In the present paper a criterion for basicity of exponential system with linear phase is obtained in Sobolev weight space Wp,ρ1(-π,π).

In solving mathematical physics problems by the Fourier method, there often arise the systems of exponents of the form
(1){ei(nt+α(t));e-i(nt+β(t))}n≥1,(2)1∪{ei(nt+α(t));e-i(nt+β(t))}n≥1,
where α(t) and β(t) are continuous or piecewise-continuous functions. Substantiation of the method requires studying the basis properties of these systems in Lebesgue and Sobolev spaces of functions. In the case when α(t) and β(t) are linear functions, the basis properties of these systems in Lp(-π,π), 1<p<+∞, were completely studied in the papers [1–9]. The weighted case of the space Lp was considered in the papers [10, 11]. The basis properties of these systems in Sobolev spaces were studied in [12–14]. It should be noted that the close problems were also considered in [15].

In the present paper we study basis properties of the systems (1) and (2) in Sobolev weight spaces when α(t)=αt, β(t)=αt, where α is a real parameter. Therewith the issue of basicity of system (2) in Sobolev spaces is reduced to the issue of basicity of system (1) in respective Lebesgue spaces.

Let Lp,ρ(-π,π) and Wp,ρ1(-π,π) be weight spaces with the norms
(3)∥u∥Lp,ρp=∫-ππ|u(θ)|pρ(θ)dθ,(4)∥u∥Wp,ρ1=∥u∥Lp,ρ+∥u′∥Lp,ρ,
respectively, where ρ(t)=∏i=1l(sin|(t-τi)/2|)βi, -π<τ1<τ2<⋯<τl<π. Denote by Lp,ρ(-π,π) the direct sum Lp,ρ=Lp,ρ⊕C, where C is the complex plane. The norm in this space is defined by the expression ∥u^∥Lp,ρ=∥u∥Lp,ρ+|λ|, where u^=(u;λ)∈Lp,ρ.

The following easily provable lemmas play an important role in obtaining the main results. It holds the following.

Lemma 1.

Let βi∈(-1,p-1), i=1,l¯; p∈(1,+∞). Then the operator
(5)A(u;λ)=λ+∫-πtu(θ)dθ
realizes an isomorphism between the spaces Lp,ρ and Wp,ρ1(-π,π); that is, the spaces Lp,ρ and Wp,ρ1 are isomorphic.

Proof.

At first prove the boundedness of the operator A. We have
(6)∥Au^∥Wp,ρ1=∥λ+∫-πtu(θ)dθ∥Lp,ρ+∥u∥Lp,ρ≤∥λ∥Lp,ρ+∥∫-πtu(θ)dθ∥Lp,ρ+∥u∥Lp,ρ=(∫-ππ|λ|pρ(θ)dθ)1/p+(∫-ππ|∫-πtu(θ)dθ|pρ(t)dt)1/p+∥u∥Lp,ρ≤|λ|∥ρ∥L11/p+(∫-ππ(∫-πt|u(θ)|dθ)pρ(t)dt)1/p+∥u∥Lp,ρ≤|λ|∥ρ∥L11/p+(∫-ππ(∫-ππ|u(θ)|dθ)pρ(t)dt)1/p+∥u∥Lp,ρ=|λ|∥ρ∥L11/p+∫-ππ|u(θ)|dθ∥ρ∥L11/p+∥u∥Lp,ρ.
Having applied the Holder inequality, hence we get
(7)∫-ππ|u(θ)|dθ=∫-ππ|u(θ)|ρ1/pρ-1/pdθ≤(∫-ππ|u(θ)|pρdθ)1/p×(∫-ππρ-q/p(θ)dθ)1/q=∥u∥Lp,ρ∥ρ1/(1-p)∥L11/q,
where
(8)1p+1q=1.
Consequently
(9)∥Au^∥Wp,ρ1≤|λ|∥ρ∥L11/p+∥u∥Lp,ρ∥ρ∥L11/p∥ρ1/(1-p)∥L11/q+∥u∥Lp,ρ=|λ|ρL11/p+(1+∥ρ∥L11/p∥ρ1/(1-p)∥L11/q+1)∥u∥Lp,ρ≤M(|λ|+∥u∥Lp,ρ)=M∥u^∥Lp,ρ,
where
(10)M=max{∥ρ∥L11/p;∥ρ∥L11/p∥ρ1/(1-p)∥L11/q+1}.
Let us show that KerA={0}. Put Au^=0; that is,
(11)λ+∫-πtu(θ)dθ=0,∀t∈(-π,π),
where λ∈C, u∈Lp,ρ. By differentiating this equality, we get u(θ)=0, a.e. on (-π,π). Hence it follows that λ=0. From (11) it directly follows that u=0 a.e. on (-π,π), and so u^=0. Show that ImA=Wp,ρ1(-π,π) (ImA is the range of values of the operator A). Let v∈Wp,ρ1(-π,π) be an arbitrary function. Let v^=(v′;v(0)). It is clear that Av^=v and v^∈Lp,ρ. Then from the Banach theorem we get that the operator A is boundedly invertible.

