Abstract
We study the system
where L and M are boundary values of some outer functions defined in the unit disc. Necessary and sufficient conditions on functions L and M are found so that the system is a Schauder basis in \(L^{p}(\mathbb {T}),\) \(1< p < \infty\).
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Introduction
We say that a system \(\{z^{m} F(z)\}_{m=0}^{\infty }\) is a Beurling system if F is an outer function. In his fundamental work [3], Beurling particularly proved that if F is an outer function from \(H^{2}(\mathbb {D})\), then the system \(\{z^{m} F(z)\}_{m=0}^{\infty }\) is complete in the space \(H^{2}(\mathbb {D})\), where \(\mathbb {D}= \{z\in \mathbb {C}: |z|<1 \}\) is the unit disc. This result can be easily extended for the spaces \(H^{p}(\mathbb {D}), 1\le p < \infty\) (see [6]). In the previous paper [12], we studied questions of representations of functions from the spaces \(H^{p}(\mathbb {D}), 1\le p < \infty\) by series with respect to Beurling systems. In the present paper, we study the following system of functions
where L and M are boundary values of some outer functions defined in \(\mathbb {D}\). We find necessary and sufficient conditions on functions L and M so that the system is a Schauder basis in \(L^{p}(\mathbb {T}),\) \(1< p < \infty\), where \(\mathbb {T} = {\mathbb {R}}/{2\pi \mathbb {Z}}\). Completeness multipliers for a complete orthonormal system and for the trigonometric system have been studied in [4, 5]. Best references for the study of the trigonometric system are [2, 20]. The paper is divided into two parts. In the first part, we adopt some results for the spaces \(H^{p}(\mathbb {T}),\) \(1\le p < \infty\), and the second part will be dedicated to the study of the systems \(e^{i kt} \Psi _{L,M}, k\in \mathbb {Z}\).
Preliminary results, definitions and notations
Let
for \(1\le p \le \infty .\) The spaces \(H^{p}(\mathbb {T}),1\le p \le \infty\) are Banach spaces of functions defined on \(\mathbb {T}.\) The convolution of functions \(g, h \in L(\mathbb {T})\) is denoted by
We would like to define on \(\mathbb {T}\) the analogue of an outer function on the unit disc \(\mathbb {D}= \{z\in \mathbb {C}: |z|<1 \}\). A holomorphic function F in \(\mathbb {D}\) is an outer function if
where \(\phi\) is a real-valued integrable function defined on \(\mathbb {T}\) [3] and
The Cauchy and the Poisson kernels are defined as follows:
An important formula for representation of positive functions is due to G. Szegö [8, 19]. In his honour, the analogue of an outer function on \(\mathbb {T}\) is called \(S-\)function. We say that a Lebesgue measurable function \(\varphi :\mathbb {T}\rightarrow \mathbb {C}\) is an \(S-\)function if \(\ln |\varphi | \in L^{1}(\mathbb {T})\) and
where \(l_{\varphi }(t)= \widetilde{\ln |\varphi |}(t)\) and the conjugate function of an integrable function g is denoted by \(\tilde{g}\). For any measurable function \(\psi :\mathbb {T}\rightarrow \mathbb {C}\) such that \(\ln |\psi | \in L^{1}(\mathbb {T})\), we put
The following statement follows from the definition.
Proposition 1.1
Let \(\varphi\) be an \(S-\)function. Then, \(S(\varphi )(t) = e^{i\alpha } \varphi (t)\) for some \(\alpha \in \mathbb {T}\).
It is easy to observe that the following properties also are true:
For a complex-valued integrable function g defined on \(\mathbb {T}\), such that \(\ln |g(t)|\) is integrable, we set
Evidently \(G_{g}(z)\) is a non-zero holomorphic function in \(z\in \mathbb {D},\) \(G_{g}\in H^{1}(\mathbb {D})\) and
We also have that
The class of \(S-\)functions in \(H^{p}(\mathbb {T}), 1\le p \le \infty\) is denoted by \(H^{p}_{\text {s}}(\mathbb {T}).\)
There is a vast literature on this topic (see, e.g. [6, 9, 13, 20] and others). It is well known (see, e.g. [9]) that \(\ln |f(t)|\in L^{1}(\mathbb {T})\) if \(f\in H^{1}(\mathbb {T}).\) The following fact is an easy consequence of well-known results that can be found in the literature which we have mentioned earlier.
