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

$$\begin{aligned} \Psi _{L,M} = \{ \overline{L(t)}e^{i nt}\}_{n=-\infty }^{-1} \cup \{ M(t) e^{i nt}\}_{n=0}^{\infty }, \end{aligned}$$
(1.1)

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

$$\begin{aligned} H^{p}(\mathbb {T})=\{\phi \in L^{p}(\mathbb {T}): \int _{\mathbb {T}} \phi (t) e^{i nt} dt = 0 \, \text {for all} \, n\in \mathbb {N}\}, \end{aligned}$$

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

$$\begin{aligned} g*h(t) = \frac{1}{2\pi } \int _{\mathbb {T}} g(\theta ) h(t-\theta ) d\theta . \end{aligned}$$

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

$$\begin{aligned} F(re^{it}) = e^{i\alpha }e^{\phi *H_r(t)}, \quad \alpha \in \mathbb {T}, \end{aligned}$$

where \(\phi\) is a real-valued integrable function defined on \(\mathbb {T}\) [3] and

$$\begin{aligned} H_r (\theta ) = \frac{1 + re^{i\theta }}{1 -re ^{i \theta }} \quad (0< r < 1, \theta \in \mathbb {T}). \end{aligned}$$

The Cauchy and the Poisson kernels are defined as follows:

$$\begin{aligned} C_{r}(\theta ) = \sum \limits _{n=0 }^{+\infty } r^{n} e^{in\theta }\quad 0<r<1, \theta \in \mathbb {T}, \end{aligned}$$
$$\begin{aligned} P_{r} (\theta ) = \sum \limits _{n=- \infty }^{+\infty } r^{|n|} e^{in \theta } = \frac{1 - r^2}{1 - 2r \cos \theta + r^2}. \end{aligned}$$

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

$$\begin{aligned} \varphi (t) = e^{i\alpha } |\varphi (t)| e^{i l_{\varphi }(t)}, \quad \text {for some}\, \alpha \in \mathbb {T}, \end{aligned}$$
(1.2)

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

$$\begin{aligned} S(\psi )(t) = |\psi (t)| e^{i l_{\psi }(t)}, \quad t\in \mathbb {T}. \end{aligned}$$

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:

$$\begin{aligned} S(\phi \psi )(t) = S(\phi )(t)\cdot S(\psi )(t); \quad S(\frac{1}{\psi })(t) = \frac{1}{S(\psi )(t)}. \end{aligned}$$
(1.3)

For a complex-valued integrable function g defined on \(\mathbb {T}\), such that \(\ln |g(t)|\) is integrable, we set

$$\begin{aligned} G_{g}(re^{it}) = e^{\ln |g|*H_r(t) }. \end{aligned}$$
(1.4)

Evidently \(G_{g}(z)\) is a non-zero holomorphic function in \(z\in \mathbb {D},\) \(G_{g}\in H^{1}(\mathbb {D})\) and

$$\begin{aligned} \lim _{r\rightarrow 1-} G_{g}(re^{it}) = S(g)(t) \quad \text {a.e. on}\, \mathbb {T}. \end{aligned}$$

We also have that

$$\begin{aligned} \frac{1}{2\pi } \int _{\mathbb {T}} S(g)(t) dt = G_{g}(0) = e^{\frac{1}{2\pi } \int _{\mathbb {T}} \ln |g(t)| dt }\ne 0. \end{aligned}$$
(1.5)

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 \(f\in H^{1}(\mathbb {T}),\)

$$\begin{aligned} f(t) = F(t)\cdot S(f)(t), \end{aligned}$$

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

$$\begin{aligned} \textbf{N}(\mathbb {T})=\bigg \{ \frac{g}{S(h)}: g,h \in H^{\infty }(\mathbb {T}) \bigg \}. \end{aligned}$$

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

$$\begin{aligned} f_{0}(t) = \left\{ \begin{array}{ll} \frac{1}{f(t)} \quad \text {if } \quad \frac{1}{|f(t)|} \le 1 \\ 1 \qquad \text {if } \quad |f(t)| < 1. \end{array}\right. \end{aligned}$$

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

$$\begin{aligned} \phi = \frac{\phi _{1}}{\phi _{2}};\quad \psi = \frac{\psi _{1}}{\psi _{2}}, \end{aligned}$$

