xi)). \end{gather}\] Thus, going back to the integral \(I\) we obtain that if \(n\) is sufficiently large, then \[\begin{gather} I =\exp\left({\rm i}F(\xi)\right)\cdot\int_{\xi-\varepsilon}^{\xi+\varepsilon} \exp\left({\rm i}\frac{(\theta-\xi)^{N}}{N!}F^{(N)}(\xi)\right)\times \\ \times\exp\left({\rm i}(\theta-\xi)F'(\xi)+{\rm i}\frac{(\theta-\xi)^{N+1}}{(N+1)!}F^{(N+1)}(\xi+t_\theta(\theta-\xi))\right)\,d\theta. \end{gather}\] Furthermore, \[\begin{gather} \exp\left({\rm i}(\theta-\xi)F'(\xi)+{\rm i}\frac{(\theta-\xi)^{N+1}}{(N+1)!}F^{(N+1)}(\xi+t_\theta(\theta-\xi))\right)\\ =1+\mathcal{O}\left(|(\theta-\xi)F'(\xi)|+\frac{|\theta-\xi|^{N+1}}{(N+1)!}|F^{(N+1)}(\xi+t_\theta(\theta-\xi))|\right)\\ =1+\mathcal{O}\left(n|\theta-\xi|\left(|f'(\xi)|+|\theta-\xi|^N\right)\right), \end{gather}\] for \(\theta\) in a small neighborhood of \(\xi\). This gives \[\begin{gather} \exp\left(-{\rm i}F(\xi)\right)I \\=\int_{\xi-\varepsilon}^{\xi+\varepsilon}\exp\left({\rm i}\frac{(\theta-\xi)^{N}}{N!}F^{(N)}(\xi)\right)\,d\theta+\mathcal{O}\left(n\varepsilon^2 |f'(\xi)|+n\varepsilon^{N+2}\right).\label{eq:I951st95decomp} \end{gather}\tag{13}\]
Let us verify that with our choice of \(\varepsilon\) we have
\(n\varepsilon^{2}|f'(\xi)|=o\left(n^{-1/N}\right)\),
\(n\varepsilon^{N+2}=o\left(n^{-1/N}\right)\),
\(n\varepsilon^{N}\rightarrow\infty\) as \(n\rightarrow\infty\).
Since \(k\le n\psi_{B}'(\xi)<k+D\), we have \(n|f'(\xi)|\lesssim 1\). Now, (i)–(iii) follow from 10 .
Thus, \[n\varepsilon^2 |f'(\xi)|+n\varepsilon^{N+2}=o(n^{-1/N}).\label{eq:O95term95I}\tag{14}\]
Next we consider \[\begin{gather} J =\int_{\xi-\varepsilon}^{\xi+\varepsilon}\exp\left({\rm i}\frac{(\theta-\xi)^{N}}{N!}F^{(N)}(\xi)\right)\,d\theta\\ \\=\int_{\xi}^{\xi+\varepsilon}\exp\left({\rm i}\frac{(\theta-\xi)^{N}}{N!}F^{(N)}(\xi)\right)\,d\theta+\int_{\xi-\varepsilon}^{\xi}\exp\left({\rm i}\frac{(\theta-\xi)^{N}}{N!}F^{(N)}(\xi)\right)\,d\theta\\ \\=J_{1}+J_{2}. \end{gather}\] First we estimate \(J_1\). Changing the variable \[u=\frac{(\theta-\xi)^{N}}{N!}F^{(N)}(\xi),\] we obtain that
\[\begin{gather} J_{1} =\frac{1}{N}\left(\frac{N!}{F^{(N)}(\xi)}\right)^{1/N}\int_{0}^{\frac{\varepsilon^{N}}{N!}F^{(N)}(\xi)}\frac{\exp\left({\rm i}u\right)}{u^{1-1/N}}\,du\\ \\=\frac{1}{N}\left(\frac{N!}{F^{(N)}(\xi)}\right)^{1/N}\left(\int_{0}^{\infty}\frac{\exp\left({\rm i}u\right)}{u^{1-1/N}}\,du-\int_{\frac{\varepsilon^{N}}{N!}F^{(N)}(\xi)}^{\infty}\frac{\exp\left({\rm i}u\right)}{u^{1-1/N}}\,du\right)\\ \\ \!=\frac{1}{N}\left(\frac{N!}{F^{(N)}(\xi)}\right)^{1/N}\!\!\left(\exp\left(\frac{{\rm i}\pi}{2N}\right)\Gamma(1/N)+\mathcal{O}\left(\frac{1}{(\varepsilon^{N}F^{(N)}(\xi))^{1-1/N}}\right)\right). \end{gather}\] Here we use that \[\int_{0}^{\infty}\frac{\exp\left({\rm i}u\right)}{u^{1-1/N}}\,du=\exp\left(\frac{{\rm i}\pi}{2N}\right)\Gamma(1/N),\] and (via integration by parts, with \(\gamma>0\)) that \[\begin{gather} -\int_\gamma^{\infty}\frac{\exp\left({\rm i}u\right)}{u^{1-1/N}}\,du =\left[{\rm i}\frac{\exp\left({\rm i}u\right)}{u^{1-1/N}}\right]_\gamma^{\infty}+\left(1-\frac{1}{N}\right){\rm i}\int_{\gamma}^{\infty}\frac{\exp\left({\rm i}u\right)}{u^{2-1/N}}\,du \\ =\mathcal{O}\left(\frac{1}{\gamma^{1-1/N}}\right). \end{gather}\] Here \(\gamma=\frac{\varepsilon^{N}}{N!}F^{(N)}(\xi)\) and thus \(\gamma^{(N-1)/N} \asymp n^{\delta(N-1)}\).
This gives \[\begin{gather} J_{1} =\frac{1}{N}\left(\frac{N!}{F^{(N)}(\xi)}\right)^{1/N}\left(\exp\left(\frac{{\rm i}\pi}{2N}\right)\Gamma(1/N)+\mathcal{O}\left(\frac{1}{n^{\delta(N-1)}}\right)\right)\\ \\=\frac{1}{N}\left(\frac{N!}{n\psi_{B}^{(N)}(\xi)}\right)^{1/N}\exp\left(\frac{{\rm i}\pi}{2N}\right)\Gamma(1/N)+o\left(\frac{1}{n^{1/N}}\right). \end{gather}\] If \(N\) is even, then \(J_2=J_1\), \[J=\frac{2}{N}\left(\frac{N!}{n\psi_{B}^{(N)}(\xi)}\right)^{1/N}\exp\left(\frac{{\rm i}\pi}{2N}\right)\Gamma(1/N)+o\left(\frac{1}{n^{1/N}}\right),\] and asymptotic formula 11 follows from the above equality combined with 13 and 14 .
