Abstract
In the present paper, we investigate under sufficient conditions the existence of frames of exponential families, where the exponents coincide with the eigenvalues of the perturbed operator
Here \(T_0\) is a closed densely defined linear operator on a separable Hilbert space \(\mathcal{H}\) with domain \(\mathcal{D}(T_0)\) having isolated eigenvalues with multiplicity one and \(T_1, T_2,\ldots \) are linear operators on \(\mathcal{H}\) having the same domain \(\mathcal{D}\supset \mathcal{D}(T_0)\) and satisfying a specific growing inequality. The obtained results are applied to a non-self adjoint problem deduced from a perturbation method for sound radiation.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Frames, which are systems that provide robust, stable and usually non-unique representations of vectors, have been well studied in literature with important applications where redundancy plays a vital and useful role. Frames were first introduced by Duffin and Schaeffer [13], reintroduced by Daubechies et al. [12] and developed later by several authors such as Christensen [8,9,10,11] and Young [28]. They have many nice properties which make them very useful in the characterization of function spaces, signal processing and many other fields. In fact, a frame can be considered as a generalized basis in the sense that every element of a separable Hilbert space \(\mathcal{H}\) can be written as a linear combination of the frame elements; while differing in a very important aspect. Indeed, they may be linearly dependent and therefore the uniqueness of representation characteristic of bases may be lost.
Actually, one has that a sequence of vectors is a Riesz basis if and only if it is a frame and is \(\omega \)-linearly independent. Nevertheless, we find that the Riesz basis property and especially Riesz basis of exponential families attracts the attention of many researchers and have many applications in mathematical physics. Among these applications, we cite the problem of radiation of a vibrating structure in a light fluid initially which was motivated by Filippi et al. [18] and widely considered in literature [5, 14,15,16,17, 20, 21]. Indeed, in [5] the authors studied the existence of a sequence of complex numbers \((\varepsilon _n)_{n\in \mathbb {N^*}}\) such that the family of exponentials associated to the eigenvalues of the operator
forms a Riesz basis in \(L^2\big (0,T\big )\), for some \(T>0\). Here K is the integral operator with kernel the Hankel function of the first kind and order 0 and \(\vert \varepsilon _n\vert <\frac{1}{\Vert K\Vert }\). More precisely, they have extended Eq. (1.1) to an abstract setting and they have considered the following operator introduced by Nagy [24]:
where \(\varepsilon \in \mathbb {C}\) and \(T_0\) is a closed densely defined linear operator on a separable Hilbert space \(\mathcal{H}\) with domain \(\mathcal{D}(T_0)\), while \(T_1, T_2,\ldots \) are linear operators on \(\mathcal{H}\) having the same domain \(\mathcal{D}\supset \mathcal{D}(T_0)\) and satisfying
where a, b and q are positive constants.
On the other hand, based on the spectral analysis developed by Nagy [24] and on a stability result of Riesz bases due to Schueller [26], Charfi et al. [5] assured, for each eigenvalue \(\lambda _n\) of \(T_0\), the existence of a sequence of complex numbers \((\varepsilon _n)_{n\in \mathbb {N}^*}\) and a sequence of eigenvalues \((\lambda _n(\varepsilon _n))_{n\in \mathbb {N}^*}\), which can be developed as entire series of \((\varepsilon _n)_{n\in \mathbb {N^*}}\), such that the system \(\{e^{i\lambda _n(\varepsilon _n)t}\}_{n\in \mathbb {N^*}}\) forms a Riesz basis in \(L^2(0,T)\).
However, the Riesz basis of exponentials hence obtained is related to the eigenvalues of a sequence of operators \((T(\varepsilon _n))_{n\in \mathbb {N}^*}\) and depend on the sequence \((\varepsilon _n)_{n\in \mathbb {N}^*}\). In order to guarantee the existence of a Riesz basis of exponentials where the exponents coincide with the eigenvalues of the perturbed operator (1.2) for a fixed complex number \(\varepsilon \), the authors in [14] used the following condition
where \(\beta \in ]0,1]\) and a, b and \(q>0\). They have proved that if the system \(\{e^{i\lambda _nt}\}_{n\in \mathbb {N^*}}\) forms a Riesz basis in \(L^2(0,T)\), then for \(\vert \varepsilon \vert \) enough small, there exists a sequence of eigenvalues \(\lambda _n(\varepsilon )\) of the perturbed operator \(T(\varepsilon )\) having the form
such that the family \(\{e^{i\lambda _n(\varepsilon )t}\}_{n\in \mathbb {N^*}}\) forms a Riesz basis in \(L^2(0,T)\). As an application, they have considered the following integro-differential operator
where \(\vert \varepsilon \vert <\frac{1}{\Vert K\Vert }\).
It is interesting to remember here that Duffin and Schaeffer [13] introduced the concept of frames in the context of systems of complex exponentials. Indeed, frames of exponentials are very special types of frames and their study is ultimately related to sampling theories tracing back to Paley–Wiener [25]. Moreover, it was shown in [4, 19, 27] that the frame properties for systems of exponentials \(\{e^{i\lambda _{n}t}\}_{n\in \mathbb {Z}}\) are closely related to density issues concerning the sequence \((\lambda _{n})_{n\in \mathbb {Z}}\). So, since frames are a much more flexible tool than orthonormal and Riesz bases, we try to extend the results established in [5] and [14] to the notion of frames. Indeed, it is very difficult to determine the density of the system of eigenvalues \((\lambda _{n}(\varepsilon ))_{n\in \mathbb {N^*}}\) of \(T(\varepsilon )\).
