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

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.

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: {sin(𝑛𝑡+𝛼(𝑡))}^∞_𝑛=1,1∪{cos(𝑛𝑡+𝛼(𝑡))}^∞_𝑛=1, (1)(2) 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 𝐿_𝑝, 1<𝑝<+∞, are completely studied in [7–12]. The case of weighted 𝐿_𝑝 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

Let 𝐿_𝑝,𝜌(0,𝜋) and 𝑊¹_𝑝,𝜌(0,𝜋) be weighted Lebesgue and Sobolev spaces with the following norms: ‖𝑢‖^𝑝_{𝐿𝑝,𝜌}=^𝜋₀|𝑢(𝜃)|^𝑝𝜌(𝜃)𝑑𝜃,‖𝑢‖_𝑊¹_𝑝,𝜌=‖𝑢‖_{𝐿𝑝,𝜌}+𝑢^′_{𝐿𝑝,𝜌}, (3) where 𝜌(𝜃)=𝜃^𝑎(𝜋−𝜃)^𝑏|𝜃−𝜃₀|^𝑐, 𝜃₀∈(0,𝜋), 𝑎,𝑏,𝑐∈(−1,+∞). Denote by ℒ_𝑝,𝜌 the following direct sum: ℒ_𝑝,𝜌=𝐿_𝑝,𝜌⊕𝐶, (4) 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)∈ℒ_𝑝,𝜌. The following easily provable lemmas play an important role in obtaining our main result.

Lemma 1. Let 𝑝∈(1,+∞),𝑎,𝑏,𝑐∈(−1,𝑝−1). Then the operator 𝐴(𝑢;𝜆)=𝜆+^𝑡₀𝑢(𝜃)𝑑𝜃 (5) performs an isomorphism between the spaces ℒ_𝑝,𝜌 and 𝑊¹_𝑝,𝜌(0,𝜋); that is, the spaces ℒ_𝑝,𝜌 and 𝑊¹_𝑝,𝜌 are isomorphic.