For odd \(N\) we obtain in an analogous way that \[J_2 =\frac{1}{N}\left(\frac{N!}{n\psi_{B}^{(N)}(\xi)}\right)^{1/N}\exp\left(-\frac{{\rm i}\pi}{2N}\right)\Gamma(1/N)+o\left(\frac{1}{n^{1/N}}\right).\]Thus, if \(N\) is odd, then \[J=\frac{2}{N}\left(\frac{N!}{n\psi_{B}^{(N)}(\xi)}\right)^{1/N}\cos\left(\frac{\pi}{2N}\right)\Gamma(1/N)+o\left(\frac{1}{n^{1/N}}\right),\] and asymptotic formula 12 follows from the above equality combined with 13 and 14 .
By 11 and 12 we obtain that \[\Sigma=\sum_{\ell\in\mathcal{L}}I_{\ell,d}=\sum_{\ell\in\mathcal{L}}c_\ell\exp(-{\rm i}d\xi_\ell),\qquad 0\le d<D.\] The
points \(\xi_\ell\) are pairwise disjoint, and hence the square matrix
\(\bigl(e^{-{\rm i}d\xi_\ell}\bigr)_{\ell\in\mathcal{L},\,0\le d<D}\) is invertible. Since it does not depend on \(n\), we conclude that \[\Omega=\max_{0\le
d<D}\Bigl | \sum_{\ell\in\mathcal{L}}I_{\ell,d} \Bigr| \gtrsim \sum_{\ell\in\mathcal{L}}|c_\ell|\asymp n^{-1/N}.\]
Step 2. Now we deal with the \(s-D\) integrals \[\int_{\xi_{\ell}-\varepsilon_{\ell}}^{\xi_{\ell}+\varepsilon_{\ell}}\exp\left({\rm i}n\,f(\theta)\right)\,d\theta,\] \(1\le \ell\le s\), \(N_\ell\neq N\) or \(\psi'_B(\xi_\ell)\neq \psi'_B(\xi_r)\), and choose \(\varepsilon_{\ell}\) in
such a way that the corresponding integrals are \(o\left(\frac{1}{n^{1/N}}\right)\). We distinguish the following two cases.
(1) If \(N_{\ell}=N\), then the integral \[\int_{\xi_{\ell}-\varepsilon_{\ell}}^{\xi_{\ell}+\varepsilon_{\ell}}\exp\left({\rm i}n\,f(\theta)\right)\,d\theta\] is treated as follows. By continuity of \(\psi_{B}'\) at the point \(\xi_{\ell}\) there exists \(\eta_{\ell}>0\) (independent of \(n\)) such that \[|\theta-\xi_{\ell}| \le \eta_{\ell}\implies |\psi_{B}'(\xi_{\ell})-\psi_{B}'(\theta)|\le |\psi_{B}'(\xi_{\ell})-\psi_{B}'(\xi_r)|/2.\] We choose \(\varepsilon_{\ell}=\eta_{\ell}\) and observe that \[n|f'_d(\theta)|\ge n|\psi_{B}'(\xi_{\ell})-\psi_{B}'(\xi_r)|/2-D\gtrsim n,\qquad |\theta-\xi_\ell| \le \varepsilon_\ell,\, 0\le d<D.\] Moreover, \(f=f_d\) is monotonic over the interval \([\xi_{\ell}-\varepsilon_{\ell},\xi_{\ell}+\varepsilon_{\ell}]\) and \(f'\) is monotonic over \([\xi_{\ell}-\varepsilon_{\ell},\xi_{\ell}]\) and \([\xi_{\ell},\xi_{\ell}+\varepsilon_{\ell}]\) (because \(f''=\psi_{B}''\) vanishes at \(\xi_{\ell}\) and nowhere else on the interval \([\xi_{\ell}-\varepsilon_{\ell},\xi_{\ell}+\varepsilon_{\ell}]\)). The assumptions of Lemma 4 are therefore satisfied and an application of this lemma gives: \[\int_{\xi_{\ell}-\varepsilon_{\ell}}^{\xi_{\ell}+\varepsilon_{\ell}}\exp\left({\rm i}n\,f_d(\theta)\right)\,d\theta=\mathcal{O}\left(\frac{1}{n}\right).\]
(2) If \(N_{\ell}<N\), then the situation is even simpler: we set \(\varepsilon_{\ell}=n^{-1/N_{\ell}}\), and estimate directly \[\biggl|\int_{\xi_{\ell}-\varepsilon_\ell}^{\xi_{\ell}+\varepsilon_\ell}\exp\left({\rm i}n\,f_d(\theta)\right)\,d\theta\biggr|\lesssim n^{-1/N_{\ell}}.\]
Step 3. We need to verify that \[\begin{gather}
\biggl|\int_{J'_\ell}\exp\left({\rm i}n\,f_d(\theta)\right)\,d\theta\biggr|+\biggl|\int_{J''_\ell}\exp\left({\rm i}n\,f_d(\theta)\right)\,d\theta\biggr|\\
=o(n^{-1/N}),\quad 1\le \ell\le s,\, 0\le d<D.
\end{gather}\] Given \(J'_\ell\) (or \(J''_\ell\) with an analogous argument), we consider the following cases.