Proposition 1.2
For any \(f\in H^{1}(\mathbb {T}),\)
where \(F\in H^{\infty }(\mathbb {T})\) and \(|F(t)|= 1\) a.e. on \(\mathbb {T}\).
Given \(f\in H^{p}_{\text {s}}(\mathbb {T})\), we can recover a non-zero holomorphic outer function in \(\mathbb {D}\) by the Poisson integral. If \(\varphi\) is a non-negative integrable function such that \(\ln \varphi \in L^{1}(\mathbb {T})\), then \(\varphi\) is the modulus of a function from \(H^{1}(\mathbb {T})\) (see, e.g. [9], p.53). If \(1<p<\infty\), we use the previous assertion for \(\varphi ^{1/p}\) to deduce a similar proposition for a function in \(H^{p}(\mathbb {T})\). The case \(p=\infty\) is studied in [9]. Thus, the following statement holds.
Proposition 1.3
Let g be a Lebesgue measurable function \(g:\mathbb {T}\rightarrow \mathbb {C}\) such that \(S(g)\in L^{p}(\mathbb {T}), 1\le p \le \infty\). Then, \(S(g)\in H^{p}(\mathbb {T}).\)
Set
It is clear that \(\ln |f(t)|\in L^{1}(\mathbb {T})\) if \(f\in \textbf{N}(\mathbb {T}).\) The analogue of the following result is well known in \(\mathbb {D}\). We give the proof because it is short.
Theorem 1.1
\(H^{1}(\mathbb {T})\subset \textbf{N}(\mathbb {T})\).
Proof
Let \(f\in H^{1}(\mathbb {T})\). Set
By Proposition 1.3, we have that \(S(f_{0})\in H^{\infty }_{\text {s}}(\mathbb {T})\). On the other hand, \(f\cdot S(f_{0}) \in H^{\infty }(\mathbb {T}).\) Hence, \(f\in \textbf{N}(\mathbb {T})\). \(\square\)
By the above theorem, we establish
Proposition 1.4
Let \(\phi \in H^{1}(\mathbb {T})\) and let \(\psi \in H^{1}_{\text {s}}(\mathbb {T})\). Then, \(\frac{\phi }{\psi }\in \textbf{N}(\mathbb {T}).\)
Proof
By Theorem 1.1, we have that
where \(\phi _{1},\psi _{1} \in H^{\infty }(\mathbb {T})\) and \(\phi _{2},\psi _{2} \in H_{\text {s}}^{\infty }(\mathbb {T})\). On the other hand, by Proposition 1.1 and (1.3), we deduce that for some \(\beta _{1}\in \mathbb {T}\)
Thus,
where \(\beta _{2}\in \mathbb {T}\). \(\square\)
We also have
Theorem 1.2
For any \(p, 1\le p \le \infty\)
Proof
Let \(\phi \in L^{p}(\mathbb {T})\) and
By Proposition 1.2, we have that \(g(t)= G(t) S(g)(t)\), where \(G\in H^{\infty }(\mathbb {T})\) and \(|G(t)|= 1\) a.e. on \(\mathbb {T}\). Hence, by (1.3), we deduce
Thus, \(S(\frac{g}{h}) \in L^{p}(\mathbb {T})\) and by Proposition 1.2 it follows that \(\phi \in H^{p}(\mathbb {T}).\) The inclusion \(H^{p}(\mathbb {T}) \subseteq L^{p}(\mathbb {T})\cap \textbf{N}(\mathbb {T})\) holds by Theorem 1.1. \(\square\)
The closed linear span in a separable Banach space \(\textbf{B}\) of a system of elements \(X=\{x_{k}\}_{x=0}^{\infty } \subset \textbf{B}\) is denoted by \(\overline{\text {span}}_{\textbf{B}}(X).\) A system \(X=\{x_{k}\}_{x=0}^{\infty }\) is complete in \(\textbf{B}\) if \(\overline{\text {span}}_{\textbf{B}}(X) = \textbf{B}.\)
Beurling’s approximation theorem [3, 6] in the \(H^{p}(\mathbb {T}),1\le p< \infty\) spaces have a simple formulation. We give its proof for the completeness of the exposition.