To grapple with such difficulties, we continue the analysis of the perturbed operator (1.2) under the new specific growing inequality for all \(\varphi \in \mathcal{D}(T_0)\):
where \(q, b_{1}, b_{2}, \ldots ,b_{N}\) are positive constants and \(\{\beta _{1}, \beta _{2}, \ldots ,\beta _{N}\}\subset ]0,1]\), with \(\beta _{i}\ne \beta _{j}\), for all \((i,j) \in \{1, \ldots , N\}^{2}.\)
Clearly, Eq. (1.6) generalizes inequalities given in Eqs. (1.3) and (1.4). On the other hand, we would like to mention that, for \(k=1\), Eq. (1.6) was recently introduced by Abdelmoumen et al. in [1] as the concept of generalized subordination that combines the two notions of relatively boundedness with order \(\beta \) and \(\beta \)-subordination used in [2, 6].
So, based on this new concept Eq. (1.6), we investigate under sufficient conditions ensuring the existence of a sequence of complex numbers \((\varepsilon _n)_{n\in \mathbb {N^*}}\) such that the family of exponentials \(\{e^{i\lambda _{n}(\varepsilon _n)t}\}_{n\in \mathbb {N^*}}\) is a frame for \(L^2(0,T)\), where the exponents coincide with the eigenvalues of the perturbed operator \(T(\varepsilon _n)\) which can be developed as entire series of \(\varepsilon _n\). Furthermore, in order to improve this result, we study the existence of a fixed complex number \(\varepsilon \) for at least the first \(M\ge 1\) vectors. More precisely, we show that for \(|\varepsilon |\) enough small the systems \(\{e^{i\lambda _n(\varepsilon ) t}\}_1^M \cup \{e^{i\lambda _n(\varepsilon _n)t}\}_{M+1}^{\infty }\) and \(\{e^{i\lambda _n(\varepsilon ) t}\}_1^M \cup \{e^{i\lambda _nt}\}_{M+1}^{\infty }\) are frames for \(L^2(0,T)\).
Clearly, the frames of exponential families that we obtain depend more or less on \(\varepsilon _n\). Actually, the exponents coincide with the eigenvalues of the perturbed operators \((T(\varepsilon _n))_{n\in \mathbb {N}^*}\) or even with the first M eigenvalues associated to the perturbed operator \(T(\varepsilon )\). It is along this line of thoughts that we try to give some supplements to these results. More accurately, we provide sufficient conditions ensuring the existence of a frame of exponentials for a fixed complex number \(\varepsilon \). Indeed, based on the spectral analysis developed in [7], we prove that for \(|\varepsilon |\) enough small, there exist a sequence of eigenvalues \((\lambda _n(\varepsilon ))_{n\in \mathbb {N^*}}\) related to the perturbed operator (1.2) having the form (1.5) such that the system \(\{e^{i\lambda _n(\varepsilon ) t}\}_{n\in \mathbb {N^*}}\) forms a frame in \(L^2(0,T)\). After that, we generalize these results using a H-lipschitz function.
This paper is organized as follows: In the next section, we state our main results concerning the existence of frames of exponentials related to the perturbed operator (1.2). Section 3 is devoted to apply the obtained results to a non-self adjoint problem of radiation of a vibrating structure.
2 Main Results
In this part, we are mainly concerned with the frames properties of the family of nonharmonic exponentials related to the perturbed operator \(T(\varepsilon )\) given by (1.2). Hence, we start with putting forward the following definition:
Definition 2.1
[8] A family \(\{\varphi _n\}_{n\in I}\) is said to be a frame for a separable Hilbert space \(\mathcal{H}\) if there exist positive constants \({\varvec{A}},{\varvec{B}}>0\) such that
where I a countable index set. The numbers \({\varvec{A}}\) and \({\varvec{B}}\) are called lower and upper frame bounds.\(\diamondsuit \)
In this paper, we consider a linear operator \(T_0\) acting on a separable Hilbert space \(\mathcal{H}\) verifying the following hypotheses:
- (H1):
-
\(T_0\) is closed with domain \(\mathcal{D}(T_0)\) dense in \(\mathcal{H}\).
- (H2):
-
The eigenvalues \((\lambda _n)_n\) of \(T_0\) are isolated, with multiplicity one and
$$\begin{aligned} \forall n\ge 1,~~|Im \lambda _{n}|\le h\quad \hbox { where }h:=\displaystyle \sup _{n}|Im\lambda _{n}|. \end{aligned}$$ - (H3):
-
The family of exponentials \(\{e^{i\lambda _nt}\}_1^{\infty }\) forms a frame in \(L^2(0,T)\), for some \(T>0\) with frame bounds A and B.
Let \(T_1, T_2, T_3,\ldots \) be linear operators on \(\mathcal{H}\) having the same domain \(\mathcal{D}\) and satisfying the hypothesis:
(H4) \(\mathcal{D}\supset \mathcal{D}(T_0)\) and there exist \(q, b_{1}, b_{2}, \ldots ,b_{N}\) positive constants and \(\{\beta _{1}, \beta _{2}, \ldots ,\beta _{N}\}\subset ]0,1]\), with \(\beta _{i}\ne \beta _{j}\), for all \((i,j) \in \{1, \ldots , N\}^{2}\), such that for all \(k\ge 1\)
Let \(\varepsilon \) be a non zero complex number and consider the eigenvalue problem
Now, we state the following results developed in [7].