(i) \(\ell\in\mathcal{L}\). Then \(\varepsilon_\ell=n^{\delta-(1/N)}\), and considering the Taylor expansion (of order \(N-2\)) of \(\psi_{B}''\) near \(\xi_{\ell}\), we obtain for large \(n\) and for \(\theta\) close to \(\xi_{\ell}\) satisfying \(|\theta-\xi_\ell|\ge\varepsilon_{\ell}\) that \[|\psi_{B}''(\theta)|\gtrsim \varepsilon_\ell^{N-2}=n^{\delta(N-2)-(N-2)/N}.\]Since \(\psi_{B}''\) does not vanish on the interval \(J'_\ell\), we conclude that \[n|f_d''(\theta)|\gtrsim n^{(2/N)+\delta(N-2)},\qquad \theta\in J'_\ell,\, 0\le d<D.\] An application of Lemma 5 gives \[\int_{J'_\ell}\exp\left({\rm i}n\,f_d(\theta)\right)\,d\theta=o(n^{-1/N}).\]
(ii) \(\ell\not\in\mathcal{L}\) and \(N_\ell=N\). Since \(\varepsilon_\ell\) does not depend on \(n\), we have \[|\psi_{B}''(\theta)|\gtrsim 1,\qquad \theta\in J'_\ell,\] and \[n|f_d''(\theta)|\gtrsim n,\qquad \theta\in J'_\ell,\, 0\le d<D.\] Therefore, by Lemma 5, we have \[\int_{J'_\ell}\exp\left({\rm i}n\,f_d(\theta)\right)\,d\theta=\mathcal{O}\left(n^{-1/2}\right).\]
(iii) \(N_\ell<N\). Here \(\varepsilon_\ell=n^{-1/N_\ell}\), and, arguing as above, we conclude that \[n|f_d''(\theta)|\gtrsim n^{2/N_\ell},\qquad
\theta\in J'_\ell,\, 0\le d<D,\] and \[\int_{J'_\ell}\exp\left({\rm i}n\,f_d(\theta)\right)\,d\theta=\mathcal{O}\left(n^{-1/N_\ell}\right).\]
Summing up, Steps 1–3 give us that \[\max_{0\le d<D}|\widehat{B^n}(k_d)|\gtrsim n^{-1/N},\] which completes the proof. ◻
Proof of Theorem 2, formula ?? .. Let \(N\ge3\) and let \(B=\prod_{1\le j\le N}b_{\lambda_j}\) be a finite Blaschke product. The argument \(\psi_{B}(\theta)\) of \(B(e^{{\rm i}\theta})\) is determined modulo \(2\pi\). Furthermore, \(\psi'_{B}\) is a real analytic \(2\pi\)-periodic function, and by 8 , we have \[\psi'_{B}(\theta)=\Re\sum_{1\le j\le N}\frac{1+\overline{\lambda_{j}}e^{{\rm i}\theta}}{1-\overline{\lambda_{j}}e^{{\rm i}\theta}}.\] Since \(\psi'_{B}\) is real analytic, the function \(\psi''_{B}\) has only finite number of zeros on \([0,2\pi]\). By Theorem 1, to obtain that \[\|\widehat{B^{n}}\|_{\ell^\infty}\asymp n^{-1/N},\] it suffices to verify that \(\psi''_{B}(0)=\ldots=\psi_{B}^{(N-1)}(0)=0\), \(\psi_{B}^{(N)}(0)\not=0\), and \(\psi''_{B}\) has no zeros of order \(N-1\).
Set \(u=\exp(\frac{2\pi {\rm i}}{N})\). For \(t\in(0,1/(2\pi))\) to be chosen later on, we set \(\zeta=t\exp(\frac{\pi {\rm i}(N-1)}{2N})\) and \[\lambda_j=\frac{\overline{\zeta u^j}}{1+\overline{\zeta u^j}}\in\mathbb{D},\qquad1\le j\le N.\] Then \[\begin{align} \psi'_{B}(\theta) & =-N+2\Re\sum_{1\le j\le N}\frac{1}{1-\overline{\lambda_{j}}e^{{\rm i}\theta}}\\ & =-N+2\Re\sum_{1\le j\le N}\frac{1+\zeta u^{j}}{1-\zeta u^{j}(e^{{\rm i}\theta}-1)}\\ & =-N+2\Re\sum_{s\ge0}\sum_{1\le j\le N}(1+\zeta u^{j})\zeta^{s}u^{js}(e^{{\rm i}\theta}-1)^{s}\\ & =N+2N\Re\sum_{k\ge1}\zeta^{kN}e^{{\rm i}\theta}(e^{{\rm i}\theta}-1)^{kN-1}\\ & =N+2N\sum_{k\ge1}t^{kN}\Re\Bigl({\rm i}^{k(N-1)}e^{{\rm i}\theta}(e^{{\rm i}\theta}-1)^{kN-1}\Bigr). \end{align}\]
Furthermore, \[\begin{align} \psi''_{B}(\theta) & =2N\Re\sum_{k\ge1}\zeta^{kN}{\rm i}e^{{\rm i}\theta}(kNe^{{\rm i}\theta}-1)(e^{{\rm i}\theta}-1)^{kN-2}\\ & =2N\sum_{k\ge1}t^{kN}\Re\Bigl({\rm i}^{k(N-1)+1}e^{{\rm i}\theta}(kNe^{{\rm i}\theta}-1)(e^{{\rm i}\theta}-1)^{kN-2}\Bigr). \end{align}\] It is clear that (independently of \(t\)) we have \(\psi''_{B}(0)=\ldots=\psi_{B}^{(N-1)}(0)=0\). Since \({\rm i}^{2N-2}\in\mathbb{R}\), for some \(c=c(t)\not=0\) we have \(\psi''_{B}(\theta)\sim c\theta^{N-2}\) at \(0\) and, hence, \(\psi_{B}^{(N)}(0)\not=0\).
Set \[\begin{align} h(\theta) & ={\rm i}^{N}e^{{\rm i}\theta}(Ne^{{\rm i}\theta}-1)(e^{{\rm i}\theta}-1)^{N-2}\\ & =(-1)^{N-1}e^{{\rm i}\theta(N+2)/2}(N-e^{-{\rm i}\theta})\cdot2^{N-2}(\sin(\theta/2))^{N-2}. \end{align}\] Then \(h(0)=0\) if and only if \(\theta\in2\pi\mathbb{Z}\), and \[\frac{h'(\theta)}{h(\theta)}=\frac{N+2}{2}{\rm i}+\frac{{\rm i}e^{-{\rm i}\theta}}{N-e^{-{\rm i}\theta}}+\frac{N-2}{2}\cdot\frac{\cos(\theta/2)}{\sin(\theta/2)}.\] Hence, \[\Im\frac{h'(\theta)}{h(\theta)}\ge\frac{N+2}{2}-\frac{1}{N-1}>0,\qquad 0<|\theta|\le \pi.\] Thus, at every point \(\theta\in]0,2\pi[\) we have \[|\Re h(\theta)|+|\Re h'(\theta)|>0.\] Since \(\Re h^{(N-2)}(0)=(-1)^{N-1}(N-1)!\not=0\), we have \[\sum_{s=0}^{N-2}|\Re h^{(s)}(\theta)|>0,\qquad |\theta|\le \pi.\] By continuity, we can find \(\delta>0\) such that \[\sum_{s=0}^{N-2}|\Re h^{(s)}(\theta)|\ge\delta,\qquad |\theta|\le \pi.\] Now we use that \[\begin{gather} \psi''_{B}(\theta)=2Nt^{N}\Re h(\theta)\\+2Nt^{2N}\sum_{k\ge2}t^{(k-2)N}\Re\Bigl({\rm i}^{N}e^{{\rm i}\theta}(kNe^{{\rm i}\theta}-1)(e^{{\rm i}\theta}-1)^{kN-2}\Bigr). \end{gather}\] Therefore, we can fix a small positive \(t\), \(t<1/(2\pi)\), such that \[\sum_{s=2}^{N}|\psi_{B}^{(s)}(\theta)| \ge\delta Nt^{N},\qquad |\theta|\le \pi.\]Now, \(\psi''_{B}\) has no zeros of order \(N-1\). Thus, by Theorem 1, for every \(N\ge3\), we have constructed a Blaschke product \(B_{N}\) of order \(N\) such that \(\|\widehat{B_N^n}\|_{\ell^\infty}\asymp n^{-1/N}\). ◻
In this section we give two explicit examples of finite Blaschke product \(B\) of degree 2 satisfying the estimate \[\|\widehat{B^n}\|_{\ell^\infty}\asymp n^{-1/5}.\]
In the first example, the zeros of \(B\) are of the same modulus, while in the second example they are on the same diameter of the unit disk.
