Continuous Approximation of a Discrete Function
For continualization we must construct a function u(x,t) representing a continuous approximation of the function of discrete argument Uj{t)
where h is the distance between points of discrete function.
Observe that it is defined with an accuracy to any arbitrary function, which equals zero in nodal points x = jh . For this reason, from a set of interpolating functions one may choose the smoothest function owing to filtration of fast oscillating terms. As it is shown in [258], the interpolating function is determined uniquely, if one requires its Fourier image
Function 
is familia normalized sine function, often called Whittaker-Kotelnikov-Shannon (WKS) interpolating function. The value of this function at the removable singularity at zero is understood to be the limit value 1. The sine function is then analytic everywhere and hence an entire function. Also this function has interesting properties: 
We note that the concept of concentrated force in the theory of elasticity with couple-stresses must be changed in comparison with classical theory of elasticity. Namely, delta-function must be replaced to the sine function due size effect caused by internal structure of material. This statement is illustrated by identity (10.83).
The kernels of integro-differential equations of the discrete and continuous systems
Let us consider the chain of material points with the equal masses m, located in equilibrium states in the points of the axis x with coordinate's jh,j = 0,±1,±2,„. and suspended by elastic couplings with stiffness c. The governing ODEs are as follows 
where y7(r) is the displacement of the y'-th material point from its static equilibrium position.
Instead of the dimensional coordinates x,y,t, we introduce dimensionless coordinates £ = x/h, X = t sJc/m,Zj —yj/h and rewrite (10.84) as 
Differential-difference Eq. (10.85) can be reduced to the following integro- differential equation [182,400]:
Equations (10.86), (10.87) describe the media with nonlocal interactions. The classical continuous approximation is:
Thus, replacement of the discrete media with the continuous one leads to two reasons for error. The first one is connected with replacement of the integral with finite limits of integration on the integral with infinite limits of integration
Such a replacement from a physical standpoint is caused due the transition from the discrete to the continuous media.
The second reason for error type is connected with approximation of the operator 4sin2 (q/2). The general approach to improvement of approximation (10.88) consists in approximation of the integral operator from the r.h.p. of Eq. (10.87) with higher- order derivatives in the assumption that nonlocal property is weak. For example, replacing integration limits in integral (10.87) on infinite ones and using truncated Maclaurin series
we obtain intermediate continuous model (Chapter 10.6):
One of the ways for improvement of the local approximation can consist in more precise approximation of the operator sin2 (|). Using for this aim PA one obtains (Chapter 10.7) 
Using Eq. (10.92) one obtains the following model: 
Two-point PA gives more precise approximation (Chapter 10.7)
Using Eq. (10.94) one obtains the following approximation:
We investigate the accuracy of the approximation of the kernel of an integro- differential equation (10.86), which describes the deformation of a discrete media. Integral (10.87) has the following asymptotics
The kernels of integro-differential equations of classical continuous approximation (10.90) Ф, (£) and intermediate continuous approximation (10.91) Фа (£) have the following expressions and asymptotics:
The kernels of integro-differential equations continuous approximation (10.93) Ф qc(E,) and improved continuous approximation (10.95) Ф9С,-(|) can be written as follows 
where
Evaluating the integral (10.109) one obtains
where Si(-) and Ci(-) are the familiar sine and cosine integrals, Ci(y) = 7 + In у + /o' /-l (cos? - l)dt, Si(y) = /o'/-1 sin/ Jt, у is the Euler constant, y = 0.5772156649... ([2], Chapter 5).
Asymptotics of kernels of continuous approximation (10.93) Ф qc{£,) and improved continuous approximation (10.95) Ф(/п- (
For a good continuous approximation is the most important as a more accurate approximation at £ -¥ oo the kernel of a integro-differential equation [258]. The best approximation gives the continuous approximation (10.95), which provides an accurate approximation up to the order 0(%~2).
Source: https://ebrary.net/182866/engineering/correspondence_functions_discrete_arguments_approximating_analytical_functions
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