Theorem 2.1
[7, Theorem 3.1] Assume that assumptions (H1) and (H4) hold. Then,
(i) For \(\vert \varepsilon \vert <q^{-1}\), the series \(\sum _{k\ge 0}\varepsilon ^kT_k \varphi \) converges for all \(\varphi \in \mathcal {D}(T_0).\) If \(T(\varepsilon )\varphi \) denotes its limit, then \(T(\varepsilon )\) is a linear operator with domain \(\mathcal {D}(T_0)\).
(ii) For \(\vert \varepsilon \vert <(q+\sum _{i=1}^N\beta _i b_i)^{-1},\) the operator \(T(\varepsilon )\) is closed.\(\diamondsuit \)
In the sequel, we designate by \(\lambda _n\) the eigenvalue number n of the operator \(T_0\). Since \((T_{0}-zI)^{-1}\) is an analytic function of z, \(\Vert (T_{0}-z)^{-1}\Vert \) is a continuous function of z. Hence, we denote by:
\(\mathcal{C}_{n}=\mathcal{C}(\lambda _{n},r_{n})\) the circle with center \(\lambda _{n}\) and with radii \(r_{n}=\frac{d_{n}}{2}\) and \(d_{n}=d(\lambda _{n},\sigma (T_{0}){\setminus }\{\lambda _{n}\})\) is the distance between \(\lambda _{n}\) and \(\sigma (T_{0}){\setminus }\{\lambda _{n}\}\).
Theorem 2.2
[7, Theorem 3.4] Assume that the hypotheses (H1), (H2) and (H4) are verified. Let \(\varphi _n\) (respectively, \(\varphi _n^{*})\) be an eigenvector of \(T_0\) (respectively, \(T_0^{*}\): the adjoint of \(T_0)\) associated to the eigenvalue \(\lambda _n\) (respectively, \(\overline{\lambda _n})\) such that \(\Vert \varphi _n\Vert =\Vert \varphi _n^{*}\Vert \) and \(\langle \varphi _n,\varphi _n^{*}\rangle =1.\) Then:
(i) For \(\vert \varepsilon \vert <\frac{1}{q+\sum _{i=1}^N\alpha _{n,i}+r_nM_n\sum _{i=1}^N\alpha _{n,i}},\)\(T(\varepsilon )\) will have a unique point of the spectrum in the neighborhood of \(\lambda _n\), and this point \(\lambda _n (\varepsilon )\) will be also with multiplicity one.
(ii) For \(\vert \varepsilon \vert <\frac{1}{q+\sum _{i=1}^N\alpha _{n,i}+\omega _n^2r_nM_n\sum _{i=1}^N\alpha _{n,i}},\)\(\lambda _n(\varepsilon )\) and the corresponding eigenvector \(\varphi _n(\varepsilon )\) of \(T(\varepsilon )\) can be developed into an entire series of \(\varepsilon {:}\)
and we have
where \(\omega _n=\Vert \varphi _n\Vert \) and \(\alpha _{n,i}:=b_iN_n^{\beta _i}M_n^{1-\beta _i}.\)\(\diamondsuit \)
Our result is formulated in the following theorem.
Theorem 2.3
Assume that hypotheses (H1)–(H4) are satisfied. Then, there exists a sequence of complex numbers \((\varepsilon _n)_{n\in \mathbb {N}^*}\) and a sequence of eigenvectors \(\{\lambda _n(\varepsilon _n)\}_{n\in \mathbb {N}^*}\) having the form
such that for \(\vert \varepsilon _n\vert <\frac{\sqrt{A}}{\pi \sqrt{\frac{2}{3}T}\omega _n^{2}r_n^{2}M_n\sum _{i=1}^N\alpha _{n,i}nte^{t(r_n+h)}+ \sqrt{A}(q+\sum _{i=1}^N\alpha _{n,i}+\omega _n^2r_nM_n\sum _{i=1}^N\alpha _{n,i})}\) the system \(\{e^{i\lambda _n(\varepsilon _n)t}\}_{n\in \mathbb {N^*}}\) forms a frame in \(L^{2}(0,T)\).\(\diamondsuit \)
To prove this result, we need the following result due to [9].