Let \(w\in\mathbb{C}\setminus\{1\}\) be such that \(\Re w>0\), \[\Re(w)=\Re(w^3)\not=\Re(w^5) \label{eq-u11}\tag{15}\] and \[\Re(w^{-1})\not=\Re(w^{-3}). \label{eq-u21}\tag{16}\] Set \(w_1=w\), \(w_2=\overline{w}\), \[\lambda_j=\frac{w_j-1}{w_j+1},\qquad j=1,2,\] and consider \[B=\prod_{1\le j\le 2}b_{\lambda_j}.\]
For example, we can choose \[w=2+{\rm i}.\] Then \[\begin{gather} \Re(w)=\Re(w^3)=2,\\ \Re(w^5)=-38,\quad \Re(w^{-1})=\frac{2}{5},\quad \Re(w^{-3})=\frac{2}{125}. \end{gather}\]
6. We have \[\|\widehat{B^{n}}\|_{\ell^\infty}\asymp n^{-1/5}.\]
Proof. As above, the argument \(\psi_{B}(\theta)\) of \(B(e^{{\rm i}\theta})\) (determined modulo \(2\pi\)) satisfies the relation \[\psi'_{B}(\theta)=\Re\sum_{1\le j\le 2}\frac{1+\lambda_je^{{\rm i}\theta}}{1-\lambda_je^{{\rm i}\theta}}.\] Furthermore, \[\begin{align} \psi''_{B}(\theta)&=2\Re\sum_{1\le j\le 2}\frac{{\rm i}\lambda_je^{{\rm i}\theta}}{(1-\lambda_je^{{\rm i}\theta})^2},\\ \psi'''_{B}(\theta)&=-2\Re\sum_{1\le j\le 2}\frac{\lambda_je^{{\rm i}\theta}+\lambda^2_je^{2{\rm i}\theta}}{(1-\lambda_je^{{\rm i}\theta})^3}. \end{align}\] Set \[e^{{\rm i}\theta}=\frac{1-{\rm i}x}{1+{\rm i}x},\qquad x\in\mathbb{R},\] and define \(h(x)=\psi''_{B}(\theta)\). Then \(h\) has a zero of order \(N\) at \(x_0\in\mathbb{R}\) if and only if \(\psi''_{B}\) has a zero of order \(N\) at \(\theta_0\), \(e^{{\rm i}\theta_0}=(1-{\rm i}x_0)/(1+{\rm i}x_0)\).
By Theorem 1, we need to verify that \[h(0)=h'(0)=h''(0)=0,\quad h'''(0)\not=0,\] \(h\) has no zeros of order \(3\) except at the origin, and \(\psi'''_{B}(\pi)\not=0\).
First, \[\begin{gather} h(x)=2\Re\sum_{1\le j\le 2}\frac{{\rm i}\frac{w_j-1}{w_j+1}\frac{1-{\rm i}x}{1+{\rm i}x}}{(1-\frac{w_j-1}{w_j+1}\frac{1-{\rm i}x}{1+{\rm i}x})^2}= \frac{1+x^2}{2}\Re\sum_{1\le j\le 2}\frac{{\rm i}(w^2_j-1)}{(1+{\rm i}w_jx)^2}\\= \frac{1+x^2}{4} \Bigl(\frac{{\rm i}(w^2-1)}{(1+{\rm i}wx)^2}+\frac{{\rm i}(\overline{w^2}-1)}{(1+{\rm i}\overline{w}x)^2}-\frac{{\rm i}(\overline{w^2}-1)}{(1-{\rm i}\overline{w}x)^2} -\frac{{\rm i}(w^2-1)}{(1-{\rm i}wx)^2} \Bigr)\\= x(1+x^2)\sum_{1\le j\le 2} \frac{w_j(w^2_j-1)}{(1+w_j^2x^2)^2}. \end{gather}\] Therefore, \(h(0)=0\). Denote \[h_1(x)=\sum_{1\le j\le 2} \frac{w_j(w^2_j-1)}{(1+w_j^2x)^2}.\] Then \(h(x)=x(1+x^2)h_1(x^2)\). We need to verify that \[h_1(0)=0,\quad h_1'(0)\not=0.\] By 15 , \[h_1(0)=\sum_{1\le j\le 2} (w^3_j-w_j)=0,\] and \[h'_1(0)=-2\sum_{1\le j\le 2} (w^5_j-w^3_j)\not=0.\]
If the function \(h\) has a zero of order \(3\) at \(x_0\not=0\), then \(h_1\) has a zero of order \(3\) at \(y_0=\sqrt{|x_0|}\not=0\). We have \[h_1(x)= \frac{Q(x)}{\prod_{1\le j\le 2}(1+w_j^2x)^2},\] where \(Q\) is a polynomial of degree at most \(2\), and, hence, \(Q\) cannot have a zero of order \(3\) at \(y_0\).
Finally, by 16 , we have \[\begin{gather} \psi'''_{B}(\pi)=2\Re\sum_{1\le j\le 2}\frac{\lambda_j-\lambda^2_j}{(1+\lambda_j)^3}=2\Re\sum_{1\le j\le 2}\frac{\frac{w_j-1}{w_j+1}(1-\frac{w_j-1}{w_j+1})}{(1+\frac{w_j-1}{w_j+1})^3}\\= \frac{1}{2}\Re\sum_{1\le j\le 2}\frac{w^2_j-1}{w^3_j}= \Re\bigl(w^{-1}-w^{-3}\bigr)\not=0. \end{gather}\] ◻
Here we give another example of a finite Blaschke product \(B\) of degree 2 such that \(\|\widehat{B^{n}}\|_{\ell^\infty}\asymp n^{-1/5}\).