The lemma is proved.

The following lemma is also valid.

Lemma 2.

Let p∈(1,+∞) and βi∈(-1,p-1), i=1,l¯. Then for all p0∈(1,α):Lp,ρ(-π,π)⊂Lp0(-π,π), where
(12)α=min{2;p;pβ1+1;pβ2+1;…;pβl+1}.

Proof.

Let f∈Lp,ρ(-π,π), p0∈(1,α). We have
(13)∫-ππ|f|p0dt=∫-ππ|f|p0ρp0/pρ-p0/pdt≤(∫-ππ|f|pρdt)p0/p(∫-ππρp0/(p0-p)dt)(p-p0)/p=∥f∥Lp,ρp0(∫-ππ∏i=1l(sin|t-τi2|)βip0/(p0-p)dt)(p-p0)/p.
Since p0<p/(βi+1) and βi>-1, then βip0/(p0-p)>-1, i=1,l¯. It is easy to see that f∈Lp0(-π,π) and moreover ∥f∥Lp0≤C∥f∥Lp,ρ.

The lemma is proved.

In obtaining the basic results we need the following main lemma.

Lemma 3.

Let f∈Lp,ρ(-π,π), ρ(t)=∏i=1l(sin|(t-τi)/2|)βi and βi∈(-1,p/q), i=1,l¯, α∈R be a real parameter, 1/p+1/q=1. Let f have the expansion
(14)f(t)=∑n≠0cnei[n+αsignn]t
in the space Lp,ρ(-π,π). Then it is valid
(15)∑|n|>|α||cnn+αsignn|<+∞.

Proof.

As it follows from Lemma 2, ∃p0∈(1,2):Lp,ρ⊂Lp0. At first consider the case when α>1/2p0-1, 1/p0+1/q0=1. In this case the system {ei(n+αsignn)t}n≠0 is minimal in Lp0(0,π) (see [4]). Then from the results of the paper [16], the Hausdorff-Young inequality is valid for this system; that is,
(16)(∑n≠0∞|cn|q0)1/q0≤M∥f∥Lp0.
Applying the Holder inequality, we obtain
(17)∑|n|>|α||cnn+αsignn|≤(∑|n|>|α||cn|q0)1/q0×(∑|n|>|α|1|n+αsignn|p0)1/p0.
If α<1/2p0-1, then ∃k∈N, α′=α+k>1/2p0-1. Then
(18)f(t)=∑n≠0cnei(n+αsignn)t=∑n=-k-1cnei(n-α)t+∑n=1kcnei(n+α)t+∑|n|≥k+1cnei(n+αsignn)t=fk(t)+∑n≠0cn+ksignnei(n+(k+α)signn)t=fk(t)+∑n≠0cn+ksignnei(n+α′signn)t,
where fk(t)=∑n=-k-1cnei(n-α)t+∑n=1kcnei(n+α)t, α′=k+α. As a result, we have
(19)f(t)-fk(t)=∑n≠0cn+ksignnei(n+α′signn)t.
Since β′>1/p0-2, then again from the results of the paper [16] it follows
(20)(∑n≠0|cn+ksignn|q0)1/q0≤M∥f-fk∥Lp0.
In the same way we establish the convergence of the series ∑|n|>|α||cn/(n+αsignn)|.