Lemma 2.1
[9, Theorem 1] Let \(\{\varphi _n\}_{n\in I}\) be a frame for \(\mathcal{H}\) with bounds \({\varvec{A}}\) and \({\varvec{B}}\) and let \(\{\psi _n\}_{n\in I}\) be a family of vectors in \(\mathcal{H}\) such that
Then, the family \(\{\psi _n\}_{n\in I}\) is a frame for \(\mathcal{H}\) with bounds \({{\varvec{A}}}\left( 1-\sqrt{\frac{M}{{{\varvec{A}}}}}\right) ^2\) and \({{\varvec{B}}}\left( 1+\sqrt{\frac{M}{{{\varvec{B}}}}}\right) ^2\). \(\diamondsuit \)
Proof of Theorem 2.3
Let \(n\in \mathbb {N}^*\) and \(\lambda _n\) the eigenvalue number n of \(T_0\). We have,
Using hypothesis (H2), we obtain
However, in view of Theorem 2.2, for \(|\varepsilon |<\frac{1}{q+\sum _{i=1}^N\alpha _{n,i}+r_{n}M_{n}\sum _{i=1}^N\alpha _{n,i}}\), the perturbed operator \(T(\varepsilon )\) will have a unique point of the spectrum in the neighborhood of \(\lambda _n\), and this point \(\lambda _{n}(\varepsilon )\) will be also with multiplicity one. Further, for \(|\varepsilon |<\frac{1}{q+\sum _{i=1}^N\alpha _{n,i}+\omega _{n}^{2} r_{n}M_{n}\sum _{i=1}^N\alpha _{n,i}}\), \(\lambda _{n}(\varepsilon )\) can be developed into an entire series of \(\varepsilon \):
where \(\lambda _{n,j}\) satisfies
For each eigenvalue \(\lambda _n\), we fix an \(\varepsilon _n\) such that
Obviously, we have
then the eigenvalue \(\lambda _n(\varepsilon _n)\) is inside \(\mathcal{C}_n\) and we get
Moreover, Eqs. (2.2) and (2.3) are satisfied. So, by combining Eqs. (2.1), (2.2), (2.3) and (2.4) we obtain
Therefore, we have
Consequently, Lemma 2.1 implies that the family \(\{e^{i\lambda _n(\varepsilon _n)t}\}_1^\infty \) forms a frame in \(L^{2}(0,T)\). \(\square \)
Now, we will show the existence of exponential frames where the first M exponents coincide with the eigenvalues of the perturbed operator \(T(\varepsilon )\) for a fixed complex number \(\varepsilon \).
Theorem 2.4
Suppose that hypotheses (H1)–(H4) hold. Then, there exists a sequence of complex numbers \((\varepsilon _n)_{n\in \mathbb {N}^*}\) and two sequences of eigenvalues \(\{\lambda _n(\varepsilon )\}_{n\in \mathbb {N}^*}\) and \(\{\lambda _n(\varepsilon _n)\}_{n\in \mathbb {N}^*}\) having the form
such that for \(\vert \varepsilon \vert <\min _{n\in [1, M]}\frac{\sqrt{A}}{\pi \sqrt{\frac{2}{3}T}\omega _n^{2}r_n^{2}M_n\sum _{i=1}^N\alpha _{n,i}nte^{t(r_{n}+h)}+\sqrt{A}(q+\sum _{i=1}^N\alpha _{n,i}+\omega _n^2r_nM_n \sum _{i=1}^N\alpha _{n,i})}\), the systems
(i) \(\{e^{i\lambda _n(\varepsilon ) t}\}_1^M \cup \{e^{i\lambda _n(\varepsilon _n)t}\}_{M+1}^{\infty }\)
(ii) \(\{e^{i\lambda _n(\varepsilon ) t}\}_1^M \cup \{e^{i\lambda _nt}\}_{M+1}^{\infty }\)
form frames in \(L^{2}(0,T)\).\(\diamondsuit \)
Proof
(i) Let \(n\in [1,M],~M\ge 1\). We reveal that
Hence, Theorem 2.2 entails the existence of a unique point of the spectrum of the perturbed operator \(T(\varepsilon )\) in the neighborhood of \(\lambda _n\), and this point \(\lambda _{n}(\varepsilon )\) will also be with multiplicity one. Also, Eqs. (2.2) and (2.3) are verified. Consequently, Eq. (2.5) yields
Then, Eq. (2.6) implies
Now, let \(n\ge M+1\). For each eigenvalue \(\lambda _n\) of \(T_0\), we fix an \(\varepsilon _n\in \mathbb {C}\) such that
Evidently, we have
Similarly as mentioned above, we prove that
Setting \(f_n\in \{e^{i\lambda _n(\varepsilon )t}\}_1^M \cup \{e^{i\lambda _n(\varepsilon _n)t}\}_{M+1}^{\infty }\). Using Eqs (2.7) and (2.8), we can see that
Hence, according to Lemma 2.1, the system \((f_n)_{n\in \mathbb {N}^{*}}\) forms a frame in \(L^{2}(0,T)\). Thus, the family \(\{e^{i\lambda _n(\varepsilon )t}\}_1^M \cup \{e^{i\lambda _n(\varepsilon _n)t}\}_{M+1}^{\infty }\) forms a frame in \(L^{2}(0,T)\). This achieves the proof of the first item.
(ii) The proof of the second item is similar to (i). \(\square \)
It is interesting to mention here that the frames of exponentials constructed in Theorems 2.3 and 2.4 rely totally on \(\varepsilon _n\) or partially on \(\varepsilon \). Hence, we intend to study the existence of a frame of exponentials related to \(T(\varepsilon )\).
Theorem 2.5
Assume that hypotheses (H1)–(H4) are verified. Suppose further that for all \(n\in \mathbb {N}^*\) there exists a sequence \((r_n)_{n\ge 1}\) in \(\mathbb {R}_+^*\) verifying
-
(i)
\(\displaystyle \{z\in \mathbb {C}~\hbox {such that}~|z-\lambda _n|\le r_n\}\cap \sigma (T_0)=\{\lambda _n\}\)
-
(ii)
\(K:=\displaystyle \sup _{n\ge 1}\left( q+\sum _{i=1}^N\alpha _{n,i}+\omega _n^{2}r_nM_n\sum _{i=1}^N\alpha _{n,i}\right) <\infty \)
-
(iii)
\(L:=\displaystyle \sum _{n=1}^{\infty }\left( \omega _n^{2}r_n^{2}M_n\sum _{i=1}^N\alpha _{n,i}\right) ^{2}\displaystyle \int _0^{T}t^2e^{2t(r_n+h)}dt<\infty \).
