Schröder's equation and functional analysis

In summary, the conversation discusses Schröder's equation, where the goal is to find a nonzero function and real number that satisfy a functional equation. The motivation behind this is to understand the dynamics of the function better. The conversation also introduces the set of functions from a subset of the real numbers to the real numbers, and a map that allows for the approximation of solutions to the functional equation. There is also discussion about fixed points and contractions, and the use of a pseudoinverse to approximate solutions. Finally, there is mention of using the projection theorem to find a function in the kernel of the map.
  • #1
phoenixthoth
1,605
2
schröder's equation is a functional equation. let's assume A is a subset of the real numbers and g maps A to itself. the goal is to find a nonzero (invertible, if possible) function f and a real number r such that [tex]f\circ g=rf[/tex].

motivation: if there is an invertible f, then the nth iterate of g is given by [tex]g^{n}=f^{-1}\left( r^{n}f\right) [/tex]. this can lead to understanding the dynamics of g better.

let V be the set of functions from A to R, the set of real numbers.

let S(r,g) denote [tex]\left\{ f\in V:f\circ g=rf\right\} [/tex]. S(r,g) is a subspace of V.

define a map k(r,g) from V to V by the following:
[tex]k\left( r,g\right) \left( f\right) =f\circ g-rf[/tex].

k(r,g) is a linear operator on V with kernel S(r,g).

putting in this context, are there any theorems (with more assumptions perhaps) that
1. give one at least a sense of when the kernel of a map is not {0}
2. can find a basis for the kernel of a map explicitly
3. can approximate solutions to k(r,g)(f)=0?

any thoughts would be helpful. if you want to get real specific, assume A=[0,1] and g is either 2^(x-1) or x^2.

as an example, [tex]x+\frac{b}{a-1}\in S\left( a,ax+b\right) [/tex] , abusing the notation a little to let ax+b represent the function g such that g(x)=ax+b.

i know that S(r,g)=fix(k(r-1,g)), where fix(h) is the set of fixed points of h. this opens me up to all the fixed point theorems in functional analysis except that i already know it has a fixed point, namely the zero function. what will be interesting is uniqueness fixed point theorems to show that 0 is the only fixed point. however, under one or two more assumptions on g and A, for example, i can turn my attention to bounded functions and use the sup norm and perhaps talk about contractions. all i can do in that case, and here it's critical that g maps A to A, is show that the norm of k(r,g) is at most 1+|r| which doesn't prove it's a contraction. it would be extremely helpful if anyone were able to tell me what the norm of this operator k(r,g) is. in other words, the sup of ||k(r,g)(f)|| where ||f||=1. then i can tell for which r k is a contraction.

also, is it true that if an operator is NOT a contraction then it doesn't have a unique fixed point or could it not be a contraction and still have a unique fixed point?

any thoughts would be helpful.
 
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  • #2
pseudoinverse

i was also thinking of using some kind of pseudoinverse.

if i can somehow define a transpose of k(r,g), [tex]k^{T}\left( r,g\right) [/tex], then one form of its pseudoinverse [tex]k^{+}[/tex]would be maybe
[tex]k^{+}\left( r,g\right) =\left( k^{T}\left( r,g\right) \circ k\left( r,g\right) \right) ^{-1}k^{T}\left( r,g\right) [/tex] and so the solution to [tex]k\left( r,g\right) \left( f\right) =0[/tex] could be approximated by [tex]k^{+}\left( r,g\right) \left( 0\right) [/tex].

two things:
1. how would i define [tex]k^{T}\left( r,g\right) [/tex] in that my space is not equipped, as far as i know, an inner product that gives rise to the sup norm and
2. ensuring that [tex]k^{T}\left( r,g\right) \circ k\left( r,g\right) [/tex] is invertible?
 
  • #3
i'm suspecting that [tex]\left\| k\left( r,g\right) \right\| =\min \left\{ 1+\left| r\right| ,\left\| g\right\| +\left| r\right| \right\} [/tex] but I'm not sure how to prove it.

in general, [tex]\left\| k\left( r,g\right) \right\| =\sup_{f}\frac{\left\| k\left( r,g\right) \left( f\right) \right\| }{\left\| f\right\| }[/tex].

it should be something that depends on g and r.
 
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  • #4
if [tex]V[/tex] is the set of all bounded continuous functions mapping [tex]A[/tex] to [tex]R[/tex], can someone exhibit an orthonormal basis [tex]B[/tex] for V? or at least confirm my suspicion that the set of power functions (restricted to A--let's say A is the unit interval even) is dense in V? if so, i can use something like grahm-schmit to find an orthonormal basis for V, right?

is it true that [tex]\left\| k\left( r,g\right) \right\| :=\sup_{f\in V}\frac{\left\| k\left( r,g\right) \left( f\right) \right\| }{\left\| f\right\| }=\sum_{\phi \in B}\left\| k\left( r,g\right) \left( \phi \right) \right\| [/tex]?
 
  • #5
possible method of attack

the projection theorem: let [tex]X[/tex] be a Hilbert space, [tex]K[/tex] a closed convex subset, and [tex]x\in X[/tex]. there there is a unique [tex]x^{\prime }\in X[/tex] such that [tex]\left\| x-x^{\prime }\right\| =\inf_{y\in K}\left\| x-y\right\| [/tex].

i want to take X to be V, the set of continuous real-valued functions from [tex][0,1][/tex] to [tex]R[/tex]. I'm not sure if this is a hilbert space. i want to take [tex]K[/tex] to be [tex]S(r,g)=ker(k(r,g))[/tex]; I'm fairly sure that's closed but I'm not sure it's convex.

this unique [tex]x^{\prime}[/tex] is often denoted [tex]P_{K}x[/tex] and referred to as the projection of [tex]x[/tex] onto [tex]K[/tex].

then what i want to do is take a seed function and apply this projection to it to get a function in the kernel and be able to specify when this projection is nonzero. i could at least maybe squeeze an existence/uniqueness theorem out of this though I'm not sure how a calculation would be done.
 
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1. What is Schröder's equation?

Schröder's equation is a linear functional equation that is used in functional analysis to describe the behavior of certain functions. It is named after the German mathematician Ernst Schröder.

2. What is functional analysis?

Functional analysis is a branch of mathematics that deals with the study of vector spaces and linear operators, with a particular focus on infinite-dimensional spaces. It is used to study functions and their properties in a more abstract and general setting.

3. How is Schröder's equation used in functional analysis?

Schröder's equation is used to study the properties of continuous functions on a given interval. It is also used to solve differential equations and to study the convergence of infinite series.

4. Are there any applications of Schröder's equation in real-world problems?

Yes, Schröder's equation has various applications in physics, engineering, and computer science. For example, it can be used to analyze the stability of dynamical systems, to model heat transfer in materials, and to design efficient algorithms for signal processing.

5. Is Schröder's equation difficult to understand?

Like any mathematical concept, Schröder's equation can be challenging to grasp at first. However, with proper study and practice, one can develop a solid understanding of its principles and applications. It is a fundamental tool in functional analysis and is worth investing time and effort to understand it.

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