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 \(\beta _{1}\in \mathbb {T}\)

$$\begin{aligned} e^{i\beta _{1}} \psi = S(\frac{\psi _{1}}{\psi _{2}}) = \frac{S(\psi _{1})}{S(\psi _{2})}. \end{aligned}$$

Thus,

$$\begin{aligned} \frac{\phi }{\psi }= & {} e^{i\beta _{1}} \frac{\phi }{S(\psi )} = e^{i\beta _{1}} \frac{\phi _{1}S(\psi _{2})}{\phi _{2}S(\psi _{1})}\\= & {} e^{i\beta _{2}}\frac{\phi _{1}S(\psi _{2})}{S(\phi _{2})S(\psi _{1})} = e^{i\beta _{2}} \frac{\phi _{1}S(\psi _{2})}{S(\phi _{2}\psi _{1})}, \end{aligned}$$

where \(\beta _{2}\in \mathbb {T}\). \(\square\)

We also have

Theorem 1.2

For any \(p, 1\le p \le \infty\)

$$\begin{aligned} L^{p}(\mathbb {T})\cap \textbf{N}(\mathbb {T}) = H^{p}(\mathbb {T}). \end{aligned}$$

Proof

Let \(\phi \in L^{p}(\mathbb {T})\) and

$$\begin{aligned} \phi = \frac{g}{S(h)}, \quad \text {where}\, g,h \in H^{\infty }(\mathbb {T}) \end{aligned}$$

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

$$\begin{aligned} \phi (t) = \frac{g(t)}{S(h)(t)}= \frac{G(t) S(g)(t)}{S(h)(t)} = G(t) S(\frac{g}{h})(t). \end{aligned}$$

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

$$\begin{aligned} \int _{\mathbb {T}} f(t) e^{int} \overline{g(t)} dt = 0 \quad \text {for all}\, n\in \mathbb {N}_{0} = \{0,1,2,\dots \}. \end{aligned}$$
(1.6)

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

$$\begin{aligned} \overline{g(t)} = \frac{\varphi (t)}{f(t)} \in H^{p'}(\mathbb {T}). \end{aligned}$$

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

$$\begin{aligned} \int _{\mathbb {T}} e^{ikt} g(t) dt = 0 \quad \text {for all}\, k\in \mathbb {N}. \end{aligned}$$

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

$$\begin{aligned} \phi _{ n}(x_{k}) = \delta _{n k} \quad (n, k \in \mathbb {N}), \end{aligned}$$

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

$$\begin{aligned} \eta (x) = 0 \quad \text {for all} \quad \eta \in \Psi \end{aligned}$$

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

$$\begin{aligned} \Vert x_{k}\Vert _{\textbf{B}} \Vert \phi _{ k}\Vert _{\textbf{B}^{*}} \le C \qquad \text {for all} \quad k\in \mathbb {N}_{0}. \end{aligned}$$

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 \(m\in \mathbb {N}\)

$$\begin{aligned} S_{m}[\varphi ](t) = \sum \limits _{j=-m}^{m} c_{j}(\varphi ) e^{ijt}, \qquad c_{j}(\varphi ) = \frac{1}{2\pi }\int _{\mathbb {T}} \varphi (\theta ) e^{-ij\theta } d\theta . \end{aligned}$$

Let \(f\in H^{1}_{\text {s}}(\mathbb {T})\) and assume without loss of generality that \(c_{0}(f) = 1.\) We set

$$\begin{aligned} F_{f}(z) = \sum \limits _{j=0}^{\infty } c_{j}(f) z^{j} \qquad z\in \mathbb {D}, \end{aligned}$$

which is a non-zero holomorphic function. Hence, \([F_{f}(z)]^{-1}\) is also a holomorphic function in \(\mathbb {D}:\)

$$\begin{aligned}{}[F_{f}(z)]^{-1} = \sum \limits _{j=0}^{\infty } b_{f,j} z^{j} \qquad z\in \mathbb {D}. \end{aligned}$$

We have that

$$\begin{aligned} \sum \limits _{j=0}^{k} c_{j}(f) b_{f,k-j} = 0 \quad \text {for all} \quad k\in \mathbb {N} \quad \text {and} \quad b_{f,0} = 1. \end{aligned}$$
(1.7)

