MULTIPLIERS OF THE TRIGONOMETRIC SYSTEM

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 ( 𝕋 ) , 1 < p < ∞ .


Introduction
We say that a system {z m F(z)} ∞ m=0 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 ( ) , then the system {z m F(z)} ∞ m=0 is complete in the space H 2 ( ) , where = {z ∈ ℂ ∶ |z| < 1} is the unit disc. This result can be easily extended for the spaces H p ( ), 1 ≤ p < ∞ (see [6]). In the previous paper [12], we studied questions of representations of functions from the spaces H p ( ), 1 ≤ p < ∞ 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 . We find necessary and sufficient conditions on functions L and M so that the system is a Schauder basis in L p ( ), 1 < p < ∞ , where = ℝ∕2 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 ( ), 1 ≤ p < ∞ , and the second part will be dedicated to the study of the systems e ikt Ψ L,M , k ∈ ℤ.

Preliminary results, definitions and notations
Let for 1 ≤ p ≤ ∞. The spaces H p ( ), 1 ≤ p ≤ ∞ are Banach spaces of functions defined on . The convolution of functions g, h ∈ L( ) is denoted by We would like to define on the analogue of an outer function on the unit disc = {z ∈ ℂ ∶ |z| < 1} . A holomorphic function F in is an outer function if where is a real-valued integrable function defined on [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 is called S−function. We say that a Lebesgue measurable function ∶ → ℂ is an S−function if ln | | ∈ L 1 ( ) and where l (t) =l n | |(t) and the conjugate function of an integrable function g is denoted by g . For any measurable function ∶ → ℂ such that ln | | ∈ L 1 ( ) , we put The following statement follows from the definition. It is easy to observe that the following properties also are true: For a complex-valued integrable function g defined on , such that ln |g(t)| is integrable, we set is a non-zero holomorphic function in z ∈ , G g ∈ H 1 ( ) and .
We also have that 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)| ∈ L 1 ( ) if f ∈ H 1 ( ). The following fact is an easy consequence of well-known results that can be found in the literature which we have mentioned earlier.
Given f ∈ H p s ( ) , we can recover a non-zero holomorphic outer function in by the Poisson integral. If is a nonnegative integrable function such that ln ∈ L 1 ( ) , then is the modulus of a function from H 1 ( ) (see, e.g. [9], p.53). If 1 < p < ∞ , we use the previous assertion for 1∕p to deduce a similar proposition for a function in H p ( ) . The case p = ∞ is studied in [9]. Thus, the following statement holds.

Proposition 1.3 Let g be a Lebesgue measurable function
The analogue of the following result is well known in . We give the proof because it is short.
Proof By Theorem 1.1, we have that We also have By Proposition 1.2, we have that The closed linear span in a separable Banach space B of a system of elements Beurling's approximation theorem [3,6] in the H p ( ), 1 ≤ p < ∞ spaces have a simple formulation. We give its proof for the completeness of the exposition.
Proof Consider the case p > 1 . We assume that there exists g ∈ H p � ( ), 1∕p + 1∕p � = 1 such that , then ∈ H 1 ( ). By Proposition 1.4 and Theorem 1.2, it follows that Hence, g is a constant function. This means that ∫ f (t)dt = 0 , which contradicts the condition (1.5). If p = 1 , then there exists g ∈ L ∞ ( ) 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. ◻ The following results were obtained by the author in the recent paper.
where nk is the Kronecker symbol ( nk = 0 if n ≠ k and kk = 1) . The system X * is called dual to X. If X is a complete and minimal system in B , then the dual system X * is unique [14].
if X is complete and minimal in B and its dual system X * is total. The following theorems were proved in [12] Theorem 1.4 Let p ∈ [1, +∞) and let f ∈ H p s ( ). The system {e int f (t)} ∞ n=0 is an M−basis in H p ( ).
That the system {e int } ∞ n=0 is minimal in H p ( , w), 1 ≤ p < ∞ was mentioned in [11] without proof. A complete and minimal system For our study, we need to know the dual system to {e int f (t)} ∞ n=0 in Theorem 1.5. By S[ ](t) , we denote the Fourier series of a function ∈ L 1 ( ) . By (1.5), we know that c 0 (f ) ≠ 0, if f ∈ H 1 s ( ). For any m ∈ ℕ Let f ∈ H 1 s ( ) 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 ∶ We have that ∫ e ikt g(t)dt = 0 for all k ∈ ℕ.
n (x k ) = nk (n, k ∈ ℕ), Set T 0 (f , t) ≡ 1 and for n ∈ ℕ By (1.7), we obtain that if j ∈ ℕ 0 and j ≤ n It is clear that the above integral is equal to zero if j > n. An integrable non-negative function on is called a weight function. We say that a weight function w is in the class A p ( ), p ≥ 1 if there exists C p > 0 such that for any interval I ⊂ Sometimes it is called Muckenhoupt's condition [15].
If f ∈ H p s ( ) for some 1 < p < ∞ and ∈ H p ( ) , as it was shown in [12], the partial sums of the expansion of with respect to the system {e int f (t)} ∞ n=0 can be represented by the formulae A function g ∈ L p ( , w), 1 ≤ p < ∞ if g ∶ → ℂ is measurable on and the norm is defined by

