# Basicity of Systems of Sines with Linear Phase in Weighted Sobolev Spaces

Authors: V. F. Salmanov; A. R. Safarova
Journal: International Journal of Engineering Mathematics
 (2013)

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## Abstract

The perturbed systems of sines, which appear when solving some partial differential equations by the Fourier method, are considered in this paper. Basis properties of these systems in weighted Sobolev spaces of functions are studied.


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## Body

## 1. Introduction

When solving many problems in mathematical physics by Fourier method (see e.g., [1–4]), there appear perturbed systems of sines and cosines of the following form:
(1){sin(nt+α(t))}n=1∞,(2)1∪{cos(nt+α(t))}n=1∞,
where α(t)=βt+γ, β,γ∈R are real parameters. Using Fourier method requires the study of basis properties of the above systems in Lebesgue and Sobolev spaces of functions. Relevant investigations date back to the well-known works by Paley and Wiener [5] and Levinson [6]. For γ=0, basis properties of these systems in spaces Lp, 1<p<+∞, are completely studied in [7–12]. The case of weighted Lp was considered by E. I. Moiseev in [13, 14]. Basis properties of some perturbed systems of exponents in Sobolev spaces are studied in [15–19]. Further references include [20–23].Our paper is devoted to the study of basis properties of these systems in weighted Sobolev spaces. Unlike previous works, we offer a different method of investigation.


## 2. Auxiliary Facts

LetLp,ρ(0,π) and Wp,ρ1(0,π) be weighted Lebesgue and Sobolev spaces with the following norms:
(3)∥u∥Lp,ρp=∫0π‍|u(θ)|pρ(θ)dθ,∥u∥Wp,ρ1=∥u∥Lp,ρ+∥u′∥Lp,ρ,
where ρ(θ)=θa(π-θ)b|θ-θ0|c, θ0∈(0,π), a,b,c∈(-1,+∞). Denote by ℒp,ρ the following direct sum:
(4)ℒp,ρ=Lp,ρ⊕C,
where C is a complex plane. The norm in this space is defined as follows: ∥u^∥ℒp,ρ=∥u∥Lp.ρ+|λ|, where u^=(u,λ)∈ℒp,ρ. The following easily provable lemmas play an important role in obtaining our main result.Lemma 1.
Letp∈(1,+∞),a,b,c∈(-1,p-1). Then the operator
(5)A(u;λ)=λ+∫0t‍u(θ)dθ
performs an isomorphism between the spaces ℒp,ρ and Wp,ρ1(0,π); that is, the spaces ℒp,ρ and Wp,ρ1 are isomorphic.Proof.
First we show the boundedness of this operator. We have the following:(6)∥Au^∥Wp,ρ=∥λ+∫0t‍u(θ)dθ∥Lp,ρ+∥u∥Lp,ρ≤∥λ∥Lp,ρ+∥∫0t‍u(θ)dθ∥Lp,ρ+∥u∥Lp,ρ=(∫0π‍|λ|pρ(θ)dθ)1/p+(∫0π‍|∫0t‍u(θ)dθ|pρ(τ)dτ)1/p+∥u∥Lp,ρ≤|λ|∥ρ∥L11/p+(∫oπ‍(∫0t‍|u(θ)|dθ)pρ(τ)dτ)1/p+∥u∥Lp,ρ≤|λ|∥ρ∥L11/p+(∫0π‍(∫0π‍|u(θ)|dθ)pρ(τ)dτ)1/p+∥u∥Lp,ρ=|λ|∥ρ∥L11/p+∫0π‍|u(θ)|dθ∥ρ∥L11/p+∥u∥Lp,ρ.
Applying Hlder's inequality, we obtain the following:
(7)∫0π‍|u(θ)|dθ=∫0π‍|u(θ)|ρ1/pρ-(1/p)≤∫0π‍|u(θ)|pρdθ(∫0π‍ρ-(q/p)(θ)dθ)1/q=∥u∥Lp,ρ∥ρ1/(1-p)∥L11/q,where1p+1q=1.
Consequently,
(8)∥Au^∥Wp,ρ≤|λ|∥ρ∥L11/p+∥u∥Lp,ρ∥ρ∥L11/p∥ρ1/(1-p)∥L11/q+∥u∥Lp,ρ=|λ|ρL11/p+(∥ρ∥L11/p∥ρ1/(1-p)∥L11/q+1)∥u∥Lp,ρ≤M(|λ|+∥u∥Lp,ρ)=M∥u^∥Lp,ρ,
where M=max{∥ρ∥L11/p;∥ρ∥L11/p∥ρ1/(1-p)∥L11/q}.
Let us show thatKerA={0}. Let Au^=0; that is,
(9)λ+∫0t‍u(θ)dθ=0,∀t∈(0,π),
where λ∈C, u∈Lp,ρ. Differentiating this equation, we obtain u(θ)=0 a.e. on (0,π).
It follows thatλ=0. From (9) it directly follows that u=0 a.e. on (0,π) and this implies that u^=0.
Show thatImA=Wp,ρ1(0,π). Let v∈Wp,ρ1(0,π) be an arbitrary function. Assume v^=(v′,v(0)). It is clear that Av^=v and v^∈Lp,ρ. Then by Banach theorem we find that the operator A has a bounded inverse. This proves Lemma 1.Now let us prove the following lemma.Lemma 2.
Letp∈(1,+∞) and a,b,c∈(-1,p-1). Then for all p0∈(1,α):Lp,ρ⊂Lp0(0,π), where α=min{p;p/(a+1);p/(b+1);p/(c+1);2}.Proof.
Letf∈Lp,ρ(0,π),  p0∈(1,α).We have the following:
(10)∫0π‍|f|p0dt=∫0π‍|f|p0ρp0/pρ-(p0/p)dt≤(∫0π‍|f|pρdt)p0/p(∫0π‍ρp0/(p0-p)dt)(p-p0)/p=∥f∥Lp,ρp0(∫0πtap0/(p0-p)|t-t0|cp0/(p0-p)(π-t)bp0/(p0-p)dt∫0πtap0/(p0-p)|t-t0|cp0/(p0-p))(p-p0)/p.
As p0<p/(a+1) and a>-1, then ap0/(p0-p)>-1. Similarly, we find that bp0/(p0-p)>-1 and cp0/(p0-p)>-1. It is easy to see that f∈Lp0(0,π) and, moreover,
(11)∥f∥p0≤C∥f∥Lp,ρ.
This proves the lemma.From results of the paper [24] it follows the validity of the following lemma.Lemma 3.
Letf∈Lp,ρ(0,π), ρ=ta|t-t0|c(π-t)b, a,b,c∈(-1,p-1), and f(t)=∑n=1∞‍cnsin(n+β/2)t in Lp,ρ(0,π). Then the series ∑n=1∞‍(cn/(n+β/2)) converges absolutely.


