Maximal generalization of Lanczos' derivative using one-dimensional integrals
aa r X i v : . [ m a t h . G M ] J un Maximal generalization of Lanczos’ derivativeusing one-dimensional integrals
Andrej Liptaj ∗ June 21, 2019
Institute of Physics, Slovak academy of SciencesDúbravská cesta 9, 845 11 Bratislava, Slovakia
Abstract
Derivative of a function can be expressed in terms of integration overa small neighborhood of the point of differentiation, so-called differen-tiation by integration method. In this text a maximal generalization ofexisting results which use one-dimensional integrals is presented togetherwith some interesting non-analytic weight functions.
Cornelius Lanczos in his work [1] published a method of differentiation by in-tegration , where the derivative of a function is approximated by an integral.The integral is performed over a small interval around the point of differentia-tion with the approximation becoming exact in the limit of the interval lengthapproaching zero. For differentiable functions one has f ′ ( x ) ≡ f ′ ( x ) | x = x = lim h → h Z h − h tf ( x + t ) dt. The expression is interesting from several aspects: it generalizes the ordinaryderivative and, also, its modifications might be useful for numerical differenti-ation (see e.g. [3, 4]).Since, the topic was addressed by several authors with noticeable growth ofinterest in the last decade [5, 6, 7, 8, 9, 10, 11, 12, 13]. The millennial work [7] ∗ [email protected] The first person to publish such method was Cioranescu [2]. The name of the method ishowever usually associated with Lanczos. Converges in situations, where the ordinary derivative is not defined.
1s probably the most interesting of them: the authors actually provide a verybroad generalization of the Lanczos’ formula for the first derivative and theirapproach can be further and straightforwardly generalized to higher orders (asdone in this article). Their text is, surprisingly, widely overlook by later workswith exception of [4, 9, 12], which, however, do not exploit the potential of it.In what follows, the second section will be dedicated to the generalizationof the Lanczos’ approach for the first derivative. The next section will covergeneralization to higher-order derivatives and, in the fourth section, a shortdiscussion will follow. A summary and conclusion will constitute the last section.Let me remark that generalizations based on multidimensional integrals canbe found in literature (e.g. formula 2.31 in [13]). Unlike other approaches, they,presumably, do not represent a special case of the generalization presented hereand remain an independent way of generalizing the Lanczos’ derivative.
Let me restate the findings from [7]. The key observation which allows for largegeneralizations is, that the approximation of the derivative can be seen as aver-aging the derivative over some small interval [ x − h, x + h ] around the pointof differentiation x . This average might be understood as weighted averagewith a weight function w h f ′ ( x ) ≈ Z x + hx − h w h ( t ) f ′ ( t ) dt, (1)where Z x + hx − h w h ( t ) dt = 1 . Negative weights cannot be excluded, yet the condition ≤ w h ( t ) might beadopted if desired. Because the weight functions are of the most interest here,a modified version of (1) will be used throughout this text f ′ ( x ) ≈ Z − w ( t ) f ′ ( x + ht ) dt, Z − w ( t ) dt = 1 , so that the weight functions are defined on a “standard” interval [ − , . Onehas w ( t ) = hw h ( x + ht ) . Using integration by partes one arrives to f ′ ( x ) ≈ h [ w ( t ) f ( x + ht )] t =1 t = − − h Z − w ′ ( t ) f ( x + ht ) dt. Two interesting observations can be done:2 If w is constant w ( t ) = 0 . then the standard definition of the derivativeis recovered f ′ ( x ) ≈ f ( x + h ) − f ( x − h )2 h . • If w ( −
