## Abstract

**Formulation of the problem**. We sometimes encounter a need to choose an analytic expression which would represent approximately a function given only by a table or a graph. A similar problem can arise for a function given by a formula, when this formula is too complicated or not appropriate to a given purpose as, for example, when the function has to be integrated while the integral cannot be expressed in terms of elementary functions. Formulas representing a functional dependence obtained from an experiment in the form of a table or a graph are called

*empirical formulas*. To represent approximately a given function

*f*(

*x*), we usually choose an

*approximating function*φ(

*x*) from among functions of a definite form; for example, we seek for a function φ(

*x*) in the form of a polynomial

$$\varphi (x)=a_{0}+a_{1}x+\cdots +a_{n}x^{n}$$

$$\varphi (x)=Ae^{rx}+Be^{sx}+\cdots$$

*x*) approaches the function

*f*(

*x*) in a certain interval

*a*⩽

*x*⩽

*b*as closely as possible. According to the manner of estimating the approximation of the function

*f*(

*x*) with φ(

*x*), we obtain various systems of parameters of the function φ(

*x*) yielding the best approximation of

*f*(

*x*).

## Keywords

Empirical Formula Uniform Approximation Interpolation Formula Interpolation Node Conditional Equation## Preview

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## Notes

- (1).For example, φ(
*x*) is a polynomial, if φ_{0}= 1, φ_{1}=*x*, …, φ_{n}=*x*^{n}, or is a trigonometric polynomial, if φ_{0}= 1, φ_{1}= cos*x*, φ_{2}= sin*x*, …, φ_{2n-l}= cos*nx*, φ_{2n}= sin*nx*.Google Scholar - (2).Two examples of orthogonal systems of functionsGoogle Scholar
- (1).
- (2).
- (1).The best approximation of
*f*(*x*) can be defined, as above (see the uniform approximation, p. 754), as a function φ(*x*) such that the maximum of [*f*(*x*_{i}) − φ(*x*_{i})] has the least value. However, determining the approximation in this way is, practically, troublesome.Google Scholar - (1).For an example of such a scheme see p. 761.Google Scholar
- (1).The decimal point is usually omitted in the table of differences and the difference is expressed in the units of the last significant figure.Google Scholar
- (1).
- (1).

## Copyright information

© Verlag Harri Deutsch, Zürich 1973