Theorem 1.3
Let \(p\in [1,+\infty )\) and let \(f\in H^{p}_{\text {s}}(\mathbb {T}).\) Then, the system \(\{f(t) e^{int}\}_{n=0}^{\infty }\) is complete in \(H^{p}(\mathbb {T}).\)
Proof
Consider the case \(p>1\). We assume that there exists \(g\in H^{p'}(\mathbb {T}),\) \(1/p +1/p' =1\) such that
If we put \(\varphi (t) = f(t) \cdot \overline{g(t)}\), then \(\varphi \in H^{1}(\mathbb {T}).\) By Proposition 1.4 and Theorem 1.2, it follows that
Hence, g is a constant function. This means that \(\int _{\mathbb {T}} f(t) dt = 0\), which contradicts the condition (1.5). If \(p=1\), then there exists \(g\in L^{\infty }(\mathbb {T})\) such that (1.6) holds and the following condition
is not true. The rest of the proof is similar to the case \(p>1\) and we skip it. \(\square\)
The following results were obtained by the author in the recent paper.
A system \(X=\{x_{k}\}_{k=0}^{\infty } \subset \textbf{B}\) is called minimal, if there exists a system \(X^{*} = \{ \phi _{ n}\}_{n=0}^{\infty } \subset \textbf{B}^{*},\) such that
where \(\delta _{n k}\) is the Kronecker symbol \((\delta _{n k} = 0 \quad \text{ if } \quad n \ne k \quad \text{ and } \quad \delta _{k k} = 1 )\). The system \(X^{*}\) is called dual to X. If X is a complete and minimal system in \(\textbf{B}\), then the dual system \(X^{*}\) is unique [14]. A set \(\Psi \subset \textbf{B}^{*}\) is called total if
if and only if \(x=\textbf{0}\). A system \(X=\{x_{k}\}_{k=0}^{\infty } \subset \textbf{B}\) is an \(M-\)basis in \(\textbf{B}\) if X is complete and minimal in \(\textbf{B}\) and its dual system \(X^{*}\) is total.
The following theorems were proved in [12]
Theorem 1.4
Let \(p\in [1,+\infty )\) and let \(f\in H^{p}_{\text {s}}(\mathbb {T}).\) The system \(\{ e^{i nt} f(t)\}_{n=0}^{\infty }\) is an \(M-\)basis in \(H^{p}(\mathbb {T}).\)
That the system \(\{ e^{int} \}_{n=0}^{\infty }\) is minimal in \(H^{p}(\mathbb {T},w),\) \(1\le p < \infty\) was mentioned in [11] without proof. A complete and minimal system \(X=\{x_{k}\}_{k=0}^{\infty } \subset \textbf{B}\) with the dual system \(X^{*} = \{ \phi _{ k} \}_{k=0}^{\infty } \subset \textbf{B}^{*}\) is uniformly minimal if there exists \(C>0\) such that
Theorem 1.5
Let \(f\in H^{p}_{\text {s}}(\mathbb {T})\) for some \(1<p< \infty .\) Then, the system \(\{ e^{i nt}f(t)\}_{n=0}^{\infty }\) is uniformly minimal in \(H^{p}(\mathbb {T}),\) if and only if \([f]^{-1}\in H^{p'}(\mathbb {T})\).
Theorem 1.6
Let \(f\in H^{1}_{\text {s}}(\mathbb {T}).\) If the system \(\{ e^{i nt}f(t)\}_{n=0}^{\infty }\) is uniformly minimal in \(H^{1}(\mathbb {T})\), then \([f]^{-1}\in H^{\infty }(\mathbb {T})\). If \([f]^{-1}\in H^{\infty }(\mathbb {T})\) and the partial sums of its Fourier series are uniformly bounded in the \(C(\mathbb {T})\) norm, then the system \(\{ e^{i nt}F^{*}(t)\}_{n=0}^{\infty }\) is uniformly minimal in \(H^{1}(\mathbb {T})\).