1) = w (+1) = 0 then a differentiation by integration method isconstructed f ′ ( x ) ≈ − h Z − w ′ ( t ) f ( x + ht ) dt. The usual Lanczos’ expression is obtained for w ( t ) = 34 (cid:0) − t (cid:1) . Indeed w ′ ( t ) = − t −→ f ′ ( x ) ≈ h Z − tf ( x + ht ) dt = 32 h Z h − h zf ( x + z ) dz. At this point one can formulate the generalization:
Any differentiable function w which satisfies Z − w ( t ) dt = 1 and w ( −
1) = w (1) = 0 can be used for differentiation by integration in the following manner f ′ ( x ) ≈ − h Z − w ′ ( t ) f ( x + ht ) dt, where its derivative w ′ appears.Let me define some useful terms: “kernel function” will from now on referto the function which is being integrated (together with function values) in the differentiation by integration procedure and let me note by small zero thoseanti-derivatives of a function k which take value zero at minus one k ( − n )0 ( t ) | t = − = 0 , ddt k ( − n )0 = k ( − n +1)0 . One can now address the question about a proper kernel function (inverse im-plication). From what was show one can deduce: k is valid kernel functioniff k ( − ( t ) | t =+1 = 0 and Z − k ( − ( t ) dt = 1 . (2)The first of the two conditions is equivalent to Z − k ( t ) dt = 0 . In case of the first derivative the kernel function is w ′ . λ = Λ ′ ) − h Z − λ ( t ) f ( x + ht ) dt = − h [ Λ ( t ) f ( x + ht )] t =1 t = − + Z − Λ ( t ) f ′ ( x + ht ) dt. If Λ ( ± = 0 , one cannot make vanish the first term on the RHS for a generalfunction f . If one takes the limit h → in the second term (using continuity of f ′ ) one arrives to lim h → Z − Λ ( t ) f ′ ( x + ht ) dt = Z − Λ ( t ) f ′ ( x ) dt = f ′ ( x ) Z − Λ ( t ) dt. One sees that a function with integral different from one provides wrong value ofthe derivative. Formulas (2) express sufficient and necessary conditions a kernelfunction has to fulfill, they represent the largest possible generalization of theLanczos’ approach.
Repeated integration by parts allows for immediate generalization Z − w ( t ) f ( n ) ( x + ht ) dt == 1 h h w ( t ) f ( n − ( x + ht ) i − − h Z − w ′ ( t ) f ( n − ( x + ht ) dt = 1 h h w ( t ) f ( n − ( x + ht ) i − − h h w ′ ( t ) f ( n − ( x + ht ) i − + 1 h Z − w ′′ ( t ) f ( n − ( x + ht ) dt = (cid:18) − h (cid:19) n Z − w ( n ) ( t ) f ( x + ht ) dt + n − X k =0 ( − k h k +1 h w ( k ) ( t ) f ( n − − k ) ( x + ht ) i − To make, for a general function f , the second term vanish, one has to require w ( k ) ( −
1) = w ( k ) (1) = 0 for all k = 0 , , . . . , n − . (3)Having this property, then, with appropriate weight function Z − w ( t ) dt = 1 , (4)4nd assuming f ( n ) is continuous, one interprets the first term as an approxima-tion of the n -th derivative lim h → (cid:18) − h (cid:19) n Z − w ( n ) ( t ) f ( x + ht ) dt = lim h → Z − w ( t ) f ( n ) ( x + ht ) dt = f ( n ) ( x ) Z − w ( t ) dt = f ( n ) ( x ) . Like at the end of the Sec. 2, one can inverse the whole procedure, start withexpression R − w ( n ) ( t ) f ( x + ht ) dt and proceed to n repeated integrations byparts (integrate w ( n ) and differentiate f ). As result one can immediately con-clude: If k is to be a valid kernel for differentiation by integration in the formula f ( n ) ( x ) ≈ (cid:18) − h (cid:19) n Z − k ( t ) f ( x + ht ) dt (5)then k ( − n )0 (1) = 0 for all n = 1 , , . . . , n and Z − k ( − n )0 ( t ) dt = 1 . With these statements valid for any weight/kernel functions for which appro-priate derivatives/integrals exist, one can claim that, for the Lanczos’ derivativewritten in the from (5), the generalization is maximal.