Proof. First we show the boundedness of this operator. We have the following: ‖𝐴𝑢‖_{𝑊𝑝,𝜌}=𝜆+^𝑡₀𝑢(𝜃)𝑑𝜃_{𝐿𝑝,𝜌}+‖𝑢‖_{𝐿𝑝,𝜌}≤‖𝜆‖_{𝐿𝑝,𝜌}+^𝑡₀𝑢(𝜃)𝑑𝜃_{𝐿𝑝,𝜌}+‖𝑢‖_{𝐿𝑝,𝜌}=^𝜋₀|𝜆|^𝑝𝜌(𝜃)𝑑𝜃^1/𝑝+^𝜋₀^𝑡₀𝑢(𝜃)𝑑𝜃^𝑝𝜌(𝜏)𝑑𝜏^1/𝑝+‖𝑢‖_{𝐿𝑝,𝜌}≤|𝜆|𝜌_𝐿^1/𝑝₁+^𝜋_𝑜^𝑡₀|𝑢(𝜃)|𝑑𝜃^𝑝𝜌(𝜏)𝑑𝜏^1/𝑝+‖𝑢‖_{𝐿𝑝,𝜌}≤|𝜆|𝜌_𝐿^1/𝑝₁+^𝜋₀^𝜋₀|𝑢(𝜃)|𝑑𝜃^𝑝𝜌(𝜏)𝑑𝜏^1/𝑝+‖𝑢‖_{𝐿𝑝,𝜌}=|𝜆|𝜌_𝐿^1/𝑝₁+^𝜋₀|𝑢(𝜃)|𝑑𝜃𝜌_𝐿^1/𝑝₁+‖𝑢‖_{𝐿𝑝,𝜌}. (6) Applying Hlder's inequality, we obtain the following: ^𝜋₀|𝑢(𝜃=)|𝑑𝜃_𝜋0|𝑢(𝜃)|𝜌^1/𝑝𝜌^−(1/𝑝)≤^𝜋₀|𝑢(𝜃)|^𝑝𝜌𝑑𝜃^𝜋₀𝜌^{−(𝑞/𝑝)}(𝜃)𝑑𝜃^1/𝑞=‖𝑢‖_{𝐿𝑝,𝜌}𝜌^1/(1−𝑝)_𝐿^1/𝑞₁1,where𝑝+1𝑞=1. (7) Consequently, ‖𝐴𝑢‖_{𝑊𝑝,𝜌}≤|𝜆|𝜌_𝐿^1/𝑝₁+‖𝑢‖_{𝐿𝑝,𝜌}𝜌_𝐿^1/𝑝₁𝜌^1/(1−𝑝)_𝐿^1/𝑞₁+‖𝑢‖_{𝐿𝑝,𝜌}=|𝜆|𝜌_𝐿^1/𝑝₁+𝜌_𝐿^1/𝑝₁𝜌^1/(1−𝑝)_𝐿^1/𝑞₁+1‖𝑢‖_{𝐿𝑝,𝜌}≤𝑀|𝜆|+‖𝑢‖_{𝐿𝑝,𝜌}=𝑀‖𝑢‖_{𝐿𝑝,𝜌}, (8) where 𝑀=max{‖𝜌‖_𝐿^1/𝑝₁;‖𝜌‖_𝐿^1/𝑝₁‖𝜌^1/(1−𝑝)‖_𝐿^1/𝑞₁}.
Let us show that Ker𝐴={0}. Let 𝐴𝑢=0; that is, 𝜆+^𝑡₀𝑢(𝜃)𝑑𝜃=0,∀𝑡∈(0,𝜋), (9) where 𝜆∈𝐶, 𝑢∈𝐿_𝑝,𝜌. Differentiating this equation, we obtain 𝑢(𝜃)=0 a.e. on (0,𝜋).
It follows that 𝜆=0. From (9) it directly follows that 𝑢=0 a.e. on (0,𝜋) and this implies that 𝑢=0.
Show that Im𝐴=𝑊¹_𝑝,𝜌(0,𝜋). Let 𝑣∈𝑊¹_𝑝,𝜌(0,𝜋) be an arbitrary function. Assume 𝑣̂=(𝑣^′,𝑣(0)). It is clear that 𝐴𝑣̂=𝑣 and 𝑣̂∈𝐿_𝑝,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. Let 𝑝∈(1,+∞) and 𝑎,𝑏,𝑐∈(−1,𝑝−1). Then for all 𝑝₀∈(1,𝛼)∶𝐿_𝑝,𝜌⊂𝐿_𝑝0(0,𝜋), where 𝛼=min{𝑝;𝑝/(𝑎+1);𝑝/(𝑏+1);𝑝/(𝑐+1);2}.

Proof. Let 𝑓∈𝐿_𝑝,𝜌(0,𝜋),  𝑝₀∈(1,𝛼). We have the following: ^𝜋₀𝑓^𝑝0=𝑑𝑡_𝜋0𝑓^𝑝0𝜌^𝑝0/𝑝𝜌^{−(𝑝0/𝑝)}≤𝑑𝑡_𝜋0𝑓^𝑝𝜌𝑑𝑡^{𝑝0/𝑝𝜋}₀𝜌^{𝑝0/(𝑝0−𝑝)}𝑑𝑡^{(𝑝−𝑝0)/𝑝}=𝑓^𝑝0_{𝐿𝑝,𝜌}^𝜋₀𝑡^{𝑎𝑝0/(𝑝0−𝑝)}𝑡−𝑡₀^{𝑐𝑝0/(𝑝0−𝑝)}(𝜋−𝑡)^{𝑏𝑝0/(𝑝0−𝑝)}𝑑𝑡^{(𝑝−𝑝0)/𝑝}. (10) 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∈𝐿_𝑝0(0,𝜋) and, moreover, 𝑓_𝑝0≤𝐶𝑓_{𝐿𝑝,𝜌}. (11) This proves the lemma.

From results of the paper [24] it follows the validity of the following lemma.

Lemma 3. Let 𝑓∈𝐿_𝑝,𝜌(0,𝜋), 𝜌=𝑡^𝑎|𝑡−𝑡₀|^𝑐(𝜋−𝑡)^𝑏, 𝑎,𝑏,𝑐∈(−1,𝑝−1), and 𝑓(𝑡)=∑^∞_𝑛=1𝑐_𝑛sin(𝑛+𝛽/2)𝑡 in 𝐿_𝑝,𝜌(0,𝜋). Then the series ∑^∞_𝑛=1(𝑐_𝑛/(𝑛+𝛽/2)) converges absolutely.