For our study, we need to know the dual system to \(\{ e^{i nt}f(t)\}_{n=0}^{\infty }\) in Theorem 1.5. By \(S[\varphi ](t)\), we denote the Fourier series of a function \(\varphi \in L^{1}(\mathbb {T})\). By (1.5), we know that \(c_{0}(f) \ne 0,\) if \(f\in H^{1}_{\text {s}}(\mathbb {T}).\) For any \(m\in \mathbb {N}\)
Let \(f\in H^{1}_{\text {s}}(\mathbb {T})\) and assume without loss of generality that \(c_{0}(f) = 1.\) We set
which is a non-zero holomorphic function. Hence, \([F_{f}(z)]^{-1}\) is also a holomorphic function in \(\mathbb {D}:\)
We have that
Set \(T_{0}(f,t)\equiv 1\) and for \(n\in \mathbb {N}\)
By (1.7), we obtain that if \(j\in \mathbb {N}_{0}\) and \(j\le n\)
It is clear that the above integral is equal to zero if \(j>n\).
An integrable non-negative function on \(\mathbb {T}\) is called a weight function. We say that a weight function w is in the class \(\mathcal {A}_{ p}(\mathbb {T}), p\ge 1\) if there exists \(C_{p} > 0\) such that for any interval \(I\subset \mathbb {T}\)
Sometimes it is called Muckenhoupt’s condition [15].
If \(f\in H^{p}_{\text {s}}(\mathbb {T})\) for some \(1<p< \infty\) and \(\varphi \in H^{p}(\mathbb {T})\), as it was shown in [12], the partial sums of the expansion of \(\varphi\) with respect to the system \(\{ e^{i nt}f(t)\}_{n=0}^{\infty }\) can be represented by the formulae
A function \(g \in L^{p}(\mathbb {T},w), 1\le p <\infty\) if \(g: \mathbb {T}\rightarrow \mathbb {C}\) is measurable on \(\mathbb {T}\) and the norm is defined by
The trigonometric system with different multipliers
The sufficiency of the following theorem was proved in [12].
Theorem 2.1
Let \(1< p<\infty\) and let \(f\in H^{p}_{\text {s}}(\mathbb {T}).\) Then, the system \(\{e^{i nt}f(t)\}_{n=0}^{\infty }\) is a Schauder basis in \(H^{p}(\mathbb {T})\) if and only if \(|f|^{p} \in \mathcal {A}_{p}(\mathbb {T}).\)
Proof
We give only the proof of the necessity. Assume that the system \(\{ e^{int}f(t)\}_{n=0}^{\infty }\) is a Schauder basis in \(H^{p}(\mathbb {T})\). Then, it is uniformly minimal in \(H^{p}(\mathbb {T})\) and by Theorem 1.5 it follows that \([f]^{-1}\in H^{p^{\prime }}(\mathbb {T})\). Thus, for some \(C_{p}>0\) independent of \(\varphi\)
The last inequality yields (see, e.g. [17], p.40)
\(\square\)
We need the following lemma for the proof.
Lemma 2.1
Let \(g\in L^{p}(\mathbb {T})\) and \(h\in H^{p^{\prime }}(\mathbb {T}),\) where \(1\le p<\infty\). Then for \(0<r<1, t\in \mathbb {T}\)
Proof
For \(z=re^{it}, |z| <1\) and any \(k\in \mathbb {N}_{0}\), we have
Hence, for any \(m\in \mathbb {N}\)
Which yields
Let \(\psi (\theta ) = \text {Re} \varphi (\theta )\). By Lemma 2.1, we easily check that
Thus, for any real-valued \(\psi \in L^{p}(\mathbb {T})\)
where \(B_{p}>0\) is independent of \(\psi .\) Afterwards, it is easy to see that the same inequality holds for any \(\psi \in L^{p}(\mathbb {T}).\) If we set \(\omega (t) = |f(t)|^{p}\), then it follows that for some \(B_{p}>0\) independent of \(g\in L^{p}(\mathbb {T},\omega )\)
By [7, 17], we finish the proof. \(\square\)
Let \(L,M\in H^{p}_{\text {s}}(\mathbb {T}), 1\le p<\infty .\) Consider the system of functions (1.1). From Theorem 1.5, we deduce
Theorem 2.2
Let \(1\le p<\infty\) and let \(L,M\in H^{p}_{\text {s}}(\mathbb {T}).\) Then, the system \(\Psi _{L,M}\) is an \(M-\)basis in \(L^{p}(\mathbb {T}).\)
Proof
Suppose that the system \(\Psi _{L,M}\) is not complete in \(L^{p}(\mathbb {T}).\) Then in the dual space \(L^{p'}(\mathbb {T})\), there exists a non-trivial \(\phi \in L^{p'}(\mathbb {T})\) such that
and
From the above relations, it follows that \(L \phi \in H^{1}(\mathbb {T})\) and \(M\overline{\phi }\) is not a constant and is a \(H^{1}(\mathbb {T})\) function. Afterwards, by Proposition 1.4 and Theorem 1.2, we obtain that \(\phi \in H^{p'}(\mathbb {T})\) and \(\overline{\phi }\in H^{p'}(\mathbb {T})\). Which can happen if and only if \(\phi \equiv const\). Thus, \(M \equiv const,\) which contradicts the condition that \(M\overline{\phi }\) is not a constant function.
