On sampling theories and discontinuous Dirac systems with eigenparameter in the boundary conditions

The sampling series (1.2) is absolutely and uniformly convergent on compact subsets of ℂ.

The WSK sampling theorem has been generalized in many different ways. Here we are interested in two extensions. The first is concerned with replacing the equidistant sampling points by more general ones, which is very important from the practical point of view. The following theorem which is known in some literature as the Paley-Wiener theorem [5] gives a sampling theorem with a more general class of sampling points. Although the theorem in its final form may be attributed to Levinson [8] and Kadec [9], it could be named after Paley and Wiener who first derived the theorem in a more restrictive form; see [6, 7, 10] for more details.

The series (1.6) converges uniformly on compact subsets of ℂ.

The sampling series (1.6) can be regarded as an extension of the classical Lagrange interpolation formula to ℝ for functions of exponential type. Therefore, (1.6) is called a Lagrange-type interpolation expansion.

Series (1.7) converges uniformly wherever ${\parallel K(\cdot ,t)\parallel }_{L^2(I)}$, as a function of t, is bounded. In this theorem, sampling representations were given for integral transforms whose kernels are more general than $exp(ixt)$. Also Kramer’s theorem is a generalization of WSK theorem. If we take $K(x,t)=e^{itx}$, $I=[-\sigma ,\sigma ]$, $t_k=\frac{k\pi }{\sigma }$, then (1.7) is (1.2).

The relationship between both extensions of WSK sampling theorem has been investigated extensively. Starting from a function theory approach, cf. [12], it was proved in [13] that if $K(x,t)$, $x\in I$, $t\in \mathbb{C}$, satisfies some analyticity conditions, then Kramer’s sampling formula (1.7) turns out to be a Lagrange interpolation one; see also [14–16]. In another direction, it was shown that Kramer’s expansion (1.7) could be written as a Lagrange-type interpolation formula if $K(\cdot ,t)$ and $t_k$ were extracted from ordinary differential operators; see the survey [17] and the references cited therein. The present work is a continuation of the second direction mentioned above. We prove that integral transforms associated with Dirac systems with an internal point of discontinuity can also be reconstructed in a sampling form of Lagrange interpolation type. We would like to mention that works in direction of sampling associated with eigenproblems with an eigenparameter in the boundary conditions are few; see, e.g., [18–20]. Also, papers in sampling with discontinuous eigenproblems are few; see [21–24]. However, sampling theories associated with Dirac systems which contain eigenparameter in the boundary conditions and have at the same time discontinuity conditions, do not exist as far as we know. Our investigation is be the first in that direction, introducing a good example. To achieve our aim we briefly study the spectral analysis of the problem. Then we derive two sampling theorems using solutions and Green’s matrix respectively.

where $\lambda \in \mathbb{C}$; the real-valued functions $p_1(\cdot )$ and $p_2(\cdot )$ are continuous in $[-1,0)$ and $(0,1]$ and have finite limits $p_1(0^{\pm }):={lim}_{x\to 0^{\pm }}{p}_1(x)$ and $p_2(0^{\pm }):={lim}_{x\to 0^{\pm }}{p}_2(x)$; $a_1,a_2,\delta \in \mathbb{R}$, $\alpha ,\beta \in [0,\pi )$; $\delta \ne 0$ and $\rho :=a_{1}cos\beta -a_{2}sin\beta >0$.

which are defined on ${\mathrm{\Gamma }}_1:=[-1,0]$ and ${\mathrm{\Gamma }}_2:=[0,1]$, respectively.

In the following lemma, we prove that the eigenvalues of problem (2.1)-(2.5) are real.

Lemma 2.1 The eigenvalues of problem (2.1)-(2.5) are real.

This contradicts the conditions ${\int }_{-1}^0({|u_1(x)|}^2+{|u_2(x)|}^2)dx+{\delta }^2{\int }_0^1({|u_1(x)|}^2+{|u_2(x)|}^2)dx>0$ and $\rho >0$. Consequently, ${\lambda }_0$ must be real. □

The operator $\mathcal{A}:\mathcal{D}(\mathcal{A})⟶\mathfrak{H}$ and the eigenvalue problem (2.1)-(2.5) have the same eigenvalues. Therefore they are equivalent in terms of this aspect.

 □

has a unique solution $\mathit{u}=\left(\begin{array}{c}{\chi }_{12}(x,\lambda )\\ {\chi }_{22}(x,\lambda )\end{array}\right)$, which is an entire function of parameter λ for each fixed $x\in [0,1]$.

has a unique solution $\mathit{u}=\left(\begin{array}{c}{\varphi }_{12}(x,\lambda )\\ {\varphi }_{22}(x,\lambda )\end{array}\right)$ for each $\lambda \in \mathbb{C}$.

By virtue of equations (2.21) and (2.23), these solutions satisfy both transmission conditions (2.4) and (2.5). These functions are entire in λ for all $x\in [-1,0)\cup (0,1]$.

for each $\lambda \in \mathbb{C}$.

Corollary 2.3 The zeros of the functions ${\omega }_1(\lambda )$ and ${\omega }_2(\lambda )$ coincide.

In the following lemma, we show that all eigenvalues of problem (2.1)-(2.5) are simple.

Lemma 2.4 All eigenvalues of problem (2.1)-(2.5) are just zeros of the function $\omega (\lambda )$. Moreover, every zero of $\omega (\lambda )$ has multiplicity one.

Proof Since the functions ${\varphi }_1(x,\lambda )$ and ${\varphi }_2(x,\lambda )$ satisfy the boundary condition (2.2) and both transmission conditions (2.4) and (2.5), to find the eigenvalues of the (2.1)-(2.5), we have to insert the functions ${\varphi }_1(x,\lambda )$ and ${\varphi }_2(x,\lambda )$ in the boundary condition (2.3) and find the roots of this equation.

contradicting the assumption $\rho >0$. The other case, when $a_2+{\lambda }^{\ast }cos\beta \ne 0$, can be treated similarly and the proof is complete. □

where $c_n\ne 0$ are non-zero constants, since all eigenvalues are simple. Since the eigenvalues are all real, we can take the eigen-vector-functions to be real-valued.