Functional equation

As a functional equation in is mathematics an equation called, one or more for their solution functions are being sought. Many functions can be defined using an underlying functional equation. Usually, only those equations are referred to as functional equations that cannot be converted to an explicitly closed form for the function (s) sought, and in which the function (s) sought occurs with different arguments .

When investigating functional equations, one is interested in all solution functions of the investigated functional space, not just one. Otherwise it is quite trivial to construct a functional equation for any given function.

“It is natural to ask what a functional equation is. But there is no easy satisfactory answer to this question. "

“It's natural to wonder what a functional equation is. But there is no satisfactory answer to this question. "

Functional equations examined by Cauchy

Augustin Louis Cauchy examined the continuous solutions of the following functional equations in Chapter 5 of his Cours d'Analyse de l'Ecole Royale Polytechnique in 1821 : ${\displaystyle \Phi }$

1)

${\displaystyle \Phi \colon \mathbb {R} \to \mathbb {R} ,\quad \Phi (x+y)=\Phi (x)+\Phi (y)\;}$

The continuous solutions of this functional equation, i.e. the solutions under the assumption that the function is continuous, are the "continuous" linear functions , where is a real constant. The term Cauchy's functional equation or Cauchy functional equation has become established for this functional equation. ${\displaystyle \Phi (x)=ax\;}$ ${\displaystyle a\;}$

2)

${\displaystyle \Phi \colon \mathbb {R} ^{+}\to \mathbb {R} ^{+},\quad \Phi (xy)=\Phi (x)\Phi (y)\;}$

The continuous solutions of this functional equation are the power functions , where is a real constant. ${\displaystyle \Phi (x)=x^{a}\;}$ ${\displaystyle a\;}$

3)

${\displaystyle \Phi \colon \mathbb {R} \to \mathbb {R} ^{+},\quad \Phi (x+y)=\Phi (x)\Phi (y)\;}$

The continuous solutions of this functional equation are the exponential functions , where is a positive real constant. ${\displaystyle \Phi (x)=a^{x}\;}$ ${\displaystyle a\;}$

4)

${\displaystyle \Phi \colon \mathbb {R} ^{+}\to \mathbb {R} ,\quad \Phi (xy)=\Phi (x)+\Phi (y)\;}$

The continuous solutions of this functional equation are the logarithmic functions , where is a positive real constant. ${\displaystyle \Phi (x)=\log _{a}(x)\;}$ ${\displaystyle a\;}$

5) Furthermore, the null function is a trivial solution to each of these functional equations.

Known functional equations of special functions

Gamma function

The functional equation

${\displaystyle \Phi \colon \mathbb {R} ^{+}\to \mathbb {R} ,\quad \Phi (x+1)=x\,\Phi (x)}$

is fulfilled by the gamma function. If one only considers functions that are logarithmically convex , then all solutions of this equation are described by, with . This is Bohr-Mollerup's theorem about the uniqueness of the gamma function as a continuation of the faculties from to . ${\displaystyle \Gamma }$ ${\displaystyle a\Gamma }$ ${\displaystyle a\in \mathbb {R} }$ ${\displaystyle \mathbb {N} _{0}}$ ${\displaystyle \mathbb {R} ^{+}}$

Furthermore, the gamma function is also a solution to the functional equation

${\displaystyle \Phi \colon \mathbb {R} \!\setminus \!\mathbb {Z} \to \mathbb {R} ,\quad \Phi (x)\,\Phi (1-x)={\frac {\pi }{\sin(\pi x)}}}$

which only represents a special kind of “reflection symmetry” , as can be seen from the substitution and subsequent logarithmizing of the new functional equation. ${\displaystyle {\tfrac {1}{2}}}$ ${\displaystyle \Phi (x)={\sqrt {\frac {\pi }{\sin(\pi x)}}}e^{\Theta (x)}}$

Polygamma functions

For are the functional equations ${\displaystyle m\in \mathbb {N} _{0}}$

${\displaystyle \Phi _{m}\colon \mathbb {R} ^{+}\to \mathbb {R} ,\quad \Phi _{m}(x+1)=\Phi _{m}(x)+{\frac {(-1)^{m}m!}{x^{m+1}}}}$

fulfilled by the polygamma functions. For solid , all continuous and monotonic solutions are represented by the functions with any . ${\displaystyle \psi _{m}}$ ${\displaystyle m}$ ${\displaystyle a+\psi _{m}}$ ${\displaystyle a\in \mathbb {R} }$

Bernoulli polynomials

For are the functional equations ${\displaystyle m\in \mathbb {N} _{0}}$

${\displaystyle \Phi _{m}\colon \mathbb {R} \to \mathbb {R} ,\quad \Phi _{m}(x+1)=\Phi _{m}(x)+mx^{m-1}}$

fulfilled by the Bernoulli polynomials . All continuous solutions of this equation are described by plus further (periodic) solutions of the homogeneous functional equation, where a is any real number. More on this in the following section. ${\displaystyle {\text{B}}_{m}}$ ${\displaystyle a+{\text{B}}_{m}}$

