Cauchy's functional equation

Cauchy's functional equation is the functional equation

Solutions to this are called additive functions. Over the rational numbers, it can be shown using elementary algebra that there is a single family of solutions, namely f(x) = cx for any arbitrary rational number c. Over the real numbers, this is still a family of solutions; however there can exist other solutions that are extremely complicated. Further constraints on f sometimes preclude other solutions, for example:

On the other hand, if no further conditions are imposed on f, then (assuming the axiom of choice) there are infinitely many other functions that satisfy the equation. This was proved in 1905 by Georg Hamel using Hamel bases. Such functions are sometimes called Hamel functions.[1]

The fifth problem on Hilbert's list is a generalisation of this equation. Functions where there exists a real number such that are known as Cauchy-Hamel functions and are used in Dehn-Hadwiger invariants which are used in the extension of Hilbert's third problem from 3-D to higher dimensions.[2]

Proof of solution over rationals

We wish to prove that any solution to Cauchy's functional equation, , takes the form .

Case 1:

Set .

.

Case 2:

By repeated application of Cauchy's equation to :

Replacing by , and multiplying by :

By the first equation:

.

Case 3:

Set .

.

Combining this with the result from case 2:

Replacing with :

Properties of other solutions

We prove below that any other solutions must be highly pathological functions. In particular, we show that any other solution must have the property that its graph is dense in , i.e. that any disk in the plane (however small) contains a point from the graph. From this it is easy to prove the various conditions given in the introductory paragraph.

Suppose without loss of generality that , and for some .

Then put .

We now show how to find a point in an arbitrary circle, centre , radius where .

Put and choose a rational number close to with:

Then choose a rational number close to with:

Now put:

Then using the functional equation, we get:

Because of our choices above, the point is inside the circle.

Proof of the existence of other solutions

The linearity proof given above also applies to any set , a scaled copy of the rationals, showing that the only solutions are linear when the domain of is restricted to such sets. However, as shown below, highly pathological solutions can be found for functions based on these linear solutions, by viewing the reals as a vector space over the field of rational numbers. Note, however, that this method is nonconstructive, relying as it does on the existence of a Hamel basis for any vector space, a theorem proved using Zorn's lemma.

Since every vector space has a basis, there is a basis for over the field , i.e. a set with the property that for any , there is a unique finite set and coefficients with such that . We note that because no explicit basis for over can ever be constructed, none of the pathological functions below can be defined explicitly.

By the argument above, for each , the restriction of to (a scaled copy of the rationals) must be a linear map, with some constant of proportionality . In other words, for , is the map . Since any can be expressed as a unique (finite) linear combination of and is additive, the function is, by necessity, extended to all of as

.

It is easy to check that is a solution to Cauchy's functional equation for any . Moreover, it is clear that every solution is of this form. In particular, the solutions of the functional equation are linear if and only if is a constant function. In this sense, despite the inability to explicitly exhibit a nonlinear solution, "most" solutions are actually nonlinear and pathological.

References

  1. Kuczma (2009), p.130
  2. V.G. Boltianskii (1978) "Hilbert's third problem", Halsted Press, Washington

External links

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