Then, for \(\vert \varepsilon \vert <\frac{\sqrt{A}}{2\sqrt{L}+ \sqrt{A}K}\) there exists a sequence of eigenvalues \((\lambda _n(\varepsilon ))_{n\in \mathbb {N}^*}\) of \(T(\varepsilon )\) that can be developed as entire series of \(\varepsilon \) such that the system \(\{e^{i\lambda _n(\varepsilon ) t}\}_1^{\infty }\) forms a frame in \(L^{2}(0,T)\).\(\diamondsuit \)
Proof
Let \(n\in \mathbb {N}^*\) and \(\lambda _n\) the eigenvalue number n of \(T_0\). Clearly,
So, it follows from Theorem 2.2 that the perturbed operator \(T(\varepsilon )\) will have a unique point of the spectrum in the neighborhood of \(\lambda _n\), denoted by \(\lambda _{n}(\varepsilon )\). Moreover, \(\lambda _{n}(\varepsilon )\) will be simple and can be developed as
with
Therefore, in view of Eq. (2.5) we get
As
we infer that
Hence, Eq. (2.9) entails the estimate
Then, it follows from Lemma 2.1 that the family \(\{e^{i\lambda _n(\varepsilon )t}\}_1^\infty \) forms a frame in \(L^{2}(0,T)\). \(\square \)
Now, we are going to generalize these results by supposing, instead of (H3), that the exponential family \(\{e^{if(\lambda _n)t}\}_1^{\infty }\) forms a frame in \(L^2(0,T)\), where f is a H-lipschitz function, i.e.,
To reach this objective, let us consider the following hypothesis:
\((H3')\) The family \(\{e^{if(\lambda _n )t}\}_1^{\infty }\) forms a frame in \(L^{2}(0,T)\) with lower frame bound A, where \(T>0\) and f is a H-lipschitz function.
Theorem 2.6
Suppose that hypotheses (H1), (H2), \((H3')\) and (H4) hold. Then,
-
(i)
For \(\vert \varepsilon _n\vert <\frac{\sqrt{6A}}{\pi \omega _n^{2}r_n^{2}M_n\sum _{i=1}^N\alpha _{n,i}n\sqrt{T}tH+\sqrt{6A}(q+\sum _{i=1}^N\alpha _{n,i}+ \omega _n^2r_nM_n\sum _{i=1}^N\alpha _{n,i})}\), the system \(\{e^{if(\lambda _n(\varepsilon _n) )t}\}_1^{\infty }\) forms a frame in \(L^{2}(0,T)\).
-
(ii)
For \(\vert \varepsilon \vert <\min _{n\in [1, M]}\frac{\sqrt{6A}}{\pi \omega _n^{2}r_n^{2}M_n\sum _{i=1}^N\alpha _{n,i}n\sqrt{T}tH+\sqrt{6A}(q+\sum _{i=1}^N\alpha _{n,i}+ \omega _n^2r_nM_n\sum _{i=1}^N\alpha _{n,i})}\), the families \(\{e^{if(\lambda _n(\varepsilon )) t}\}_1^M \cup \{e^{if(\lambda _n(\varepsilon _n))t}\}_{M+1}^{\infty }\) and \(\{e^{if(\lambda _n(\varepsilon )) t}\}_1^M \cup \{e^{if(\lambda _n)t}\}_{M+1}^{\infty }\) are frames in \(L^{2}(0,T)\).
-
(iii)
If we assume further that for all \(n\in \mathbb {N}^*\), there exists a sequence \((r_n)_{n\ge 1}\) in \(\mathbb {R}_+^*\) such that:
$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \{z\in \mathbb {C}~\hbox {such that}~|z-\lambda _n|\le r_n\}\cap \sigma (T_0)=\{\lambda _n\}\\ S:=\displaystyle \sup _{n\ge 1}\left( q+\sum _{i=1}^N\alpha _{n,i}+\omega _n^{2}r_nM_n\sum _{i=1}^N\alpha _{n,i}\right)<\infty \\ Q:=\displaystyle \sum _{n=1}^{\infty }\left( \omega _n^{2}r_n^{2}M_n\sum _{i=1}^N\alpha _{n,i}\right) ^{2}<\infty , \end{array}\right. \end{aligned}$$
hence for \(\vert \varepsilon \vert <\frac{\sqrt{A}}{H\sqrt{\frac{QT^3}{3}} +S\sqrt{A}}\) the system \(\{e^{if(\lambda _n(\varepsilon ) )t}\}_1^{\infty }\) forms a frame in \(L^{2}(0,T)\).\(\diamondsuit \)
Proof
Let \(n\in \mathbb {N}^*\), \(\lambda _n\) the eigenvalue number n of \(T_0\). Clearly, we have
As f is a H-lipschitz function, we obtain
The rest of the proof of (i) (respectively, (ii) and (iii)) is similar to that of Theorem 2.3 (respectively, Theorems 2.4 and 2.5). \(\square \)
Remark 2.1
It should be noted here that our results extend the outcomes developed in [5] and [14] since Riesz bases are frames with the property of \(\omega \)-linearly independent.\(\diamondsuit \)
3 Application to a Problem of Radiation of a Vibrating Structure in a Light Fluid
In the domain \(-a<x<a\) of the plan \(y=0\), we consider an elastic membrane excited by a harmonic force \(\exp (-iwt)\) with an amplitude f(x) which is independent of the third space variable. This is embedded along the two straights \(x=-a\) and \(x=a\), in the two half-planes perfectly rigid \((x<-a,\ y=0)\) and \((x>a,\ y=0)\). The two half-spaces \(y<0\) and \(y>0\) are filled with gas. The fluid motion is described by a Helmholtz equation in \(\mathbb {R}^2\) whereas the motion’s equation of the membrane is reduced to the equation of the vibrant cord. Let u denote the displacement of the membrane, the problem is described by the boundary value problem:
\(\forall -a<x<a,\)
and satisfying \(u(-a)=u(a)=0.\)
Here where \(H_0(z)=J_0(z)+iY_0(z)\) is the Hankel function of the first kind and order 0, while the physical characteristics of the system are \(\rho _0\) the fluid density, \(c_0\) the fluid sound speed, \(\rho _1\) and \(T_1\) are respectively the surface density and the tightness of the membrane. \(k_0=\displaystyle {\frac{\omega }{c_0}}\) is the wave number of the fluid. The flexion wave speed in the membrane is characterized by \(c_1\): \(=(\frac{\rho _1}{T_1})^{-\frac{1}{2}}\). We denote by p the acoustic pressure in the fluid and satisfies
The problem (3.1) satisfies the following conditions:
where \(P(x)=\displaystyle \lim _{y\rightarrow 0^{+}} p(x,y)-\lim _{y\rightarrow 0^{-}}p(x,y).\)
In the sequel, we shall need the following operators:
and
Now, we state a useful result from [3].