Consider the case when α=1/2p0-1. Then ∃p′∈(p0,2):α>1/2p′-1. In this case the system {ei(n+αsignn)t}n≠0 is minimal in Lp′, and, consequently, from the results of the paper [16] it holds the Hausdorff-Young inequality; that is,
(21)(∑n≠0|cn|q′)1/q′≤M∥f∥Lp′,
where 1/p′+1/q′=1.

From the previous reasonings we get the absolute convergence of the series ∑|n|>|α||cn/(n+αsignn)|. The lemma is proved.

Theorem 4.

Let p∈(1,+∞), ρ(t)=∏i=1l(sin|(t-τi)/2|)βi be a weight function, α(t)≡αt, β(t)≡αt, α∈R be a real parameter, and the inequalities -1<βi<p/q, i=1,l¯; 1/2p-1/2>α>1/2p-1 hold. Then the following statements are equivalent:

system (2) forms a basis for Wp,ρ(-π,π);

system (1) forms a basis for Lp,ρ(-π,π).

Proof.

At first, suppose that the system (1) forms a basis for Lp,ρ(-π,π). Let us show that the system {u^n}n=0∞ forms a basis for Lp,ρ, where u^0=(0;1), u^n=(i(n+αsignn)ei(n+αsignn)t;e-i(n+αsignn)π), n≠0.

It is enough to prove that the arbitrary element u^=(u;λ) of Lp,ρ has the unique expansion
(22)u^=∑n≠0λnu^n;
that is,
(23)u(t)=∑n≠0i(n+αsignn)λnei(n+αsignn)t,(24)λ=λ0+∑n≠0λn.
Since the system (1) forms a basis for Lp,ρ(-π,π), then the expansion (23) holds and the coefficients λn(n≠0) are uniquely determined. By Lemma 3 the series ∑n≠0λn absolutely converges. Then it is clear that the number λ0 from (24) is uniquely determined. This means that the system {u^n}n=∞∞ forms a basis for Lp,ρ. Consider the system {vn}n=-∞∞, where vn=Au^n, A[(u,λ)]=λ+∫-πtu(τ)dt, u^=(u;λ)∈Lp,ρ. It is not difficult to see that
(25)v0=1,vn=ei(n+αsignn)t,n≠0.

Now, let us prove the converse. Assume that the system (2) forms a basis for Wp1(-π,π). Consider the system u^n=A-1vn, n∈Z. It is easy to see that the inverse operator is determined as A-1v=(v′;v(-π)). It is obvious that the system {u^n}-∞∞ forms a basis for Lp,ρ. We have
(26)u^0=(0;1),u^n=(i(n+αsignn)ei(n+αsignn)t;e-i(n+αsignn)π).
Consequently, u^=(u;λ)∈Lp,ρ has a unique expansion (22) in Lp,ρ. As a result, we obtain that each u∈Lp,ρ has a unique expansion of the form (23). Indeed, let there exist another expansion for u(t) in Lp,ρ:
(27)u(t)=∑n≠0i(n+αsignn)μnei(n+αsignn)t.
The absolute convergence of the series ∑n≠0μn follows from Lemma 3. Put μ0=λ-∑n≠0μn. It is clear that the biorthogonal coefficients of the element u^=(u;λ) are (μ-;μ0), where μ-={μn}n≠0. From the basicity of the system {u^n}n=-∞∞ in Lp,ρ we obtain that μn=λn, n∈Z. This contradicts our conjecture. The theorem is proved.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors would like to express their sincere gratitude to Professor Bilal T. Bilalov for his attention to the paper and for valuable advice.

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