With the acquired knowledge one can propose some new, potentially interestingkernels and weight functions. Idea of universality might be a compelling one,by which I mean the independence on the order of the derivative (from now onnoted n ). Kernels have to be n -dependent , but one can look for n -independentweight functions. Such a universal weight function has to fulfill condition (3) forall derivatives, yet it cannot be zero so as to respect the condition (4). Thereforeit must be non-analytic at -1 and 1.As first example I propose w e = 1 K exp (cid:18) x − (cid:19) with K ≈ . . With no explicit n -dependence in the weight function, this dependence comesfrom differentiation f ( n ) ( x ) ≈ (cid:18) − h (cid:19) n K Z − dtf ( x + ht ) d n dt n exp (cid:18) t − (cid:19) The LHS of (5) is n -dependent, so has to be the RHS. But, with the exception of (cid:0) − h (cid:1) n ,there are no other explicit n -dependent factors on the RHS, thus the dependence must behidden in k ( t ) . f ′ ( x ) ≈ hK Z − dtf ( x + ht ) t ( t − ( t + 1) exp (cid:18) t − (cid:19) f ′′ ( x ) ≈ h K Z − dtf ( x + ht ) 3 t − t − ( t + 1) exp (cid:18) t − (cid:19) f ′′′ ( x ) ≈ h K Z − dtf ( x + ht ) x (cid:0) t + 3 t − t + 3 (cid:1) ( t − ( t + 1) exp (cid:18) t − (cid:19) Even more interesting example is a one with shifted Fabius function [14] w F b ( t ) = F b ( t + 1) . The Fabius function (which I note
F b ) is non-analytic for all ≤ x and itsbehavior with respect to the conditions (3,4) can be deduced from differentialfunctional equation F b ′ ( x ) = 2 F b (2 x ) . (6)One has Z F b ( x ) dx x =2 z = Z F b (2 z ) dz = Z F b ′ ( z ) dz = [ F b ( z )] z =1 z =0 = 10 = F b (0) = 12
F b ′ (0) = 12 14 F b ′′ (0) = . . .F b ( n ) (2) = 12 n +1 F b ( n +1) (1) = 0 , where the very last equality (all derivatives vanishing at x = 1 ) is consequence ofthe symmetry condition F b (1 − x ) = 1 − F b ( x ) and the behavior of derivativesat x = 0 . When shifting Fabius function to the interval [ − , all mentionedproperties remain conserved (on the shifted the interval). Equation (6) allowsus to formulate corresponding kernel functions in a very elegant way, where theexplicit dependence on derivatives is not present f ( n ) ( x ) ≈ (cid:18) − h (cid:19) n n ( n +1) Z − F b [2 n ( t + 1)] f ( x + ht ) dt. Value of the Fabius function for < x can be very easily related to the valueof this function on the interval [0 , . Using an efficient method for its eval-uation on the interval [0 , , one achieves an effective method for computingkernel function values and thus the whole integral, and this for any order of thederivative. One can notice that the expression is defined for any real value of n . Use of tabulated values, or recipes from [15, 16, 17]. Discussion
One of the most cited results [8] generalizes the Lanczos’ derivative by usingLegendre polynomials . It might be interesting to check its behavior from theperspective of presented results. The authors of [8] propose (among others) thefollowing form of the kernel function k n ( x ) = ( − n n + 1)!! P n ( x ) , with P n ( x ) being the Legendre polynomials. The latter can be defined byRodrigues’ formula P n ( x ) = 12 n n ! d n dx n (cid:0) x − (cid:1) n . Observing the inner bracket (going to zero for x = ± ) being raised to the n -thpower, one immediately sees that the condition (3) is obeyed. Next, one canstudy the integral of the weight function ( − n n + 1)!! 12 n n ! Z − (cid:0) x − (cid:1) n dx. With partial results [18] Z − (cid:0) x − (cid:1) n dx = √ π ( − n n ! Γ (cid:0) n + (cid:1) and Γ (cid:18) n + 32 (cid:19) = √ π (2 n + 1)!!2 ( n +1) one finds that also the condition (4) is respected.Several other realizations for differentiation by integration can be found inthe literature, most of them with higher technical complexity then the pre-vious one. From what was shown, all of these representation (based on one-dimensional integrals) have to comply with the restrictions (3) and (4).This text focuses on the main result of generalizing the Lanczos’ derivativeand does not address specific issues of precision and rapidity of convergence incase of a numerical implementation and related questions of kernel function pref-erence. With kernel function being completely general (possibly non-analyticeverywhere) one can hardly rely on standard tools for error estimates (i.e. Tay-lor series). In any specific context the recipes exiting in the literature are to beused. In this text the result published in [7] was generalized to higher-order derivativesand, assuming pattern (5), this generalization is maximal. Restrictions (3) and(4) allow for a very broad family of functions, which might make the search forwell performing kernels for numerical purposes more efficient. A similar result was in the same year published by [9]. Factor ( − n is here to cancel the same factor in (5) from in front of the integral. eferenceseferences