Lemma 3.1
[3, Lemma 4.1] The following assertions hold:
-
(i)
\(T_0\) is a self-adjoint operator with dense domain.
-
(ii)
The injection from \(\mathcal{D}(T_0)\) into \(L^2(-a,a)\) is compact.
-
(iii)
The resolvent set of \(T_0\) is not empty. In fact, \(0\in \rho (T_0)\).
-
(iv)
The spectrum of \(T_0\) is constituted only of point spectrums which are positive, denumerable and of which the multiplicity is one and which have no finite limit points and satisfies
$$\begin{aligned} 0<\lambda _1\le \lambda _2\le \cdots \le \lambda _n\rightarrow +\infty . \end{aligned}$$Further,
$$\begin{aligned} \lambda _n=\left( \displaystyle \frac{n\pi }{2a}\right) ^2. \end{aligned}$$
\(\diamondsuit \)
Remark 3.1
In view of Lemma 3.1 the operator \(T_0\) is self-adjoint with compact resolvent. Then, it follows from [23, p. 20] that its spectral decomposition is given by
where \(\lambda _n=\left( \frac{n\pi }{2a}\right) ^2\) is the eigenvalue number n of \(T_0\) associated to the eigenvector \(\varphi _n(x)=\mu e^{i\sqrt{\lambda _n}x}+\eta e^{-i\sqrt{\lambda _n}x}\).\(\diamondsuit \)
In the sequel, we consider the following operator:
and the following eigenvalue problem:
Find the values \(\lambda (\varepsilon )\in \mathbb {C}\) for which there is a solution \(u\in H_0^1(-a,a)\cap H^2(-a,a),\)\(u\ne 0\) for the equation
where \(\displaystyle {\lambda =\frac{\omega ^2 \rho _1}{T_1}}\) and \(\displaystyle {\varepsilon =\frac{2\rho _0}{\rho _1}}\).
Note that \(\lambda \) and \(\varphi \) each depends on the value of \(\varepsilon \). So, we designate by \(\lambda :=\lambda (\varepsilon )\) and \(\varphi :=\varphi (\varepsilon )\).
We remark that for \(|\varepsilon |<\frac{1}{\Vert K\Vert }\), the operator \(I+\varepsilon K\) is invertible. Then, the problem (3.2) becomes:
Find the values \(\lambda (\varepsilon )\in \mathbb {C}\) for which there is a solution \(u\in H_0^1(-a,a)\cap H^2(-a,a),\)\(u\ne 0\) for the equation
The problem (3.3) is equivalent to:
Find the values \(\lambda (\varepsilon )\in \mathbb {C}\) for which there is a solution \(u\in H_0^1(-a,a)\cap H^2(-a,a),\)\(u\ne 0\) for the equation
where \(T_n:=(-1)^nK^nB\) with \(\mathcal{D}(T_n)=H^1(-a,a)\) for all \(n\ge 1\).
In order to prove our theorem, we shall prove some auxiliary results.