Periodic functions

The functional equation

${\displaystyle \Phi \colon \mathbb {R} \to \mathbb {R} ,\quad \Phi (x+1)=\Phi (x)}$

represents the homogeneous part of the solution of the above functional equations, since one can simply add their solution to a solution of any inhomogeneous functional equation and thus obtain a new one, as long as no further restrictive conditions are violated. If one considers all holomorphic functions in their entirety , then all solution functions are represented by ${\displaystyle \mathbb {C} }$

Linear combinations of with . ${\displaystyle e^{2\pi inx}}$ ${\displaystyle n\in \mathbb {Z} }$

This knowledge is a basis of the Fourier analysis . With the exception of the case n = 0, all of these functions are neither convex nor monotonic.

Zeta function

The functional equation

${\displaystyle \Phi \colon \mathbb {R} \to \mathbb {R} ,\quad \Phi (x+1)=-2\,(2\pi )^{x}\cos \left({\frac {\pi x}{2}}\right)\Gamma (-x)\Phi (-x)}$

is fulfilled by the Riemann zeta function. denotes the gamma function . ${\displaystyle \zeta }$ ${\displaystyle \Gamma }$

Note: Through the substitution

${\displaystyle \Phi (x)={\frac {2\pi ^{x/2}\,\Theta (x+{\tfrac {1}{2}})}{x(x-1)\,\Gamma \left({\tfrac {x}{2}}\right)}}}$

and subsequent algebraic simplification, this functional equation for is converted into a new one for which ${\displaystyle \Phi }$ ${\displaystyle \Theta }$

${\displaystyle \!\ \Theta ({\tfrac {1}{2}}+x)=\Theta ({\tfrac {1}{2}}-x)}$

reads. May thus be placed by transforming a shape the original functional equation that only an even function to calls. The Riemann zeta function transformed in this way is known as the Riemann Xi function . ${\displaystyle {\tfrac {1}{2}}}$ ${\displaystyle \xi }$

Even and odd functions

The two functional equations

${\displaystyle \Phi \colon \mathbb {R} \to \mathbb {R} ,\quad \Phi (x)=\pm \Phi (-x)}$

are fulfilled by all even or odd functions. Another "simple" function equation is

${\displaystyle \Phi \colon I\!\subset \!\mathbb {R} \to \mathbb {R} ,\quad \Phi (\Phi (x))=x}$

so all functions that are their own inverse function on the interval describe their solution set. With these three functional equations, however, the focus is more on the question of how their solutions can be meaningfully characterized. ${\displaystyle I}$

"Real" iterates of a function

Given an analytical , bijective function , then Schröder's functional equation reads ${\displaystyle f\colon I\subset \mathbb {R} \to J\subset \mathbb {R} }$

${\displaystyle \Phi \colon I\cap J\to \mathbb {R} ,\quad \Phi (f(x))=c\,\Phi (x)}$

with a fixed to be determined . If one applies the inverse function of on both sides of this equation , then one can generalize this to the definition of ${\displaystyle c\in \mathbb {R} }$ ${\displaystyle \Phi }$

${\displaystyle \forall \,t\in \mathbb {R} \colon \quad f_{t}(x)=\Phi ^{-1}(c^{t}\Phi (x))}$

and for any fixed t this function behaves like a t-times iterated function . A simple example: given the general power function for on for fixed . In this case the solution is Schröder's equation and the result is becomes. ${\displaystyle f_{t}}$ ${\displaystyle f}$ ${\displaystyle a\in \mathbb {R} ^{+}}$ ${\displaystyle x^{a}}$ ${\displaystyle f}$ ${\displaystyle \mathbb {R} ^{+}}$ ${\displaystyle \Phi (x)=\ln(x)}$ ${\displaystyle c=a}$ ${\displaystyle f_{t}(x)=x^{a^{t}}}$

Modular shapes

The functional equation

${\displaystyle \Phi \colon \mathbb {C} \to \mathbb {C} ,\quad \Phi \left({az+b \over cz+d}\right)=(cz+d)^{k}\Phi (z)\quad {\text{ mit }}\quad ad-bc=1}$

where are given is used in the definition of modular forms . ${\displaystyle a,b,c,d,k\in \mathbb {Z} }$

Wavelets and approximation theory

For and defines the functional equation ${\displaystyle d\in \mathbb {N} }$ ${\displaystyle a_{-d},a_{-d+1},\ldots ,a_{d}\in \mathbb {R} }$

${\displaystyle \Phi \left({\frac {x}{A}}\right)=a_{-d}\Phi (x-d)+\cdots +a_{0}\Phi (x)+\cdots +a_{d}\Phi (x+d)=\sum _{k=-d}^{d}a_{k}\Phi (x-k)\quad {\text{ mit }}\quad A:=\sum _{k=-d}^{d}a_{k}}$

in the theory of wavelet bases the scaling function of a multiscale analysis . The B-splines important in approximation theory and computer graphics are solutions to such a refinement equation; further solutions including the coefficients can be found under Daubechies wavelets . There are extensions with vector-valued solution functions f and matrices as coefficients.

