Differentiation by integration using orthogonal polynomials, a survey <s> Lanczos' 1956 work <s> Generalizations of the Lanczos derivative provide a basis for certain grid-free Finite Interpolation Methods (FIMs) which appear to have advantages over alternatives such as the standard Finite Difference Methods (FDMs) or Finite Element Methods (FEMs) for certain problems, e.g., hypervelocity impact problems in computational material dynamics. <s> BIB001 </s> Differentiation by integration using orthogonal polynomials, a survey <s> Lanczos' 1956 work <s> The method of differentiation by integration due to Lanczos is generalized to cover derivatives of arbitrary order. Umbral versions and further extensions are indicated. <s> BIB002 </s> Differentiation by integration using orthogonal polynomials, a survey <s> Lanczos' 1956 work <s> Everyone knows that the derivative of a function at a point measures the slope, or instantaneous rate of change, of the function at that point; this follows from the usual difference-quotient definition. However, various alternate formulations of the derivative have proved valuable. Many of these are discussed in Bruckner's text [1]. The Lanczos derivative is one alternate definition not mentioned by Bruckner, which has been a focus of recent attention [3, 5, 10]. Attributed to the Hungarian-born mathematician Cornelius Lanczos (pronounced Lan'tsosh) [7], this derivative is actually defined in terms of a definite integral! The Lanczos derivative of a function f at x is given by <s> BIB003
In a book published in 1956 Lanczos [42, (5-9.1)], apparently unaware of the earlier work by Cioranescu  and Haslam-Jones  , rediscovered formula (3.7). He called this differentiation by integration. His work got quite a lot of citations, see for instance  ,  , BIB001 , BIB002 , BIB003 ,  ,  . The name Lanczos derivative, notated as f L (x), became common for a value obtained from the right-hand side of (3.7). In BIB002 and BIB003 also (3.6) (the Legendre case for general n) was rediscovered. As an important new aspect Lanczos observed that 3 2δ as an approximation of f (x), is the limit as N → ∞ of the quotient of Riemann sums We have seen this last expression already in (3.5) (the special case of (3.1) with a centered Gram polynomial of degree 1). The expression approximates f (x) for N −1 δ small. In fact, (3.5) was the starting point of Lanczos, see [42, (5-8.4) ].