3. Main Result

Theorem 4. Let 𝑝∈(1,+∞), 𝑎,𝑏,𝑐∈(−1,𝑝−1). Then system (2) forms a basis for 𝑊¹_𝑝,𝜌(0,𝜋) if and only if system (1) forms a basis for 𝐿_𝑝,𝜌(0,𝜋), where 𝛼(𝑡)=(𝛽/2)𝑡,𝛽≠−2𝑘, 𝑘=1,2,3,….

Proof. First let us assume that the system 𝛽sin𝑛+2𝑡^∞_𝑛=1 (12) forms a basis for 𝐿_𝑝,𝜌(0,𝜋). Show that the system {𝑢_𝑛(𝑡)}^∞_𝑛=0 forms a basis for ℒ_𝑝,𝜌, where 𝑢₀=(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 ℒ_𝑝,𝜌 can be expanded in a unique way as a series of the following: 𝑢=^∞_𝑛=0𝜆_𝑛𝑢_𝑛; (13) that is, 𝑢(𝑡)=−^∞_𝑛=1𝛽𝑛+2𝜆_𝑛𝛽sin𝑛+2𝑡,𝜆=𝜆₀+^∞_𝑛=1𝜆_𝑛. (14)(15) Since system (1) forms a basis for 𝐿_𝑝,𝜌(0,𝜋), we have expansion (14), and the coefficients {𝜆_𝑛}^∞_𝑛=1 are determined uniquely. By Lemma 3, the series ∑^∞_𝑛=1𝜆_𝑛 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 ℒ_𝑝,𝜌.
Consider the system {𝜈_𝑛}^∞_𝑛=0, where 𝜈_𝑛=𝐴𝑢_𝑛,𝑛≥0, 𝐴𝑢=𝐴[(𝑢;𝜆)]=𝜆+∫^𝑡₀𝑢(𝜏)𝑑𝜏, and 𝑢=(𝑢;𝜆)∈ℒ_𝑝,𝜌. By Lemma 1, the operator 𝐴 is an isomorphism in 𝐿(𝑊¹_𝑝,𝜌;ℒ_𝑝,𝜌). Then the system {𝜈_𝑛}^∞_𝑛=0 forms a basis for 𝑊¹_𝑝,𝜌(0,𝜋). It is easy to see that 𝜈₀=1,𝜈_𝑛𝛽=cos𝑛+2𝑡,𝑛≥1. (16) Now we prove the converse. Assume that the system 1∪{cos(𝑛+𝛽/2)𝑡}^∞_𝑛=1 forms a basis for 𝑊¹_𝑝,𝜌(0,𝜋).
Consider the system 𝑢_𝑛=𝐴⁻¹𝜈_𝑛,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 ℒ_𝑝,𝜌 and 𝑢₀=(0,1),𝑢_𝑛=−𝛽𝑛+2𝛽sin𝑛+2𝑡;1. (17) Consequently, 𝑢=(𝑢;𝜆)∈ℒ_𝑝,𝜌has a unique expansion of the form (14), (15) in ℒ_𝑝,𝜌. As a result, we obtain that for all 𝑢∈𝐿_𝑝,𝜌 has an expansion with respect to system (1) in 𝐿_𝑝,𝜌. Suppose there exists another expansion of 𝑢 in 𝐿_𝑝,𝜌: 𝑢(𝑡)=^∞_𝑛=1𝛽𝑛+2𝜇_𝑛𝛽sin𝑛+2𝑡. (18)
Then Lemma 3 implies the absolute convergence of series ∑^∞_𝑛=1𝜇_𝑛.
Assume 𝜇₀=𝜆−∑^∞_𝑛=1𝜇_𝑛. It is clear that the biorthogonal coefficients of the element 𝑢=(𝑢;𝜆) are (𝜇;𝜇₀), where 𝜇={𝜇_𝑛}^∞_𝑛=1.
Then, from basicity of system {𝑢_𝑛}^∞_𝑛=0 in ℒ_𝑝,𝜌, we obtain that 𝜇_𝑛=𝜆_𝑛, for all  𝑛≥0. So we came upon a contradiction which proves the theorem.