Sine and cosine

Considering the functional equation , the exponential function over the complex numbers satisfied, and divides the range of values into real and imaginary parts on so , and further restricts the definition range on a, we obtain two functional equations in two unknown functions, namely ${\displaystyle \Phi (x+y)=\Phi (x)\Phi (y)\!}$ ${\displaystyle \Phi (x)=\Theta (x)+\mathrm {i} \Omega (x)\!}$ ${\displaystyle \mathbb {R} }$

${\displaystyle \Theta (x+y)=\Theta (x)\Theta (y)-\Omega (x)\Omega (y)\!}$

and

${\displaystyle \Omega (x+y)=\Theta (x)\Omega (y)+\Omega (x)\Theta (y)\!}$

which correspond to the addition theorems and can be understood as a functional equation system for the real sine and cosine functions.

More examples of general functional equations

Recursion equations

A simple class of functional equations consists of the recursion equations above . From a formal point of view, an unknown function is sought. ${\displaystyle \mathbb {Z} }$ ${\displaystyle f\colon \mathbb {Z} ^{k}\to \mathbb {Z} }$

A very simple example of such a recursion equation is the linear equation of the Fibonacci sequence :

${\displaystyle f(n+2)=f(n+1)+f(n)}$.

This can of course also be considered embedded in the set of real numbers, so here

${\displaystyle \Phi \colon \mathbb {R} \to \mathbb {R} ,\quad \Phi (x+2)=\Phi (x+1)+\Phi (x)}$

whose analytical solutions then all have the form

${\displaystyle \Phi (x)=a\left({\frac {1+{\sqrt {5}}}{2}}\right)^{x}+b\left({\frac {1-{\sqrt {5}}}{2}}\right)^{x}}$

have with any . Only as a function can all of its solution functions, e.g. B. as ${\displaystyle a,b,\in \mathbb {R} }$ ${\displaystyle f\colon \mathbb {Z} \to \mathbb {Z} }$

${\displaystyle f(n)={\frac {1}{\sqrt {5}}}\left(f(1)-{\tfrac {1-{\sqrt {5}}}{2}}f(0)\right)\left({\frac {1+{\sqrt {5}}}{2}}\right)^{n}+{\frac {1}{\sqrt {5}}}\left({\tfrac {1+{\sqrt {5}}}{2}}f(0)-f(1)\right)\left({\tfrac {1-{\sqrt {5}}}{2}}\right)^{n}}$

specify. Although irrational numbers appear in this representation, there is an integer value for each as long as are. ${\displaystyle n}$ ${\displaystyle f(0),f(1)\in \mathbb {Z} }$

Laws of Calculation

Calculation laws such as commutative law , associative law and distributive law can also be interpreted as functional equations.

Example associative law: a set is given . For their binary associative link  or two-parameter function apply to all ${\displaystyle M}$ ${\displaystyle \times \,\colon M^{2}\to M}$ ${\displaystyle f\colon M^{2}\to M}$ ${\displaystyle a,b,c\in M}$

Infix notation : ${\displaystyle (a\times b)\times c=a\times (b\times c)}$

and in

Prefix notation : ${\displaystyle f(f(a,b),c)=f(a,f(b,c))}$

where is identified. ${\displaystyle f(a,b){\text{ mit }}a\times b}$

If you designate the binary logic function of the 2nd level (e.g. multiplication) and the logic function of the 1st level (e.g. addition), then a distributive law would be written as a functional equation ${\displaystyle g}$ ${\displaystyle f}$

${\displaystyle g(a,f(b,c))=f(g(a,b),g(a,c))}$ for all ${\displaystyle a,b,c\in M}$

ring.

Remarks

All examples have in common that two or more known functions (multiplication by a constant, addition, or just the identical function) are used as arguments of the unknown function.

When searching for all solutions to a functional equation, additional conditions are often set, for example, in the Cauchy equation mentioned above, continuity is required for reasonable solutions. Georg Hamel showed, however, in 1905 that, given the axiom of choice , discontinuous solutions also exist. These solutions are based on a Hamel basis of the real numbers as a vector space over the rational numbers and are primarily of theoretical importance.

literature

• Janos Aczel: Lectures on Functional Equations and Their Applications , Dover 2006, ISBN 0486445232

Web links

• Cauchy's equation in Mathworld (Engl.)

Individual evidence

1. ↑ Pl. Kannappan, Functional Equations and Inequalities with Applications , Springer 2009, ISBN 978-0-387-89491-1 , preface
2. ↑ visualiseur.bnf.fr
3. ↑ G. Hamel: A basis of all numbers and the discontinuous solutions of the functional equation f (x + y) = f (x) + f (y). Math. Ann. 60, 459-462, 1905.