Differentiation by integration using orthogonal polynomials, a survey <s> Smoothing filters <s> In this paper we are mainly concerned with two kinds of linear transformations, the sequence convolution transformation ::: ::: $$ {y_n} = \sum\limits_{v = - 8}^\infty {{a_{n - v}}{x_v}} $$ ::: ::: (1) ::: ::: and the integral convolution transformation ::: ::: $$ g\left( x \right) = \int_{ - \infty }^\infty {\Lambda \left( {x - 1} \right)f\left( t \right)} dt, $$ ::: ::: (2) ::: ::: where the sequence a n and the function Δ(x) are thought of as given. In §2 we also consider the ordinary linear transformation ::: ::: $$ {y_i} = \sum\limits_{k = 1}^n {{a_{ik}}{x_k}} \quad \quad \left( {i = 1, \cdots ,m} \right). $$ ::: ::: (3) ::: ::: The loosely connected topics to be discussed concerning these transformations are perhaps best brought together under the general subject of smoothing operations. <s> BIB001 </s> Differentiation by integration using orthogonal polynomials, a survey <s> Smoothing filters <s> The principal result of this paper is the following theorem. Theorem 1. | C{6)\ <1 if 89*0, and C(0) = 1. Wilf [l] has previously shown that there exists an integer ft0 such that Theorem 1 holds if ft^ft0, and has conjectured that ft0 = 0. The present proof, which is different from Wilf's, establishes the theorem for all ft^O. If a weighting function w has a transform (2) which satisfies the inequality of Theorem 1, it is said to be stable, for reasons previously discussed by Schoenberg and DeForest, as described in [l] and its references. In addition, if one thinks of w(x), — l^x^Sl, as the weighting function of a filter with input /(*) = 5(«) + N(x), where S is a polynomial of degree not greater than 2ft+ 1, and N is a zero mean stationary random process with spectral density \p(8), then the output of the filter is <s> BIB002
For m = 0 the filters given by the left-hand sides of (4.1) and (4.6) are examples of smoothing filters. These have a very long history, see Schoenberg  , BIB001 and references given there, which go back as far as De Forest's work in 1878. We put M := −N in (4.7) and M = −N = −1 in (4.8): and we usually take ρ symmetric: ρ(x) = ρ(−x). We say that (4.12) or (4.13) is exact for the degree j (where j < 2N in case of (4.12)) if g = f whenever f is a polynomial of degree ≤ j, but g = f for some polynomial f of degree j + 1. Because of symmetry of ρ, such j will always be odd. Exactness of (4.12) for degree at least 2n + 1 < 2N can equivalently be stated as where y → t 2k (y, 2N + 1) is a Gram polynomial (see (2.4) ) and h 2k is the corresponding constant given by (2.6). For such ρ the sum of squares  where K 2n is the Christoffel-Darboux kernel of degree 2n (see (2.14)) for the orthogonal polynomials y → t k (y, 2N + 1). Similarly, in case of (4.13) and assuming that ρ is a polynomial, the requirements that the formula is exact for the degree 2n + 1 and that where K 2n is the Christoffel-Darboux kernel of degree 2n for the Legendre polynomials P k . More generally than (4.14) we can work with orthogonal polynomials p n satisfying (2.2) or (2.3) for equidistant points running over a symmetric set and with symmetric weights: where w(x) = w(−x). In terms of these polynomials p n exactness of (4.12) for degree at least 2n + 1 < 2N can equivalently be stated as In particular, the choice with K 2n the Christoffel-Darboux kernel and with (2.15) used for the second equality, will make (4.12) exact for the degree 2n + 1. The characteristic function for (4.12) respectively (4.13) is defined by (4.9) with M = −N respectively (4.10) with M = −N = −1. The condition that (4.12) or (4.13) is exact for the degree 2n + 1 is equivalent with the condition that φ has power series of the form with a = 0. An m-fold iteration of (4.12) (with N = ∞ for convenience) yields where ρ m = ρ * . . . * ρ is an m-fold convolution product. De Forest (1878) raised the question for which choices of ρ the asymptotic behaviour of ρ m (x) for large m can be described. Schoenberg holds with a > 0. For ρ given by (4.18) we see from (4.11) that a > 0 is satisfied for any choice of the weights and that a is explicitly given by The stability condition (4.20) can also be considered for the continuous smoothing formula (4.13), where now ω ∈ R\{0} in (4.20) . For the Legendre case where ρ is given by (4.15), stability was proved by Trench BIB002 and Lorch & Szego  .