Let us show that the system
is dual to \(\Psi _{L,M}\). We have that for any \(j\in \mathbb {N}\)
Thus from (1.7), it follows that for any \(k\in \mathbb {N}\)
For any \(j\in \mathbb {N}\) by (1.9), we obtain that
The rest of the proof of the minimality of the system \(\Psi _{L,M}\) is clear. It remains to check that the system (2.1) is total. Assume that it is not true. Then for some non-trivial \(\psi \in L^{p}(\mathbb {T})\) such that
and
The relations on the last line are equivalent to the following conditions: \(\int _{\mathbb {T}} \psi (t) e^{-i jt} dt = 0\) for all \(j\in \mathbb {N}_{0}.\) Using the fact that \(\int _{\mathbb {T}} \psi (t) dt = 0\), we deduce that \(\int _{\mathbb {T}} \psi (t) e^{i kt} dt = 0\quad \forall k\in \mathbb {N}.\) Hence, \(\psi (t) =0\) a.e. on \(\mathbb {T}.\) \(\square\)
By the above result and Theorem 1.5, we obtain
Theorem 2.3
Let \(1< p<\infty\) and let \(L,M\in H^{p}_{\text {s}}(\mathbb {T}).\) Then, the system \(\Psi _{L,M}\) is uniformly minimal in \(L^{p}(\mathbb {T}),\) if and only if the functions \([L]^{-1}, [M]^{-1}\in H^{p'}(\mathbb {T})\).
From Theorems 2.2 and 2.3, we easily obtain
Theorem 2.4
Let \(1\le p<\infty\), \(k\in \mathbb {Z}\) and let \(L,M\in H^{p}_{\text {s}}(\mathbb {T}).\) Then, the system
is an \(M-\)basis in \(L^{p}(\mathbb {T}).\) Moreover, if \(1\le p<\infty\), the system \(e^{i kt} \Psi _{L,M}\) is uniformly minimal in \(L^{p}(\mathbb {T}),\) if and only if the functions \([L]^{-1}, [M]^{-1}\in H^{p'}(\mathbb {T})\).
When \(p=1\) by Theorem 1.6, we have
Theorem 2.5
Let \(L,M\in H^{1}_{\text {s}}(\mathbb {T})\) and let \(k\in \mathbb {Z}\). If the system \(e^{i kt} \Psi _{L,M}\) is uniformly minimal in \(L^{1}(\mathbb {T})\), then \([L]^{-1},[M]^{-1}\in H^{\infty }(\mathbb {T})\). Moreover, if \([L]^{-1},[M]^{-1}\in H^{\infty }(\mathbb {T})\) and the partial sums of the Fourier series of the functions \([L]^{-1},[M]^{-1}\) are uniformly bounded in the \(C(\mathbb {T})\) norm, then the system \(e^{i kt} \Psi _{L,M}\) is uniformly minimal in \(L^{1}(\mathbb {T})\).
The following lemma was proved in [12]
Lemma 2.2
Let \(f\in H^{p}(\mathbb {T})\) and \(g\in H^{p'}(\mathbb {T}),\) where \(1\le p<\infty\) and \(\frac{1}{p} + \frac{1}{p'} =1,\) \(p'=\infty\) if \(p=1\). Then, \(S_{n}[fg](t) = {\sum }_{j=0}^{n} c_{j}(f) e^{ijt} S_{n-j}[g](t)\) for any \(n\in \mathbb {N}_{0}\).