Set \(T_{0}(f,t)\equiv 1\) and for \(n\in \mathbb {N}\)

$$\begin{aligned} T_{n}(f,t) = e^{i nt} + \sum \limits _{\nu =0}^{n-1} \bar{b}_{f,n-\nu } e^{i \nu t} \qquad t\in \mathbb {T}. \end{aligned}$$
(1.8)

By (1.7), we obtain that if \(j\in \mathbb {N}_{0}\) and \(j\le n\)

$$\begin{aligned} \frac{1}{2\pi } \int _{\mathbb {T}} e^{i j t} f(t) \overline{T_{n}(f,t)} dt = \frac{1}{2\pi } \int _{\mathbb {T}} \sum \limits _{k=0}^{n-j} c_{k}(f) e^{i(j+k) t} \overline{T_{n}(f,t)} dt \end{aligned}$$
$$\begin{aligned} = \sum \limits _{k=0}^{n-j} c_{k}(f) b_{f,n-j-k} = \delta _{j n}. \end{aligned}$$
(1.9)

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 \(I\subset \mathbb {T}\)

$$\begin{aligned} \frac{1}{|I|} \int _{I} w(t)dt \left[ \frac{1}{|I|} \int _{I} w(t)^ {-\frac{1}{p-1}}dt \right] ^{p-1} \le C_{ p}. \end{aligned}$$

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

$$\begin{aligned} \sigma _{m}[\varphi ](t) = f(t) S_{m}[\varphi f^{-1}](t) \qquad m\in \mathbb {N}_{0}. \end{aligned}$$
(1.10)

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

$$\begin{aligned} \Vert g \Vert _{L^{p}(\mathbb {T},w)}= \left( \int _{\mathbb {T}} |g(t)|^{p} w(t) dt\right) ^{\frac{1}{p}} < +\infty . \end{aligned}$$

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 \(\varphi\)

$$\begin{aligned} \Vert f \cdot S_{m}[\varphi f^{-1}] \Vert _{L^{p}} = \Vert \sigma _{m}[\varphi ] \Vert _{H^{p}} \le C_{p} \Vert \varphi \Vert _{H^{p}} \qquad \forall m\in \mathbb {N}_{0}. \end{aligned}$$

The last inequality yields (see, e.g. [17], p.40)

$$\begin{aligned} \Vert f \cdot (\varphi f^{-1})*P_{r} \Vert _{L^{p}} \le C_{p} \Vert \varphi \Vert _{H^{p}} \qquad \text {for}\, 0<r<1. \end{aligned}$$

\(\square\)

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 \(0<r<1, t\in \mathbb {T}\)

$$\begin{aligned} \frac{1}{2\pi } \int _{\mathbb {T}} g(\theta ) h(\theta ) P_{r}(t-\theta ) d\theta = \frac{1}{2\pi } \int _{\mathbb {T}} g(\theta ) P_{r}(t-\theta ) d\theta \frac{1}{2\pi } \int _{\mathbb {T}} h(\theta ) P_{r}(t-\theta ) d\theta . \end{aligned}$$

Proof

For \(z=re^{it}, |z| <1\) and any \(k\in \mathbb {N}_{0}\), we have

$$\begin{aligned} \frac{1}{2\pi } \int _{\mathbb {T}} g(\theta ) e^{ik\theta } P_{r}(t-\theta ) d\theta = \sum \limits _{j=-\infty }^{+\infty } c_{j}(g) z^{j+k} = z^{k}\sum \limits _{j=-\infty }^{+\infty } c_{j}(g) z^{j}. \end{aligned}$$

Hence, for any \(m\in \mathbb {N}\)

$$\begin{aligned} \frac{1}{2\pi } \int _{\mathbb {T}} g(\theta ) \sum \limits _{k=0}^{m} c_{k}(h) e^{ik\theta } P_{r}(t-\theta ) d\theta = \sum \limits _{k=0}^{m} c_{k}(h) z^{k} \sum \limits _{j=-\infty }^{+\infty } c_{j}(g) z^{j}. \end{aligned}$$

Which yields

$$\begin{aligned} \frac{1}{2\pi } \int _{\mathbb {T}} g(\theta ) h(\theta ) P_{r}(t-\theta ) d\theta= & {} \underset{m\rightarrow +\infty }{\lim } \frac{1}{2\pi } \int _{\mathbb {T}} g(\theta ) \sum \limits _{k=0}^{m} c_{k}(h) e^{ik\theta } P_{r}(t-\theta ) d\theta \\= & {} \sum \limits _{k=0}^{+\infty } c_{k}(h) z^{k} \sum \limits _{j=-\infty }^{+\infty } c_{j}(g) z^{j} = h*P_{r}(t) \cdot g*P_{r}(t). \end{aligned}$$