The trigonometric system with different multipliers
The sufficiency of the following theorem was proved in [12]. Proof We give only the proof of the necessity. Assume that the system {e int f (t)} ∞ n=0 is a Schauder basis in H p ( ) . Then, it is uniformly minimal in H p ( ) and by Theorem 1.5 it follows that [f ] −1 ∈ H p � ( ) . Thus, for some C p > 0 independent of The last inequality yields (see, e.g. [17], p.40)

◻
We need the following lemma for the proof.
For z = re it , |z| < 1 and any k ∈ ℕ 0 , we have Hence, for any m ∈ ℕ

Which yields
Let ( ) = Re ( ) . By Lemma 2.1, we easily check that Thus, for any real-valued ∈ L p ( ) where B p > 0 is independent of . Afterwards, it is easy to see that the same inequality holds for any ∈ L p ( ). If we set (t) = |f (t)| p , then it follows that for some B p > 0 independent of g ∈ L p ( , ) By [7,17], we finish the proof. Proof Suppose that the system Ψ L,M is not complete in L p ( ). Then in the dual space L p � ( ) , there exists a non-trivial ∈ L p � ( ) such that .  .7), it follows that for any k ∈ ℕ For any j ∈ ℕ by (1.9), we obtain that The rest of the proof of the minimality of the system Ψ 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 ∈ L p ( ) such that and The relations on the last line are equivalent to the following conditions: ∫ (t)e −ijt dt = 0 for all j ∈ ℕ 0 . Using the fact that ∫ (t)dt = 0 , we deduce that ∫ (t)e ikt dt = 0 ∀k ∈ ℕ. Hence, (t) = 0 a.e. on . ◻ By the above result and Theorem 1.5, we obtain   (2.1) The following lemma was proved in [12] Lemma 2.2 L et f ∈ H p ( ) a n d g ∈ H p � ( ), w h e re 1 ≤ p < ∞ a n d 1 p + 1 p � = 1, p � = ∞ i f p = 1 . Then, S n [fg](t) = ∑ n j=0 c j (f )e ijt S n−j [g](t) for any n ∈ ℕ 0 . Theorem 2.6 Let 1 < p < ∞ , k ∈ ℤ and let L, M ∈ H p s ( ). Then the system e ikt Ψ L,M is a Schauder basis in L p ( ) if and only if |L| p , |M| p ∈ A p ( ).
Proof By Theorem 2.4, we know that the system e ikt Ψ L,M is complete and minimal in L p ( ) . For any n ∈ ℕ , we set Let g ∈ L p ( ) be a real-valued function and let g H be its projection on H p ( ) . By Theorem 2.2, it follows that Hence, for any n ∈ ℕ We have that