You are presented with multiple paper abstracts, alongside the associated literature review and chapter titles. Given the provided information, you are going to do a literature review. Please generate a coherent literature review that synthesizes the content, draws connections between different works, highlights major findings, and identifies any gaps in the existing research.

Literature Review Title: Differentiation by integration using orthogonal polynomials, a survey 
Chapter Title:  Generalized Taylor series 
1. Abstract of Cited Paper (BIB001):  The device to control the direction of a light beam comprises mainly the following elements substantially aligned in the direction of a laser beam emitted by a source: a low-deflection, controlled deflection device; a holographic multiplexer comprising several recorded zones and giving several possible deflections depending on the direction of the beam that it receives; a system of shutters enabling the choice of the direction of the laser beam to be transmitted.

Literature Review:
Rewrite (3.1) as Here µ δ (E) := µ(δ −1 E). Then the polynomials x → p n (δ −1 x) are orthogonal with respect to the measure µ δ . The formal Taylor series can accordingly be seen as a termwise limit of the formal generalized Fourier series Indeed, use (3.19) and the limit δ n p n (δ −1 η)/k n → η n as δ ↓ 0. While for a big class of orthogonal polynomials and for f moderately smooth, the series (3.21) converges with sum f (x + η) because of equiconvergence theorems given in Szegő [64, Ch. 9 and 13], we will need analyticity of f in a neighbourhood of x for convergence of (3.20)  . Then an open disk around x of radius less than the convergence radius of (3.20) is in the interior of the ellipse of convergence (3.21) for δ small enough. This was discussed for Legendre polynomials by Fishback BIB002 . A different limit from orthogonal polynomials to monomials is discussed by Askey & Haimo BIB001 . This involves Gegenbauer polynomials, which we write as Jacobi polynomials P Consider now the formal expansion of f (x + η) for η ∈ [−1, 1] in terms of the polynomials P (α,α) n (η): where we used the identitity for δ = 1 in (3.10). The last form of the above formal series tends termwise to the formal Taylor series (3.20) . This is seen from (3.22) and the fact that the