In this paper we give a sampling expansion for integral transforms whose kernels arise from Green’s function of differential operators in a space of vector-functions. The differential operators are in a space of dimension m and consist of systems of m equations in m unknowns. We assume the simplicity of the eigenvalues. | Turk J Math (2017) 41: 67 – 79 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Sampling theorem by Green’s function in a space of vector-functions Hassan Atef HASSAN∗ Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt Received: • Accepted/Published Online: • Final Version: Abstract: In this paper we give a sampling expansion for integral transforms whose kernels arise from Green’s function of differential operators in a space of vector-functions. The differential operators are in a space of dimension m and consist of systems of m equations in m unknowns. We assume the simplicity of the eigenvalues. Key words: Sampling theory, vector-functions, Green’s function, boundary value problems 1. Introduction The use of Green’s function in sampling theory is due to the work of Zayed [8], where a sampling theorem for integral transform whose kernel includes Green’s function of not necessarily self-adjoint problems of the form n ∑ a ⩽ x ⩽ b, pk (x)y (n−k) (x) = λy, λ ∈ C, k=0 n ∑ () αji y (j−1) (a) + βji y (j−1) (b) = 0, i = 1, 2, · · · , n, j=1 is introduced. This theorem can be stated as follows. Let H(x, ξ, λ) be the Green’s function of the problem () and ∏ ( ) ∞ 1− λ , k=1 λk ) ( P (λ) = ∏ λ ∞ 1 − λ , k=2 λk if all λk ̸= 0, () if one of λk , say λ1 = 0, where {λk }∞ k=1 are the eigenvalues of (). If this product is not convergent, then a multiplication by exp(λ/λk ) is needed. For some fixed ξ0 ∈ [a, b], put Φ(x, λ) = P (λ) H(x, ξ0 , λ). Hence, the sampling theorem of [8] reads: ∗Correspondence: hassanatef1@ Department of Mathematics, Faculty of Basic Education, Public Authority for Applied Education and Training, Ardiya, Kuwait 2010 AMS Mathematics Subject Classification: 94A20, 26B12, 34B27, 34K10. 67 HASSAN/Turk J Math Theorem Let ∫ f (λ) = b g(x)Φ(x, λ) dx, g ∈ L2 (a, .