Let \(w_1,w_2\in(0,\infty)\setminus\{1\}\) be such that \[w_1+w_2=w^3_1+w^3_2\not=w^5_1+w^5_2 \label{eq-u12}\tag{17}\] and \[w^{-1}_1+w^{-1}_2\not=w^{-3}_1+w^{-3}_2. \label{eq-u22}\tag{18}\] Set \[\lambda_j=\frac{w_j-1}{w_j+1},\qquad j=1,2,\] and consider \[B=\prod_{1\le j\le 2}b_{\lambda_j}.\]
For example, we can choose \[w_1=\frac{1}{2},\quad w_2=\frac{1+\sqrt{13}}{4}.\] Then \[\begin{gather} w_1+w_2=w^3_1+w^3_2=\frac{3+\sqrt{13}}{4},\quad w^5_1+w^5_2=\frac{63+19\sqrt{13}}{64},\\ w^{-1}_1+w^{-1}_2=\frac{5+\sqrt{13}}{3},\quad w^{-3}_1+w^{-3}_2=\frac{176+16\sqrt{13}}{27}. \end{gather}\]
7. We have \[\|\widehat{B^n}\|_{\ell^\infty}\asymp n^{-1/5}.\]
Proof. As above, \[\begin{align} \psi'_{B}(\theta)&=\Re\sum_{1\le j\le 2}\frac{1+\lambda_je^{{\rm i}\theta}}{1-\lambda_je^{{\rm i}\theta}},\\ \psi''_{B}(\theta)&=2\Re\sum_{1\le j\le 2}\frac{{\rm i}\lambda_je^{{\rm i}\theta}}{(1-\lambda_je^{{\rm i}\theta})^2},\\ \psi'''_{B}(\theta)&=-2\Re\sum_{1\le j\le 2}\frac{\lambda_je^{{\rm i}\theta}+\lambda^2_je^{2{\rm i}\theta}}{(1-\lambda_je^{{\rm i}\theta})^3}. \end{align}\] Set \[e^{{\rm i}\theta}=\frac{1-{\rm i}x}{1+{\rm i}x},\qquad x\in\mathbb{R},\] and define \(h(x)=\psi''_{B}(\theta)\).
As above, by Theorem 1, we need to verify that \[h(0)=h'(0)=h''(0)=0,\quad h'''(0)\not=0,\] \(h\) has no zeros of order \(3\) except at the origin, and \(\psi'''_{B}(\pi)\not=0\).
First, \[\begin{gather} h(x)=2\Re\sum_{1\le j\le 2}\frac{{\rm i}\frac{w_j-1}{w_j+1}\frac{1-{\rm i}x}{1+{\rm i}x}}{(1-\frac{w_j-1}{w_j+1}\frac{1-{\rm i}x}{1+{\rm i}x})^2}= \frac{1+x^2}{2}\Re\sum_{1\le j\le 2}\frac{{\rm i}(w^2_j-1)}{(1+{\rm i}w_jx)^2}\\= \frac{1+x^2}{4} \Bigl(\frac{{\rm i}(w_1^2-1)}{(1+{\rm i}w_1x)^2}+\frac{{\rm i}(w_2^2-1)}{(1+{\rm i}w_2x)^2}-\frac{{\rm i}(w_1^2-1)}{(1-{\rm i}w_1x)^2} -\frac{{\rm i}(w_2^2-1)}{(1-{\rm i}w_2x)^2} \Bigr)\\= x(1+x^2)\sum_{1\le j\le 2} \frac{w_j(w^2_j-1)}{(1+w_j^2x^2)^2}. \end{gather}\] Therefore, \(h(0)=0\). Denote \[h_1(x)=\sum_{1\le j\le 2} \frac{w_j(w^2_j-1)}{(1+w_j^2x)^2}.\] As above, \(h(x)=x(1+x^2)h_1(x^2)\), and we need to verify that \[h_1(0)=0,\quad h_1'(0)\not=0.\] By 17 , \[h_1(0)=\sum_{1\le j\le 2} (w^3_j-w_j)=0,\] and \[h'_1(0)=-2\sum_{1\le j\le 2} (w^5_j-w^3_j)\not=0.\]
If the function \(h\) has a zero of order \(3\) at \(x_0\not=0\), then \(h_1\) has a zero of order \(3\) at \(y_0=\sqrt{|x_0|}\not=0\). We have \[h_1(x)= \frac{Q(x)}{\prod_{1\le j\le 2}(1+w_j^2x)^2},\] where \(Q\) is a polynomial of degree at most \(2\), and, hence, \(Q\) cannot have a zero of order \(3\) at \(y_0\).
Finally, by 18 , we have \[\begin{gather} \psi'''_{B}(\pi)=2\sum_{1\le j\le 2}\frac{\lambda_j-\lambda^2_j}{(1+\lambda_j)^3}=2\sum_{1\le j\le 2}\frac{\frac{w_j-1}{w_j+1}(1-\frac{w_j-1}{w_j+1})}{(1+\frac{w_j-1}{w_j+1})^3}\\= \frac{1}{2}\sum_{1\le j\le 2}\frac{w^2_j-1}{w^3_j}= \frac{1}{2}\sum_{1\le j\le 2}\bigl(w_j^{-1}-w_j^{-3}\bigr)\not=0. \end{gather}\] ◻
Here we give an explicit example of a finite Blaschke product \(B\) of degree 4 such that \(\|\widehat{B^n}\|_{\ell^\infty}\asymp n^{-1/7}\).