## 3. Main Result

Theorem 4.
Letp∈(1,+∞), a,b,c∈(-1,p-1). Then system (2) forms a basis for Wp,ρ1(0,π) if and only if system (1) forms a basis for Lp,ρ(0,π), where α(t)=(β/2)t,β≠-2k, k=1,2,3,….Proof.
First let us assume that the system(12){sin(n+β2)t}n=1∞
forms a basis for Lp,ρ(0,π). Show that the system {u^n(t)}n=0∞ forms a basis for ℒp,ρ, where u^0=(0;1), u^n=(-(n+β/2)sin[(n+β/2)t];1), for n≥1. It suffices to show that every element u^=(u;λ) of space ℒp,ρ can be expanded in a unique way as a series of the following:
(13)u^=∑n=0∞‍λnu^n;
that is,
(14)u(t)=-∑n=1∞‍(n+β2)λnsin[(n+β2)t],(15)λ=λ0+∑n=1∞‍λn.
Since system (1) forms a basis for Lp,ρ(0,π), we have expansion (14), and the coefficients {λn}n=1∞ are determined uniquely. By Lemma 3, the series ∑n=1∞‍λn converges absolutely. Then, it is clear that the number λ0 in (15) is determined uniquely. And this implies that the system {u^n}n=0∞ forms a basis for ℒp,ρ.
Consider the system{νn}n=0∞, where νn=Au^n,n≥0, Au^=A[(u;λ)]=λ+∫0t‍u(τ)dτ, and u^=(u;λ)∈ℒp,ρ. By Lemma 1, the operator A is an isomorphism in L(Wp,ρ1;ℒp,ρ). Then the system {νn}n=0∞ forms a basis for Wp,ρ1(0,π). It is easy to see that
(16)ν0=1,νn=cos(n+β2)t,n≥1.
Now we prove the converse. Assume that the system 1∪{cos(n+β/2)t}n=1∞ forms a basis for Wp,ρ1(0,π).
Consider the systemu^n=A-1νn,n≥0. The inverse operator A-1 is defined as A-1ν=(ν′;ν(0)). It is absolutely clear that the system {u^n}n=0∞ forms a basis for ℒp,ρ and
(17)u^0=(0,1),u^n=(-(n+β2)sin(n+β2)t;1).
Consequently, u^=(u;λ)∈ℒp,ρhas a unique expansion of the form (14), (15) in ℒp,ρ. As a result, we obtain that for all u∈Lp,ρ has an expansion with respect to system (1) in Lp,ρ. Suppose there exists another expansion of u in Lp,ρ:
(18)u(t)=∑n=1∞‍(n+β2)μnsin(n+β2)t.
Then Lemma3 implies the absolute convergence of series ∑n=1∞‍μn.
Assumeμ0=λ-∑n=1∞‍μn. It is clear that the biorthogonal coefficients of the element u^=(u;λ) are (μ¯;μ0), where μ¯={μn}n=1∞.
Then, from basicity of system{u^n}n=0∞ in ℒp,ρ, we obtain that μn=λn, for all  n≥0. So we came upon a contradiction which proves the theorem.



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*Source: 612472-2013-05-18.xml*