Interpolation: Second Edition

§1. Introduction

1.The theory of interpolation may, with certain reservations, be said to occupy itself with that kind of information about a function which can be extracted from a table of the function. If, for certain values of the variable x0, x1, x2, …, the corresponding values of the function f(x0), f(x1), f(x2), … are known, we may collect our information in the table

and the first question that occurs to the mind is, whether it is possible, by means of this table, to calculate, at least approximately, the value of f(x) for an argument not found in the table.

It must be admitted at once, that this problem—interpolation in the more restricted sense of the term—cannot be solved by means of these data alone. The table contains no other information than a correspondence between certain numbers, and if we insert another argument x and a perfectly arbitrarily chosen number f(x) as corresponding to that argument, we in no way contradict the information contained in the table.

2.It follows that, if the problem of interpolation is to have a definite, even if approximate, solution, it is absolutely indispensable to possess, beyond the data contained in the table, at least a certain general idea of the character of the function. In practice, it is a very general custom to derive formulas of interpolation on the assumption that the function with which we have to deal is a polynomial of a certain degree. The formula will, then, produce exact results if applied to a polynomial of the same or lower degree; but if it is applied to a polynomial of higher degree or to a function which is not a polynomial, nothing whatever is known about the accuracy obtained. In order to justify the application to such cases, it is customary to refer to the fact that most functions can, at least within moderate intervals, be approximated to by polynomials of a suitable degree. But this is only shirking the real difficulty; for if we have to deal with a numerical calculation, it is not sufficient to know that an approximation is obtainable; what we want to know, is how close is the approximation actually obtained.

3.A more fertile assumption, the correctness of which can often be ascertained, is that f(x) possesses, in a certain interval, a continuous differential coefficient of a certain order k. It will be shown later, that any such function can be represented as the sum of a polynomial and a remainder-term which is of such a simple nature, that if two numbers are known between which the differential coefficient of order k is situated, we can find limits to the error committed by neglecting the remainder-term in the interpolation.

In order to avoid having continuously to revert to the question of the assumptions made about f(x), let it be stated here once and for all, that f(x), where nothing else is expressly said, means a real, single-valued function, continuous in a closed interval, say a ≤ x ≤ b, and possessing in this interval a continuous differential coefficient of the highest order of which use is made in deriving each formula under consideration. If this order be k, f(k +¹) (x) need not exist at all; much less is there any necessity for assuming that f(x) is a polynomial or at least an analytical function.

Making such liberal assumptions about f(x), we will, nevertheless, be able to solve, in a simple and satisfactory manner, not only the interpolation problem in the restricted sense, but a great number of problems of a similar nature, which, taken together, form the contents of the theory of interpolation in the wider sense of the word.

4.While I assume that the reader is familiar with the notion of a continuous function and with the definitions of a differential coefficient and an integral, I make otherwise little use of analysis in this book. Two theorems belonging to elementary mathematical analysis are, however, so frequently referred to, that I find it practical to state them here, together with their proofs:

1.Rolle’s Theorem. If f(x) is continuous in the closed interval a ≤ x ≤ b and differentiable in the open interval a < x < b, and if f(a) = f(b) = 0, it is possible to find at least one point ξ inside the interval, such that

Proof. If f(x) is not identically zero in the interval (in which case the theorem is obvious), let m and M be the smallest and largest values it attains. As f(x) is a continuous function, at least one of these values must be attained for some argument ξ situated between a and b, as f(a) = f(b). For this value ξ we must have f′(ξ) = 0, as otherwise f(x) would be increasing or decreasing in the neighbourhood of ξ and assume values larger and smaller than f(ξ) which is impossible.

2. The Theorem of Mean Value. Let f(x) and φ(x) be integrable functions of which f(x) is continuous in the closed interval a ≤ x ≤ b, while φ(x) does not change sign in the interval. There exists, then, at least one point ξ inside the interval such that

Proof. Let φ(x) be positive (otherwise we may consider − φ(x)), and let m ≤ f(x) ≤ M. Then

There must, therefore, exist a number μ, intermediate between m and M, such that

and as f(x) is continuous, μ may be replaced by f(ξ).

The Theorem of Mean Value is easily extended to double integrals and to sums; we need not go into details.

Occasionally use has been made of results, borrowed from the theory of the Gamma-Function;¹ but the student who is not familiar with that function can, without any inconvenience, leave out those paragraphs, which have, therefore, been printed in smaller type.

5.Finally, we make considerable use of a simple theorem, belonging to the theory of series. Let us put

an equation which serves to define Rn + 1, if S, U0, U1, … Un are given. But we may also look upon (2) as an expansion for S with its remainder-term Rn + 1. In that case, let the first non-vanishing term after Un be Un + s. We have, then,

From this is immediately seen, that if Rn + 1 and Rn + s + 1 have opposite signs, then Un + s must have the same sign as Rn +1 and be numerically larger. This theorem, to which we shall refer as the Error-Test, expresses, then, that if Rn + 1 and Rn + s + 1 have opposite signs, then the remainder-term is numerically smaller than the first rejected, non-vanishing term, and has the same sign.

