interpolation

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interpolation

[in‚tər·pə′lā·shən]
(mathematics)
A process used to estimate an intermediate value of one (dependent) variable which is a function of a second (independent) variable when values of the dependent variable corresponding to several discrete values of the independent variable are known.

interpolation

see MEASURES OF CENTRAL TENDENCY.

Interpolation

 

an insertion or correction in an original text made by someone other than the author.

Interpolations played a pivotal role in the texts by Roman jurists that are compiled in the Digest. They were made to eliminate contradictions in the texts as well as the statutes and attitudes that were inappropriate for the Justinian era. Various kinds of interpolations were made, including specification and substitutions of the rule of law, substitution or elimination of terms, and lexical changes. The medieval humanists were the first to discover the interpolations in the Digest.


Interpolation

 

in mathematics and statistics, the process of finding values of a quantity between some of its known values. An example is finding values of the function f(x) at points x lying between the points (nodes of interpolation) x0 < x1 < … < xn by means of the known values yi = f(x1), where i = 0, 1, …, n. In the case when x lies outside the interval included between x0 and xn, the analogous problem is called an extrapolation problem.

In the simplest case, linear interpolation, the value of f(x) at a point x satisfying the inequality x0 < x1, is taken to be equal to the value

of the linear function coinciding with f(x) at the points x = x0 and x = x1. The interpolation problem is undefined from a strict mathematical viewpoint: if nothing is known about the function f(x) except its values at the points x0, x1, …, xn, then its value at a point x, which is different from all these points, remains completely arbitrary. The interpolation problem acquires a definite meaning if the function f(x) and its derivatives are subject to certain inequalities. If, for example, the values f(x0) and f(x1) are given and it is known that for xo < x < x1 the inequality | f”(x) | ≤ M is fulfilled, then the error of the formula (*) may be estimated with the aid of the inequality

It makes sense to use more complex interpolation formulas only in the case when it is certain that the function is sufficiently “smooth,” that is, when it has a sufficient number of derivatives that do not increase rapidly.

In addition to the computation of values of functions, interpolation has numerous other applications (for example, approximate integration, approximate solution of equations, and, in statistics, the smoothing of distribution series with the aim of eliminating random distortions).

REFERENCES

Goncharov, V. L. Teoriia interpolirovaniia i priblizheniia funktsii, 2nd ed. Moscow, 1954.
Krylov, A. N. Lektsii o pribiizhennykh vychisleniiakh, 6th ed. Moscow, 1954.
Yule, G. U., and M. G. Kendall. Teoriia statistiki, 14th ed. Moscow, 1960. (Translated from English.)

interpolation

interpolation

In computer graphics, it is the creation of new values that lie between known values. For example, when objects are rasterized into two-dimensional images from their corner points (vertices), all the pixels between those points are filled in by an interpolation algorithm, which determines their color and other attributes (see graphics pipeline).

Another example is when a video image in a low resolution is upscaled to display on a monitor with a higher resolution, the missing lines are created by interpolation. In a digital camera, the optical zoom is based on the physical lenses, but the digital zoom is accomplished by algorithms (see interpolated resolution).
References in periodicals archive ?
m]), [absolute value of x] [right arrow] [infinity], x [member of] R is crucial for exact interpolating recovery in [E.
In a fundamental paper [17] the authors study in more general settings the problem of sampling and interpolating in Paley-Wiener classes by using as a basic tool De Brange's theory of entire functions [4].
s[sigma],p], 1 [less than or equal] p < [infinity], s [greater than 1 integer, can be exactly recovered by using interpolating formulas based on sampling of the given function and its derivatives up to order s - 1 at a rate k[pi]/[sigma], k [member of] Z, i.
The idea of blending interpolants can be extended further to constructions of interpolating recoveries of entire functions of exponential type based on blending of infinite interpolating procedures.
The representation formula (4) for entire functions of exponential type is interpolating and it is constructed by using the following interpolating data:
The representation formula (4) yields explicit interpolating recoveries of entire functions of exponential type 0.
Denote by h(z) the right-hand side of the interpolating formula (4) and consider the auxiliary entire function
By using the interpolating formula (1) we construct oscillating entire functions of exponential type.
Oscillating algebraic polynomials can be easily constructed by using Lagrange interpolating formula based oil a finite number of interpolating data.

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