Let \(\psi (\theta ) = \text {Re} \varphi (\theta )\). By Lemma 2.1, we easily check that

$$\begin{aligned} |(\psi f^{-1})*P_{r}(t)| \le |(\varphi f^{-1})*P_{r}(t)| \qquad 0<r<1, t\in \mathbb {T}. \end{aligned}$$

Thus, for any real-valued \(\psi \in L^{p}(\mathbb {T})\)

$$\begin{aligned} \Vert f \cdot (\psi f^{-1})*P_{r} \Vert _{L^{p}} \le B_{p} \Vert \psi \Vert _{L^{p}} \qquad \text {for}\, 0<r<1, \end{aligned}$$

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 \(g\in L^{p}(\mathbb {T},\omega )\)

$$\begin{aligned} \Vert g*P_{r} \Vert _{L^{p}(\mathbb {T},\omega )} \le B_{p} \Vert g \Vert _{L^{p}(\mathbb {T},\omega )} \qquad \text {for}\, 0<r<1, \end{aligned}$$

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

$$\begin{aligned} \int _{\mathbb {T}} \overline{L(t)} e^{-i nt} \overline{\phi (t)} dt = 0\quad n\in \mathbb {N} \end{aligned}$$

and

$$\begin{aligned} \int _{\mathbb {T}} M(t) e^{i nt} \overline{\phi (t)} dt = 0\quad n\in \mathbb {N}_{0}. \end{aligned}$$

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

$$\begin{aligned} \{\overline{T_{-n}(L,t)} \}_{n=-\infty }^{-1} \cup \{ T_{n}(M,t)\}_{n=0}^{\infty } \end{aligned}$$
(2.1)

is dual to \(\Psi _{L,M}\). We have that for any \(j\in \mathbb {N}\)

$$\begin{aligned} \int _{\mathbb {T}} e^{i jt} M(t) e^{i nt} dt = 0\quad \text {for all}\, n\in \mathbb {N}_{0}. \end{aligned}$$

Thus from (1.7), it follows that for any \(k\in \mathbb {N}\)

$$\begin{aligned} \int _{\mathbb {T}} T_{k}(L,t) M(t) e^{i nt} dt = 0\quad \text {for all}\, n\in \mathbb {N}_{0}. \end{aligned}$$

For any \(j\in \mathbb {N}\) by (1.9), we obtain that

$$\begin{aligned} \frac{1}{2\pi } \int _{\mathbb {T}} e^{ -i j t} \overline{L(t)}T_{n}(L,t) dt = \frac{1}{2\pi } \overline{\int _{\mathbb {T}} e^{i j t} L(t) \overline{ T_{n}(L,t)} dt } = \delta _{j n} \quad \forall n\in \mathbb {N}. \end{aligned}$$

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

$$\begin{aligned} \int _{\mathbb {T}} \psi (t) T_{n}(L,t) dt = 0\quad n\in \mathbb {N} \end{aligned}$$

and

$$\begin{aligned} \int _{\mathbb {T}} \psi (t) \overline{T_{n}(M,t)} dt = 0\quad n\in \mathbb {N}_{0}. \end{aligned}$$

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

$$\begin{aligned} e^{i kt} \Psi _{L,M} = \{ \overline{L(t)}e^{i (n+k)t}\}_{n=-\infty }^{-1} \cup \{ M(t) e^{i (n+k)t}\}_{n=0}^{\infty } \end{aligned}$$

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

$$\begin{aligned} B^{*}_{L,n}(t,\theta )= & {} \overline{L(t)} \sum \limits _{k=-n}^{-1} e^{i kt} T_{-k} (L,\theta ) \\= & {} \overline{L(t)} \sum \limits _{k= 1}^{n} e^{-i kt} \sum \limits _{j=0}^{k} \overline{b_{L,k-j}} e^{i j\theta } \\= & {} \overline{L(t)} \sum \limits _{k=1}^{n}\overline{b_{L,k}} e^{-i kt}\\{} & {} + \overline{L(t)} \sum \limits _{j=1}^{n} e^{i j\theta } \sum \limits _{k=j}^{n} \overline{b_{L,k-j}}e^{-i kt} \\= & {} \overline{L(t)}\, \overline{S_{n}[L^{-1}-1](t)} \\{} & {} + \overline{L(t)} \sum \limits _{j=1}^{n} e^{i j(\theta -t)} \overline{S_{n-j}[L^{-1}](t)} \end{aligned}$$