Let \(w_1,w_2\in\mathbb{C}\setminus\{1\}\) be such that \(\Re w_1,\Re w_2>0\), \[\Re(w_1+w_2)=\Re(w^3_1+w^3_2)=\Re(w_1^5+w_2^5)\not=\Re(w^7_1+w^7_2) \label{eq-u1}\tag{19}\] and \[\Re(w^{-1}_1+w^{-1}_2)\not=\Re(w^{-3}_1+w^{-3}_2). \label{eq-u2}\tag{20}\] Set \[\begin{gather} w_{j+2}=\overline{w_j},\qquad j=1,2,\\ \lambda_j=\frac{w_j-1}{w_j+1}, \qquad 1\le j\le 4, \end{gather}\] and consider \[B=\prod_{1\le j\le 4}b_{\lambda_j}.\]
For example, we can choose \[w_1=1+\frac{2{\rm i}}{\sqrt{3}},\quad w_2=2+\frac{{\rm i}}{\sqrt{3}}.\] Then \[\begin{gather} \Re(w_1+w_2)=\Re(w^3_1+w^3_2)=\Re(w_1^5+w_2^5)=3,\\ \Re(w^7_1+w^7_2)=-\frac{421}{9},\quad \Re(w^{-1}_1+w^{-1}_2)=\frac{81}{91},\\ \Re(w^{-3}_1+w^{-3}_2)=-\frac{122391}{753571}. \end{gather}\]
8. We have \[\|\widehat{B^{n}}\|_{\ell^\infty}\asymp n^{-1/7}.\]
Proof. As above, \[\begin{align} \psi'_{B}(\theta)&=\Re\sum_{1\le j\le 4}\frac{1+\lambda_je^{{\rm i}\theta}}{1-\lambda_je^{{\rm i}\theta}},\\ \psi''_{B}(\theta)&=2\Re\sum_{1\le j\le 4}\frac{{\rm i}\lambda_je^{{\rm i}\theta}}{(1-\lambda_je^{{\rm i}\theta})^2},\\ \psi'''_{B}(\theta)&=-2\Re\sum_{1\le j\le 4}\frac{\lambda_je^{{\rm i}\theta}+\lambda^2_je^{2{\rm i}\theta}}{(1-\lambda_je^{{\rm i}\theta})^3}. \end{align}\] Set \[e^{{\rm i}\theta}=\frac{1-{\rm i}x}{1+{\rm i}x},\qquad x\in\mathbb{R},\] and define \(h(x)=\psi''_{B}(\theta)\). Then \(h\) has a zero of order \(N\) at \(x_0\in\mathbb{R}\) if and only if \(\psi''_{B}\) has a zero of order \(N\) at \(\theta_0\), \(e^{{\rm i}\theta_0}=(1-{\rm i}x_0)/(1+{\rm i}x_0)\).
We need to verify that \[h(0)=h'(0)=h''(0)=h'''(0)=h^{(4)}(0)=0,\quad h^{(5)}(0)\not=0,\] \(h\) has no zeros of order \(5\) except at the origin, and \(\psi'''_{B}(\pi)\not=0\).
As above, \[\begin{gather} h(x)=2\Re\sum_{1\le j\le 4}\frac{{\rm i}\frac{w_j-1}{w_j+1}\frac{1-{\rm i}x}{1+{\rm i}x}}{(1-\frac{w_j-1}{w_j+1}\frac{1-{\rm i}x}{1+{\rm i}x})^2}= \frac{1+x^2}{2}\Re\sum_{1\le j\le 4}\frac{{\rm i}(w^2_j-1)}{(1+{\rm i}w_jx)^2}\\= \frac{1+x^2}{4}\sum_{1\le j\le 2}\Bigl(\frac{{\rm i}(w^2_j-1)}{(1+{\rm i}w_jx)^2}+\frac{{\rm i}(\overline{w^2_j}-1)}{(1+{\rm i}\overline{w_j}x)^2}-\frac{{\rm i}(\overline{w^2_j}-1)}{(1-{\rm i}\overline{w_j}x)^2} -\frac{{\rm i}(w^2_j-1)}{(1-{\rm i}w_jx)^2} \Bigr)\\= x(1+x^2)\sum_{1\le j\le 4} \frac{w_j(w^2_j-1)}{(1+w_j^2x^2)^2}. \end{gather}\] Therefore, \(h(0)=0\). Denote \[h_1(x)=\sum_{1\le j\le 4} \frac{w_j(w^2_j-1)}{(1+w_j^2x)^2}.\] Then \(h(x)=x(1+x^2)h_1(x^2)\). We need to verify that \[h_1(0)=h_1'(0)=0,\quad h_1''(0)\not=0.\] By 19 , \[h_1(0)=\sum_{1\le j\le 4} (w^3_j-w_j)=0.\] Furthermore, \[h'_1(x)=-2\sum_{1\le j\le 4} \frac{w^3_j(w^2_j-1)}{(1+w_j^2x)^3}.\] Again by 19 , \[h'_1(0)=-2\sum_{1\le j\le 4} (w^5_j-w^3_j)=0.\] Next, \[h''_1(x)=6\sum_{1\le j\le 4} \frac{w^5_j(w^2_j-1)}{(1+w_j^2x)^4},\] and by 20 , \[h''_1(0)=6\sum_{1\le j\le 4} w^5_j(w^2_j-1)\not=0.\]
If the function \(h\) has a zero of order \(5\) at \(x_0\not=0\), then \(h_1\) has a zero of order \(5\) at \(y_0=\sqrt{|x_0|}\not=0\). We have \[h_1(x)= \frac{Q(x)}{\prod_{1\le j\le 4}(1+w_j^2x)^2},\] where \(Q\) is a polynomial of degree at most \(6\). Since \(h_1(0)=h_1'(0)=0\), \(Q\) cannot have a zero of order \(5\) at \(y_0\).