Proposition 3.1
There exist positive constants \(b_1=q=\Vert K\Vert ,~\beta _1=\frac{1}{2}\) such that
for all \(\varphi \in \mathcal{D}(T_0)\) and for all \(k\ge 1.\)\(\diamondsuit \)
Proof
Let \(\varphi \in \mathcal{D}(T_0)\). We have \(\mathcal {D}(T_0)\subset \mathcal {D}(B)\) and for every \(\varphi \in \mathcal {D}(T_0)\)
Using Hölder’s inequality, we get
Hence,
However, we have
for all \(\varphi \in \mathcal {D}(T_0)\) and \(k\ge 1.\) Hence, it suffices to take \(N=1\), \(b_1=q=\Vert K\Vert \) and \(\beta _1=\frac{1}{2}\). \(\square \)
Proposition 3.2
For \(\vert \varepsilon \vert <\frac{1}{\Vert K\Vert },\) the series \(\sum _{k\ge 0} \varepsilon ^kT_k \varphi \) converges for all \(\varphi \in \mathcal{D}(T_0).\) If we designate its sum by \(T(\varepsilon )\varphi , \) we define a linear operator \(T(\varepsilon )\) with domain \(\mathcal{D}(T_0).\) For \(\vert \varepsilon \vert < \frac{2}{3\Vert K\Vert }, \) the operator \(T(\varepsilon )\) is closed.\(\diamondsuit \)
The main result of this section asserts:
Proposition 3.3
The family \(\{e^{i\sqrt{\lambda _n}t}\}_1^{\infty }\) forms a frame in \(L^2(0,a_1), a_1<4a\).\(\diamondsuit \)
Proof
In view of [5, Theorem 4.1], the family \(\{e^{i\sqrt{\lambda _n}t}\}_1^{\infty }\) forms a Riesz basis in \(L^2(0,4a)\). Then, there exist positive constants C and D such that
Given \(f\in L^2(0,a_1)\), extend it to (0, 4a) by setting \(f(x)=0\) for \(a_1<x< 4a\). Then, we can apply Eq. (3.5), but because of the zero extension, the norm and inner product are from \(L^2(0,a_1)\). So, \(\{e^{i\sqrt{\lambda _n}t}\}_1^{\infty }\) forms a frame in \(L^2(0,a_1)\). \(\square \)
Remark 3.2
In [5, Theorem 4.1], it was shown that \(\{e^{i\sqrt{\lambda _n}t}\}_1^{\infty }\) forms a Riesz basis in \(L^2(0,4a)\). However, if we reduce the interval (0, 4a), the Riesz basis properties fail to exist. Indeed, the situation is more complicated since the set \(\{e^{i\sqrt{\lambda _n}t}\}_1^{\infty }\) is over complete on an interval less than 4a.\(\diamondsuit \)
Now, we are ready to state the objective of this section.
Theorem 3.1
For \(\vert \varepsilon _n\vert \) enough small and \(\vert \varepsilon \vert \) enough small there exist two sequences of eigenvalues \(\{\lambda _n(\varepsilon _n)\}_{n\in \mathbb {N^*}}\) and \(\{\lambda _n(\varepsilon )\}_{n\in \mathbb {N^*}}\) of \(T(\varepsilon )\) having the form
such that the systems
(i) \(\{e^{i\sqrt{|\lambda _n(\varepsilon _n)|}t}\}_{1}^{\infty }\)
(ii) \(\{e^{i\sqrt{|\lambda _n(\varepsilon )|}t}\}_1^M \cup \{e^{i\sqrt{|\lambda _n(\varepsilon _n)|}t}\}_{M+1}^{\infty }~~~~ \)
(iii) \(\{e^{i\sqrt{|\lambda _n(\varepsilon )|}t}\}_1^M \cup \{e^{i\sqrt{\lambda _n}t}\}_{M+1}^{\infty }\)
are frames in \(L^2(0,a_1).\)\(\diamondsuit \)
Proof
The results follow from Theorem 2.6, Lemma 3.1 and Propositions 3.1 and 3.3. \(\square \)
Theorem 3.2
There exists a sequence of eigenvalues \((\lambda _n(\varepsilon ))_{n\in \mathbb {N}^*}\) of \(T(\varepsilon )\) having the form
such that for \(|\varepsilon |\) enough small the family \(\{e^{i\sqrt{|\lambda _n(\varepsilon )|}t}\}_1^\infty \) forms a frame in \(L^2(0,a_1)\).\(\diamondsuit \)
Proof
Let \(n\in \mathbb {N}^*\), \(\lambda _n\) the nth eigenvalue of \(T_0\) and \(r_n=\displaystyle \frac{\lambda _{n+1}-\lambda _n}{2}\).
We have \(\{z\in \mathbb {C},~|z-\lambda _n|\le r_n\}\cap \sigma (T_0)=\{\lambda _n\}\). Let \(\mathcal{{C}}_n=\mathcal{{C}}(\lambda _n,r_n)\) the closed circle with center \(\lambda _n\) and radius \(r_n\) and \(z\in \mathcal{{C}}_n\).
Since \(T_0\) is self-adjoint, then it follows from [22] that
\(\Vert R_z\Vert =\Vert (T_0-zI)^{-1}\Vert =\displaystyle \frac{1}{d(z,\sigma (T_0))}.\)
So,
Further, as \(z\in \mathcal{{C}}_n\) then \(z=\lambda _n+r_ne^{it}\) with \(t\in [0,2\pi [\). Hence, \(|z|\le \lambda _n+r_n\).
Consequently,
entails the estimate
Since
Eqs. (3.6) and (3.7) imply that
As
we obtain
So,
Then, Eqs. (3.8) and (3.9) yield
Consequently,
Now, if we set \(\widetilde{\varphi _n}(z)=\displaystyle \frac{1}{(n+1)^2}\varphi _n(z)\), we get that \((\widetilde{\varphi _n})_n\) is a system of eigenvectors of \(T_0\) associated to the eigenvalues \((\lambda _n)_n\) and \(\widetilde{\omega _n}=\Vert \widetilde{\varphi _n}\Vert =\displaystyle \frac{1}{(n+1)^2}\).