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

$$\begin{aligned} \int _{\mathbb {T}} g_{H}(\theta ) B^{*}_{L,n}(t,\theta ) d\theta = 0\quad n\in \mathbb {N}. \end{aligned}$$

Hence, for any \(n\in \mathbb {N}\)

$$\begin{aligned} \Lambda ^{*}_{L,n}[g](t) = \frac{1}{2\pi } \int _{\mathbb {T}} g(\theta ) B^{*}_{L,n}(t,\theta ) d\theta = \Lambda ^{*}_{L,n}[g- g_{H}](t). \end{aligned}$$

We have that

$$\begin{aligned} \Lambda ^{*}_{L,n}[g- g_{H}](t)= & {} \overline{L(t)} \sum \limits _{j=1}^{n} c_{-j}(g) e^{-i jt} \overline{S_{n-j}[L^{-1}](t)} \\= & {} \overline{L(t)} \sum \limits _{j=1}^{n} \overline{c_{j}(g)} e^{-i jt} \overline{S_{n-j}[L^{-1}](t)}\\= & {} \overline{L(t)} \sum \limits _{j=1}^{n} \overline{c_{j}(\overline{g-g_{H}})} e^{-i jt} \overline{S_{n-j}[L^{-1}](t)}\\= & {} \overline{L(t)}\, \overline{S_{n }[\overline{(g-g_{H})}L^{-1}](t)}. \end{aligned}$$

The last equation is obtained by Lemma 2.2. In a similar way, we deduce (see also [12])

$$\begin{aligned} B_{M,n}(t,\theta )= & {} M(t) \sum \limits _{k=0}^{n} e^{i kt} \overline{ T_{k} (M,\theta )}\\= & {} M(t)\sum \limits _{k=0}^{n} e^{i kt} \sum \limits _{j=0}^{k} b_{M,k-j}e^{-i j\theta }\\= & {} M(t)\sum \limits _{j=0}^{n} e^{-i j\theta } \sum \limits _{k=j}^{n} b_{M,k-j}e^{i kt}\\= & {} M(t)\sum \limits _{j=0}^{n} e^{i j(t-\theta )} S_{n-j}[M^{-1}](t). \end{aligned}$$

We set

$$\begin{aligned} \Lambda _{M,n}[g](t)= & {} \frac{1}{2\pi } \int _{\mathbb {T}} g(\theta ) B_{M,n}(t,\theta ) d\theta \\= & {} M(t) \sum \limits _{j=0}^{n} c_{j}(g_{H}) e^{ijt} S_{n-j}[M^{-1}](t)\\= & {} M(t) S_{n}[g_{H}M^{-1}](t) = M(t) S_{n}[g_{H}M^{-1}](t), \end{aligned}$$

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

$$\begin{aligned} \int _{\mathbb {T}} |\Lambda ^{*}_{L,n}[g](t)|^{p} dt \le C_{p} \int _{\mathbb {T}}|g(t)|^{p} dt, \end{aligned}$$

where \(C_{p}>0\) is independent of g. Indeed,

$$\begin{aligned} \int _{\mathbb {T}} |\Lambda ^{*}_{L,n}[g](t)|^{p} dt= & {} \int _{\mathbb {T}} |\Lambda ^{*}_{L,n}[g-g_{H}](t)|^{p} dt\\= & {} \int _{\mathbb {T}}| L(t)\, S_{n }[\overline{(g-g_{H})}L^{-1}](t)|^{p} dt\\= & {} \int _{\mathbb {T}}|S_{n }[\overline{(g-g_{H})}L^{-1}](t)|^{p} u(t) dt \\\le & {} A_{p} \int _{\mathbb {T}}|[g(t)- g_{H}(t)] [L(t)]^{-1}|^{p} u(t) dt\\= & {} A_{p} \int _{\mathbb {T}}|g(t)- g_{H}(t)|^{p} dt\\\le & {} C_{p} \int _{\mathbb {T}}|g(t)|^{p} dt. \end{aligned}$$

The proof of the necessity follows by Theorem 2.1. \(\square\)