Finally, by 20 , we have \[\begin{gather} \psi'''_{B}(\pi)=2\Re\sum_{1\le j\le 4}\frac{\lambda_j-\lambda^2_j}{(1+\lambda_j)^3}=2\Re\sum_{1\le j\le 4}\frac{\frac{w_j-1}{w_j+1}(1-\frac{w_j-1}{w_j+1})}{(1+\frac{w_j-1}{w_j+1})^3}\\= \frac{1}{2}\Re\sum_{1\le j\le 4}\frac{w^2_j-1}{w^3_j}= \Re\sum_{1\le j\le 2}\bigl(w^{-1}_j-w^{-3}_j\bigr)\not=0. \end{gather}\] ◻
We recall that \[e_{nm} = \frac{(1 - |\lambda_m|^2)^{1/2}}{z-\lambda_m} B^n.\] We set \(g=S^{-1}e_{nm}\) so that \(g = g(0)+ z e_{nm}\). Then
\[\label{etoile}
\|S^{-1}\|_{\ell^\infty_A\to\ell^\infty_A} \ge \frac{\|\widehat{S^{-1}e_{nm}}\|_{\ell^\infty}}{\|\widehat{e_{nm}}\|_{\ell^\infty}}
= \frac{\|\widehat{g}\|_{\ell^\infty}}{\|\widehat{e_{nm}}\|_{\ell^\infty}}
\ge \frac{|g(0)|}{\|\widehat{e_{nm}}\|_{\ell^\infty}}.\tag{21}\] Let us first concentrate on estimating \(\|\widehat{e_{nm}}\|_{\ell^\infty}\). We follow the same steps as in the proof of Theorem 1. We set \(d_{nm}= \frac{e_{nm}}{(1-|\lambda_m|^2)^{1/2}}\). We have \[\widehat{d_{nm}}(k) = \frac{1}{2\pi}
\int_0^{2\pi} \frac{B^n(e^{{\rm i}\theta})}{e^{{\rm i}\theta} - \lambda_m} e^{-{\rm i}k\theta} \,d\theta = \frac{1}{2\pi} \int_0^{2\pi} \frac{e^{{\rm i}n\,f(\theta)}}{e^{{\rm i}\theta} - \lambda_m} \,d\theta,\]where \(f(\theta) = \psi(\theta) - \frac{k}{n} \theta\) and \(\psi(\theta)=\arg(B(e^{{\rm i}\theta}))\). Using the same notation as in Theorem 1, we consider the sequence \((\xi_{\ell})_{\ell=1}^s\) of consecutive zeros of \(\psi''\) on \([0,2\pi)\) with
respective multiplicities \((N_\ell-2)_{\ell=1}^s\), \(N_\ell \ge 3\). We have: \[\begin{gather}
\widehat{d_{nm}}(k) =\sum_{\ell=1}^{s}\int_{\xi_{\ell}-\varepsilon}^{\xi_{\ell}+\varepsilon}\frac{e^{{\rm i}n\,f(\theta)}}{e^{{\rm i}\theta} - \lambda_m}\,d\theta +\sum_{\ell=1}^{s-1}\int_{\xi_{\ell}+\varepsilon}^{\xi_{\ell+1}-\varepsilon}\frac{e^{{\rm
i}n\,f(\theta)}}{e^{{\rm i}\theta} - \lambda_m}\,d\theta\\ +\int_{0}^{\xi_{1}-\varepsilon}\frac{e^{{\rm i}n\,f(\theta)}}{e^{{\rm i}\theta} - \lambda_m}\,d\theta +\int_{\xi_{s}+\varepsilon}^{2\pi}\frac{e^{{\rm i}n\,f(\theta)}}{e^{{\rm i}\theta} -
\lambda_m}\,d\theta.
\end{gather}\] From now on we put \[\varepsilon=n^{-1/N}\rightarrow0,\qquad n\rightarrow\infty,\] where \(N = \max_{1 \le \ell \le s} N_\ell\). Clearly \[\left|\int_{\xi_{\ell}-\varepsilon}^{\xi_{\ell}+\varepsilon}\frac{e^{{\rm i}n\,f(\theta)}}{e^{{\rm i}\theta} - \lambda_m}\,d\theta\right|\le \frac{2\varepsilon}{\bigl|1-|\lambda_m|\bigr|},\] and therefore the first sum is
bounded from above as \[\left|\sum_{\ell=1}^{s}\int_{\xi_{\ell}-\varepsilon}^{\xi_{\ell}+\varepsilon}\frac{e^{{\rm i}n\,f(\theta)}}{e^{{\rm i}\theta} - \lambda_m}\,d\theta\right|\lesssim n^{-1/N}.\] Furthermore, \(f''=\psi''\) does not vanish on the intervals \((0,\xi_{1}-\varepsilon)\), \((\xi_{s}+\varepsilon,2\pi)\) and \((\xi_{\ell}+\varepsilon,\xi_{\ell+1}-\varepsilon)\) for \(\ell=1,\ldots, s-1\). Writing the Taylor expansion (of order \(N_{\ell}-2\)) of \(\psi''\) near \(\xi_{\ell}\) we find \[\psi''(\theta)=\frac{\psi^{(N_{\ell})}(\xi_{\ell})}{(N_{\ell}-2)!}\left(\theta-\xi_{\ell}\right)^{N_{\ell}-2}\left(1+\mathcal{O}\left(\theta-\xi_{\ell}\right)\right),\] and it follows that if \(\theta\) is close to \(\xi_\ell\) but satisfies \(|\theta - \xi_\ell| \ge \varepsilon\), then \[|\psi''_B(\theta)| \gtrsim
n^{-(N_\ell-2)/N},\] for \(n\) large enough. Below we apply Lemma 5 to the remaining integrals: \[\label{integral}
\int_{\xi_{l}+\varepsilon}^{\xi_{l+1}-\varepsilon} \frac{e^{{\rm i}n\,f(\theta)}}{e^{{\rm i}\theta}-\lambda_m}\,d\theta,\quad
\int_{0}^{\xi_{1}-\varepsilon} \frac{e^{{\rm i}n\,f(\theta)}}{e^{{\rm i}\theta}-\lambda_m}\,d\theta,\quad
\int_{\xi_{s}+\varepsilon}^{2\pi}\frac{e^{{\rm i}n\,f(\theta)}}{e^{{\rm i}\theta}-\lambda_m}\,d\theta.\tag{22}\] We describe how to get an upper estimate on the first integral in 22 , the argument for two others
being similar. We have \[\begin{gather}
\biggl|\int_{\xi_{l}+\varepsilon}^{\xi_{l+1}-\varepsilon} \frac{e^{{\rm i}n\,f(\theta)}}{e^{{\rm i}\theta}-\lambda_m}\,d\theta\biggr|\\ \le
\biggl|\int_{\xi_{l}+\varepsilon}^{\xi_{l+1}-\varepsilon} \Re\biggl(\frac{e^{{\rm i}n\,f(\theta)}}{e^{{\rm i}\theta}-\lambda_m}\biggr)\,d\theta\biggr|+
\biggl|\int_{\xi_{l}+\varepsilon}^{\xi_{l+1}-\varepsilon} \Im\biggl(\frac{e^{{\rm i}n\,f(\theta)}}{e^{{\rm i}\theta}-\lambda_m}\biggr)\,d\theta\biggr|.
\end{gather}\] We set \(\lambda_m=\rho_m\exp({\rm i}\theta_m)\) and \[G(\theta)=\Re \left(\frac{1}{e^{{\rm i}\theta}-\lambda_m}\right) = \frac{\cos{\theta}-\rho_m \cos
\theta_m}{(\cos{\theta}-\rho_m \cos \theta_m)^2+(\sin{\theta}-\rho_m \sin \theta_m)^2}.\] The function \(G\) is bounded on \([0, 2\pi]\) and it vanishes twice on \([0, 2\pi)\). Furthermore, \(G'\) has a finite number of zeros on \([0, 2\pi)\). Therefore, we can split every interval of integration into a (uniformly
bounded in \(n\)) number of intervals so that \(G\) is of constant sign and monotonic on each of them. Then \[|F''|\gtrsim n^{2/N}\] on these
intervals and therefore, applying Lemma 5, we obtain that the corresponding integrals are \(\mathcal{O}\left(n^{-1/N}\right)\). Applying the same
steps with \(G(\theta)= \Bigl(\dfrac{1}{e^{{\rm i}\theta}-\lambda_m}\Bigr)\), and estimating in the same way the remaining integrals in 22 , we conclude that \(\|\widehat{e_{nm}}\|_{\ell^\infty} \lesssim n^{-1/N}\).