Theorem 2.6
Let \(1< p<\infty\), \(k\in \mathbb {Z}\) and let \(L,M\in H^{p}_{\text {s}}(\mathbb {T}).\) Then the system \(e^{i kt} \Psi _{L,M}\) is a Schauder basis in \(L^{p}(\mathbb {T})\) if and only if \(|L|^{p}, |M|^{p} \in \mathcal {A}_{p}(\mathbb {T}).\)
Proof
By Theorem 2.4, we know that the system \(e^{i kt} \Psi _{L,M}\) is complete and minimal in \(L^{p}(\mathbb {T})\). For any \(n\in \mathbb {N}\), we set
Let \(g\in L^{p}(\mathbb {T})\) be a real-valued function and let \(g_{H}\) be its projection on \(H^{p}(\mathbb {T})\). By Theorem 2.2, it follows that
Hence, for any \(n\in \mathbb {N}\)
We have that
The last equation is obtained by Lemma 2.2. In a similar way, we deduce (see also [12])
We set
Let \(u(t) = |L(t)|^{p},\) and let \(v(t) = |M(t)|^{p}\), then by a well-known weighted norm inequality [7] (see also [10]) we finish the proof of sufficiency because the conditions of Banach’s theorem [1] hold in our case. We will give the proof of the inequality
where \(C_{p}>0\) is independent of g. Indeed,
The proof of the necessity follows by Theorem 2.1. \(\square\)
References
S. Banach, Théorie des opérations linéaires, Chelsea Publ. Co. (New York, 1955).
N. K. Bary, A Treatise on Trigonometric Series, GIFML, Moscow (1961). English trans.:Pregamon Press, New York (1964).
A. Beurling, On two problems concerning linear transformations in Hilbert space, Acta Math. 81 (1949), 239–255.
J.S. Byrnes, D.J. Newman, Completeness preserving multipliers, Proc. Amer. Math. Soc., 21(1969), 445–450.
J.S. Byrnes, Complete multipliers, Trans. Amer. Math. Soc., 172(1972), 399–403.
P.L. Duren, Theory of \(H^{p}\) spaces, New-York: Academic Press (1970).
R. Hunt, B. Muckenhoupt and R.L. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc., 176 (1973), 227–251.
U. Grenander and G. Szegö, Toeplitz forms and their applications, University of California Press (1958).
K. Hoffman, Banach spaces of analytic functions, Englewood Cliffs: Prentice-Hall (1962).
K.S. Kazarian, On bases and unconditional bases in the spaces \(L^{p} p (d\mu ),\)\(1\le p < \infty\), Studia Math. 71 (1982), 227–249.
K. S. Kazarian, Summability of generalized Fourier series and Dirichlet’s problem in \(L^p(d\mu )\) and weighted \(H^p\)-spaces \((p>1)\), Analysis Mathematica, 13 (1987) 173–197.
K.S. Kazarian, On representations by Beurling systems , presented for publication (2022).
P. Koosis, Introduction to \(H_{p}\) spaces, Cambridge University Press (1980).
J. Lindenstrauss and L. Tzafriri, Classical Banach spaces I, Springer (1977).
B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., 165(1972), 207–226.
F. and R. Nevanlinna, Über die Eigenschafen analytischer Funktionen in der Umgebung einer singulären Stelle oder Linie, Acta Soc. Sci. Fenn., 50, 5 (1922).
M. Rosenblum, Summability of Fourier series in \(L^{p}(d\mu )\), sTrans. Amer. Math. Soc., 105 (1962), 32–42.
I. Singer, Bases in Banach spaces II, Springer-Verlag, Berlin Heidelberg New York (1981).
G. Szegö, Über die Randwerte analytischer Funktionen, Math. Ann., 84(1921), 232–244.
A. Zygmund, Trigonometric series, v. 1 and 2, Cambridge Univ. Press (1959).
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Kazarian, K.S. MULTIPLIERS OF THE TRIGONOMETRIC SYSTEM. J Math Sci 271, 7–17 (2023). https://doi.org/10.1007/s10958-022-06248-2
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DOI: https://doi.org/10.1007/s10958-022-06248-2