Differentiation by integration using orthogonal polynomials, a survey <s> Generalized Taylor series <s> In his response, in which he considers functions restricted to the reals, he points out that the infinite differentiability of a function does not itself assure that its power series will converge to that function, whereas mere periodicity and a little smoothness are enough to have the Fourier series converge uniformly to the function. Further, the terms of the Fourier series describe simple harmonic motion so that the function may be considered as a linear combination of harmonic motions, whereas the terms of a power series have no such physical interpretation. While calling attention to such differences is instructive, an interesting and deeper question might also be posed. <s> BIB001 </s> Differentiation by integration using orthogonal polynomials, a survey <s> Generalized Taylor series <s> 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. <s> BIB002
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 