It would seem at first, that nothing much is gained by this theorem, as properties of the remainder-term are expressed by other properties belonging to the same; but we shall see later on that the sign of the remainder-term can often be easily determined by means of the known properties of the function to be developed, so that the theorem is of considerable practical value.

It may be noted that it is also seen from (3) that the condition that the two remainder-terms shall have opposite signs, is not only sufficient but also necessary, in order that the theorem shall hold.

¹See, for instance, Whittaker and Watson: Modern Analysis, third ed., Chapter XII.

§2. Displacement-Symbols and Differences.

6.In the theory of interpolation it is convenient to make use of certain symbols, denoting operations. Some of these symbols are important analogues of the symbols, known from the differential and integral calculus, denoting differentiation and integration.

In dealing with finite differences, we must first mention the symbol of displacement Ea. If this symbol is prefixed to a function f(x) or, as we shall say for brevity, if Ea is applied to f(x), we mean, that f(x) is changed into f(x + a). We have therefore, according to definition,

The letter a stands for any real number (positive, zero or negative). We have evidently E⁰ = 1. If a = 1, the exponent is usually left out, so that E = E¹.

The practical utility of the displacement-symbol depends on the fact that it obeys certain fundamental laws, and that it is in several respects permissible to operate with it, as if it were a number.

Thus, this symbol possesses the distributive property which is expressed in the equation

According to this relation, the correctness of which is obvious, the symbol Ea can be applied to the sum of two functions by applying it to each of the functions separately and adding the result.

Two displacement-symbols are said to be multiplied with each other, if one is applied after the other to the same function. In this respect, too, these symbols resemble numbers, as the order in which the factors are taken is immaterial. It is, for instance, evident that

This property is called the commutative property; it also holds with respect to a constant k, as

The exponents a, b, etc., resemble real exponents in that

in words, we shall say that the displacement-symbol obeys the index law. It follows that the nth power of the symbol denotes the operation repeated n times.

The product of two displacement-symbols (Ea Eb) may be looked upon as a single operation Ea + b, and it is easily seen that

a relation which expresses the so-called associative property, another property which our symbol has in common with numbers.

A linear function of displacement-symbols, and consequently any polynomial in such symbols, may be considered as an operation, e.g.

and this operation has, like the displacement-symbol itself, the distributive, commutative and associative properties. The component parts of the compound operation possess the same properties; thus

The reader will, finally, easily ascertain that two linear functions of displacement-symbols can be multiplied together according to ordinary rules, for instance

It follows from all these properties, taken together, that polynomials in displacement-symbols can be formed according to the same rules as are valid for numbers.

On the other hand it is necessary to call attention to the fact, that while division with Ea may without ambiguity be interpreted as multiplication with E− a, it is not yet allowed to divide by a polynomial in displacement-symbols, nor is it permitted to employ infinite series of such symbols. The examination of these questions follows in §18.

In many cases where confusion is not to be feared, the expression of the function f(x) is left out, and the calculation performed with the symbols alone. In that case we write, for instance, k + Ea instead of kf(x) + f(x + a). This often means a considerable economy in writing.

In dealing with functions of two or more variables (or functions) it is often necessary to introduce symbols each acting on one of the variables (or functions) and not on the others. These symbols must then be distinguished from each other, e.g., by denoting them by , , etc. After what has been said, the reader will have no difficulty in satisfying himself that polynomials in and can be formed as if and were numbers.

7.The simplest, and at the same time most important, linear functions of displacement-symbols are the differences. In particular, we note the three kinds of differences defined by the equations

or, in symbolical form,

They are called respectively the descending, the ascending and the central difference. The reason for this is made clear by a consideration of the following three difference-tables. In these, the numerical values are the same in corresponding places in all the three tables, so that only the notation differs.

It is seen that for constant n, are always found on the same descending line, f(n), ∇f(n), ∇²f(n), . . . . on the same ascending line, while f(n), δ²f(n), δ⁴f(n) … are found on a horizontal line, the same applying to

8.The three kinds of differences are analogous to the symbol of differentiation D, defined by Df(x) = f′(x). We introduce the three polynomials of degree n

They are called respectively the descending, the ascending and the central factorial. For n = 0 we assign to them the value 1 (which is also obtained if they are expressed by Gamma-Functions). For n > 0 they all contain x as a factor, and therefore vanish for x = 0. We now find

further

finally

We have, therefore, proved the following important properties:

these relations are quite analogous to the formula Dxn = nxn − ¹, well known from the differential calculus.

The central factorials of even and odd order may respectively be written

which shows that x[2v] is an even, x[2v +¹] an odd function of x.

On some occasions the following notation will be found useful:

The former of these functions is an odd, the latter an even function of x, so that, as in the case of (5) the symbolical exponent is odd when the function is odd, and even when it is even.

9.It is easily