Moreover,
Consequently, according to Theorem 2.6, Lemma 3.1 and Propositions 3.1 and 3.3 the family \(\{e^{i\sqrt{|\lambda _n(\varepsilon )|}t}\}_1^\infty \) forms a frame in \(L^2(0,a_1)\). \(\square \)
References
Abdelmoumen, B., Lafi, A.: On an unconditional basis with parentheses for generalized subordinate perturbations and application to Gribov operators in Bargmann space. Indag. Math. (N.S.) 28, 1002–1018 (2017)
Aimar, M.-T., Intissar, A., Jeribi, A.: On an unconditional basis of generalized eigenvectors of the nonself-adjoint Gribov operator in Bargmann space. J. Math. Anal. Appl. 231, 588–602 (1999)
Ali, N.Ben, Jeribi, A.: On the Riesz basis of a family of analytic operators in the sense of Kato and application to the problem of radiation of a vibrating structure in a light fluid. J. Math. Anal. Appl. 320, 78–94 (2006)
Casazza, P.G., Christensen, O., Li, S., Lindner, A.: Density results for frames of exponentials. In: Harmonic Analysis and Applications. Applied Numerical Harmonic Analysis, pp. 359–369. Birkhäuser Boston, MA (2006)
Charfi, S., Jeribi, A., Walha, I.: Riesz basis property of families of nonharmonic exponentials and application to a problem of a radiation of a vibrating structure in a light fluid. Numer. Funct. Anal. Optim. 32(4), 370–382 (2011)
Charfi, S., Damergi, A., Jeribi, A.: On a Riesz basis of finite-dimensional invariant subspaces and application to Gribov operator in Bargmann space. Linear Multilinear Algebra 61, 1577–1591 (2013)
Charfi, S., Ellouz, H.: Riesz basis of eigenvectors for analytic families of operators and application to a non-symmetrical Gribov operator. Preprint
Christensen, O.: Frames and the projection method. Appl. Comput. Harmon. Anal. 1, 50–53 (1993)
Christensen, O.: Frame perturbations. Proc. Am. Math. Soc. 123, 1217–1220 (1995)
Christensen, O.: Frames containing a Riesz basis and approximation of the frame coefficients using finite-dimensional methods. J. Math. Anal. Appl. 199, 256–270 (1996)
Christensen, O.: Frames, Riesz bases, and discrete Gabor/wavelet expansions. Bull. Am. Math. Soc. (N.S.) 38, 273–291 (2001). (electronic)
Daubechies, I., Grossmann, A., Meyer, Y.: Painless nonorthogonal expansions. J. Math. Phys. 24, 1271–1283 (1986)
Duffin, R.J., Schaeffer, A.C.: A class of nonharmonic Fourier series. Trans. Am. Math. Soc. 72, 341–366 (1952)
Ellouz, H., Feki, I., Jeribi, A.: On a Riesz basis of exponentials related to the eigenvalues of an analytic operator and application to a non-selfadjoint problem deduced from a perturbation method for sound radiation. J. Math. Phys. 54, 112101 (2013)
Feki, I., Jeribi, A., Sfaxi, R.: On an unconditional basis of generalized eigenvectors of an analytic operator and application to a problem of radiation of a vibrating structure in a light fluid. J. Math. Anal. Appl. 375, 261–269 (2011)
Feki, I., Jeribi, A., Sfaxi, R.: On a Schauder basis related to the eigenvectors of a family of non-selfadjoint analytic operators and applications. Anal. Math. Phys. 3, 311–331 (2013)
Feki, I., Jeribi, A., Sfaxi, R.: On a Riesz basis of eigenvectors of a nonself-adjoint analytic operator and applications. Linear Multilinear Algebra 62, 1049–1068 (2014)
Filippi, P.J.T., Lagarrigue, O., Mattei, P.O.: Perturbation method for sound radiation by a vibrating plate in a light fluid: comparison with the exact solution. J. Sound Vib. 177, 259–275 (1994)
Jaffard, S.: A density criterion for frames of complex exponentials. Mich. Math. J. 38, 339–348 (1991)
Jeribi, A.: Denseness, bases and frames in Banach spaces and applications. de Gruyter, Berlin (2018)
Jeribi, A., Intissar, A.: On an Riesz basis of generalized eigenvectors of the nonselfadjoint problem deduced from a perturbation method for sound radiation by a vibrating plate in a light fluid. J. Math. Anal. Appl. 292, 1–16 (2004)
Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1980)
Markus, A.S.: Introduction to the Spectral Theory of Polynomial Operator Pencils. Translations of Mathematical Monographs, vol. 71. American Mathematical Society, Providence, RI (1988)
Sz, B.: Nagy, Perturbations des transformations linéaires fermées. Acta Sci. Math. Szeged 14, 125–137 (1951)
Paley, R.E.A.C., Wiener, N.: Fourier Transforms in the Complex Domain. Colloquium Publications, New York (1934)
Schueller, A.: Uniqueness for near-constant data in fourth-order inverse eigenvalue problems. J. Math. Anal. Appl. 258, 658–670 (2001)
Seip, K.: On the connection between exponential bases and certain related sequences in \(L^2(-\pi,\pi )\). J. Funct. Anal. 130, 131–160 (1995)
Young, R.M.: An Introduction to Nonharmonic Fourier Series. Academic Press, London (1980)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Daniel Aron Alpay.
Rights and permissions
About this article
Cite this article
Charfi, S., Ellouz, H. Frame of Exponentials Related to Analytic Families Operators and Application to a Non-self Adjoint Problem of Radiation of a Vibrating Structure in a Light Fluid. Complex Anal. Oper. Theory 13, 839–858 (2019). https://doi.org/10.1007/s11785-018-0807-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11785-018-0807-4