Now we return to (21 ). It remains to estimate \(|g(0)|\). We use that \(g =S^{-1}e_{nm} \in K_B\), and that \[K_B = \Biggl\{
\frac{P(z)}{\prod\limits_{i=1}^m (1-\bar{\lambda}_i z)^n}: \deg P \le nm - 1\Biggr\}.\] Thus we can determine \(g(0)\) by the relation \[\label{g40041}
g(0) + (1-|\lambda_m|^2)^{1/2} \frac{z(z-\lambda_m)^{n-1}}{(1-\bar{\lambda}_m z)^n} \prod_{j=1}^{m-1} \left(\frac{z-\lambda_j}{1-\bar{\lambda}_j z}\right)^n \in K_B.\tag{23}\] Since \[\frac{z-\lambda}{1-\bar{\lambda}z} = -\frac{1}{\bar{\lambda}}+ \frac{(1/\bar{\lambda})-\lambda}{1-\bar{\lambda}z},\qquad
\frac{z}{1-\bar{\lambda}z} = -\frac{1}{\bar{\lambda}}+ \frac{1/\bar{\lambda}}{1-\bar{\lambda}z},\] relation 23 implies that \[g(0) + (1-|\lambda_m|^2)^{1/2}\prod_{j=1}^m
\frac{1}{(-\bar{\lambda}_j)^n} + \frac{P(z)}{\prod_{j=1}^m (1 - \bar{\lambda}_j z)^n} \in K_B,\] where \(\deg P \le nm-1\). Therefore, the condition \(g \in K_B\) implies that \[g(0) = - (1-|\lambda_m|^2)^{1/2}\prod_{j=1}^m \frac{1}{(-\bar{\lambda}_j)^n}.\] Thus, 21 yields \[\label {result th3}
\|S^{-1}\|_{\ell^\infty_A\to\ell^\infty_A} \gtrsim \frac{n^{1/N}}{\prod_{j=1}^m |\lambda_j|^n},\] which completes the proof of part (i) because of 7 .
To prove part (ii), we use the analytic expression of \(\Phi\) established by Gluskin–Meyer–Pajor [13]: \[\Phi(\lambda_1,\dots,\lambda_1,\dots,\lambda_m,\dots,\lambda_m) = \sup \{ |\det T|\cdot \|T^{-1}\| \},\] where we take the supremum by all norms in \(\mathbb{C}^{nm}\) and all \(T\) such that \(\|T\|\le 1\) in the induced norm and \(\sigma_T=\{\lambda_1, \ldots, \lambda_m\}\) with multiplicity \(n\) of
every eigenvalue. We conclude by setting \(T=S\) and using 7 and [result th3]. 0◻
9. Here we establish part (ii) of Theorem 3 as a direct application of Theorem 1. To this aim we reproduce and adapt the duality method used to prove the main result of [11]. We recall the definition of the Wiener algebra, which is the subset of \(H(\mathbb{D})\) of absolutely convergent Fourier series, \[\begin{align} W:=\Bigl\{f=\sum_{k\ge0}\hat{f}(k)z^{k}:\|f\|_W:=\sum_{k\ge0}|\hat{f}(k)|<\infty\Bigr\}. \end{align}\] Let \(B\) be the finite Blaschke product with simple zeros at \(\lambda_{1},\dots,\lambda_{m}\). It is easily verified that if \(f\) is in \(W\) and \(f(\lambda)=0\) with \(\lambda\in\mathbb{D}\), then \[\Bigl\|\frac{f}{z-\lambda}\Bigr\|_W\le\frac{\|f\|_W}{1-|\lambda|}.\] In particular, \(\frac{f}{z-\lambda}\) is also in \(W\). Therefore, for any \(h\in W\) with \[h(\lambda_j)=h'(\lambda_j)=\ldots=h^{(n-1)}(\lambda_j)=0,\qquad j=1,\dots,m,\] we have \(h/B^n\in W\). With this observation we rewrite the Gluskin–Meyer–Pajor expression for \(\Phi\) as follows: \[\begin{gather} \label{star2} \Phi(\lambda_1,\dots,\lambda_1,\dots,\lambda_m,\dots,\lambda_m)\\ =\inf\biggl\{ \|h\|_W-|h(0)| : h\in B^{n}W,\, h(0)=\prod_{j=1}^{m}\lambda_j^n\biggr\}. \end{gather}\tag{24}\] Let \(L^{2}(\partial\mathbb{D})\) be the usual \(L^{2}\) space on the unit circle \(\partial\mathbb{D}\), equipped with the standard scalar product \[\left\langle f,g\right\rangle :=\int_{-\pi}^{\pi}f(e^{{\rm i}\varphi})\overline{g(e^{{\rm i}\varphi})}\,\frac{d\varphi}{2\pi}.\] For \(f=\sum_{k}\hat{f}(k)z^{k}\), \(g=\sum_{k}\hat{g}(k)z^{k}\in H^2\), it is well known [18] that the \(L^{2}(\partial\mathbb{D})\) scalar product can be written as \[\left\langle f,\,g\right\rangle =\sum_{j\ge0}\hat{f}(j)\overline{\hat{g}(j)}.\] Let \(h=B^ng\) with \(g\in W\) and \(h(0)=\prod_{j=1}^m\lambda_j^n\). Then \[\|h\|_W\cdot\|B^n\|_{\ell^\infty_A}\ge \bigl|\bigl\langle h,B^{n}\bigr\rangle\bigr| =\bigl|\bigl\langle g,1\bigr\rangle\bigr| =|g(0)|=\Bigl|\frac{h(0)}{B^{n}(0)}\Bigr|=1.\]It follows that any candidate function \(h\) in 24 satisfies \[\|h\|_W\ge\frac{1}{\|B^{n}\|_{\ell^\infty_A}},\] and consequently \[\Phi\left(\lambda_1,\dots,\lambda_1,\dots,\lambda_m,\dots,\lambda_m\right)\ge\frac{1}{\|B^n\|_{\ell^\infty_A}}-\prod_{j=1}^{m} |\lambda_j|^n.\] It remains to apply Theorem 1 to obtain that \[\Phi(\lambda_1, \dots, \lambda_1, \dots, \lambda_m,\dots, \lambda_m) \gtrsim n^{1/N},\qquad n\to\infty.\]