Wikipedia / Cauchy's functional equation
Wikipedia / Cauchy's functional equation
Cauchy's functional equation is the functional equation
that solves this equation is called an additive function. Over the rational numbers, it can be shown using elementary algebra Over the real numbers, the family of linear maps an arbitrary real constant, is likewise a family of solutions; however there can exist other solutions not of this form that are extremely complicated. However, any of a number of regularity conditions, some of them quite weak, will preclude the existence of these pathological is linear if:
is continuous (Cauchy to be continuous at one point (Darboux, 1875).
for all .
is bounded above or below on any interval.
and some positive integer .
is not dense in .
On the other hand, if no further conditions are imposed on , 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.
The fifth problem on Hilbert's list 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 3D to higher dimensions.
This equation is sometimes referred to as Cauchy's additive functional equation to distinguish it from the other functional equations introduced by Cauchy in 1821, the exponential functional equation , the logarithmic functional equation , and the multiplicative functional equation .
Solutions over the rational numbers
are vector spaces over an extension field.
Theorem: -linear.
Proof:.
.
.
we know , therefore . Thus far we have proved
.
and hence .
and , so, putting things together,
, q.e.d.
Properties of nonlinear solutions over the real numbers
We prove below that any other solutions must be highly pathological functions. In particular, it is shown that any other solution must have the property that its graph is dense in , that is, 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.
. From this, one can construct an orthonormal basis can be approximated.
Existence of nonlinear solutions over the real numbers
The linearity proof given above also applies to is a scaled copy of the rationals. This shows that only linear solutions are permitted when the domain and 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 statement proved using Zorn's lemma. (In fact, the existence of a basis for every vector space is logically equivalent to the axiom of choice.) There exist models such as the Solovay model where all sets of reals are measurable which are consistent with ZF + DC, and therein all solutions are linear.
over the field can be expressed uniquely as is a finite subset of is in can be written down, the pathological solutions defined below likewise cannot be expressed explicitly.
must be a linear map for each for is the map can be expressed as a unique (finite) linear combination of the
on the basis elements, . 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 constant over all . Thus, in a sense, despite the inability to exhibit a nonlinear solution, "most" (in the sense of cardinality) solutions to the Cauchy functional equation are actually nonlinear and pathological.
See also
References
External links
Solution to the Cauchy Equation Rutgers University