Convergence of sequences of functions with differing domains?

[Stephen Tashi]

TL;DR Summary

Is there a type of convergence defined for sequences of functions that have different domains?

Of the various notions of convergence for sequences of functions (e.g. pointwise, uniform, convergence in distribution, etc.) which of them can describe convergence of a sequence of functions that have different domains? For example, let $F_n(x)$ be defined by $F_n(x) = 1 + h$ where $h = 1/n$ and F_n(x) has the domain $\{h,2h,3h,..1\}$ Intuitively, the sequence of functions $F_n, n = 1,2,3,...$ converges to the function $F(x) = 1$ defined on $(0,1]$.

Which definitions handle the above type of convegence without assuming the definition of each function in the sequence is extended to have the same domain as the function that is the limit? ( For example, for convergence in $L_2$ we could stipulate that the domain of each $F_n(x)$ is extended to $(0,1]$ by defining $F_n(x)$ to be a step function over intervals $(kh, (k+1)h)$ Then the same notion of integration would apply to both $F_n(x)$ and $F(x)$. Can we avoid making such stipulations?)

 

[fresh_42]

Convergence is a topological term. So if you have another topological space for each function, convergence makes no sense.

But your functions do not depend on the variable, so their domain is $\mathbb{R}$ and all functions are in $C^\infty(\mathbb{R})$ converging to $1$.

 

[Stephen Tashi]

But your functions do not depend on the variable, so their domain is $\mathbb{R}$

I agree we can define new functions by extending the domain of the functions in the example, but the domain of a function $F_n$ is given in it's definition. For example, nothing in the definition of $F_2(x)$ says that $F_2$ can be evaluated at $x = 1/10$.

To make an example using non-constant functions, Define $G_n(x) = x^2 + h$ for $h = 1/n$ and $x \in \{h,2h,3h,...1\}$.

 

[fresh_42]

Doesn't matter, you cannot define convergence across different spaces.

 

[Stephen Tashi]

Doesn't matter, you cannot define convergence across different spaces.

That suggests considering the graphs of $F$ and $F_n$ each to be subsets of the metric space $\mathbb{R}^2$. So one aspect of the convergence of $F_n$ to $F$ can handled by the a definition for what it means for a sequence of sets to converge to a set.

 

[fresh_42]

In this case you have to define a metric on $\mathcal{P}(\mathbb{R}^2)$ or define a topology otherwise which allows you to define "nearby". Maybe the concept of nets is what you were looking for.

 

[Stephen Tashi]

I found this paper:

https://projecteuclid.org/download/pdf_1/euclid.rae/1229619414

ON THE GRAPH CONVERGENCE OF SEQUENCES OF FUNCTIONS
by Zbigniew Grande

Let $(X,T_X)$ and $(Y,T_Y)$ be topological spaces and let $\mathbb{R}$ be the set of all reals considered with the natural topology $T_e$. Denote by $T_X \times T_Y$ the product topology in $X×Y$. We will say that a sequence of functions $f_n:X→Y$ graph converges to a function $f:X→Y$ if for each set $U∈T_X×T_Y$ containing the graph $Gr(f)$ of the function $f$ there is a positive integer $k$ such that $Gr(f_n)⊂U$ for all $n k$.

I think "$nk$" is a misprint for $n>k$.

That definition is relevant to the OP if the notation "$f_n:X \rightarrow Y$" allows that the domain of $f_n$ might be a proper subset of $X$ and the co-domain might be a proper subset of $Y$.

It's interesting that meaning of "$Gr(f)$" is taken to be self-evident. (I assume "graph of a function" in this context is a separate concept from "graph" as studied in Graph Theory. )

 

[fresh_42]

The graph of a function $f:\mathbb{R}\to \mathbb{R}$ is the set $\{(x,f(x))\,|\,x\in D\subseteq \mathbb{R}\}$ with domain $D$. It's a subset of $\mathbb{R}^2$. You can have convergence on this set, here as the Euclidean product topology. In this sense your graphs converge. It's another question whether this is useful for anything. You have a lot of artificial gaps, so I can't see how this could be useful.

In case your functions are linear operators on complex Hilbert spaces, this type of convergence is called graph convergence. (At least in my book.) They are useful in case of self adjoint operators. However, in this case we have a sequence of functions and a sequence of operators such that $(f_n,T_nf_n)\to (f,g).$

 

[Stephen Tashi]

You have a lot of artificial gaps, so I can't see how this could be useful.

I'm considering questions about when a sequence of solutions to recurrence relations defined on smaller and smaller intervals will converge to a unique function defined everywhere on some subinterval of the reals. Since the solutions to recurrence relations are (in general) only defined on discrete values ${h, 2h, 3h,...}$ a notion of convergence of a sequence of functions is needed that applies to this case.

In case your functions are linear operators on complex Hilbert spaces, this type of convergence is called graph convergence. (At least in my book.) They are useful in case of self adjoint operators. However, in this case we have a sequence of functions and a sequence of operators such that $(f_n,T_nf_n)\to (f,g).$

I'll have to think about whether operators bear on the topic of recurrence relations. Anyway, the recurrence relations are a topic for a different thread.

The definition given by Zbigniew Grande (ZB) defines convergence for sequences of graphs of a function to a graph of a function. It doesn't define the more general notion of the convergence of sequences of graphs of a function to a set S (which may or may not be the graph of a function). Using $S$ to denote a subset of $T_x \times T_y$, we can generalize ZB's definition to define what it means for $S$ to be the limit, by using $S$ in place of $Gr(f)$. I'll call this generalization a "Generalized Zibigniew Grande Limit" (GZGL)


Natural questions about that generalization about GZGLs are

1. If a GZGL exists, is it unique?

2. If the GZGL limit of sequence of functions exists, is that limit also a function?

I think the paper answers "yes" to both questions in the case of $X = \mathbb{R}$ and $Y = \mathbb{R}$ since it gets those results in the cases where the product topology is "T 1" and Hausdorff.

 

[fresh_42]

Think about your first example. Let's define $g(x)=f(x)\equiv 1$ with $D(f)=[0,1]$ and $D(g)=[0,1]\cap \mathbb{Q}$. Then we have two different limits.

 

[Stephen Tashi]

Think about your first example. Let's define $g(x)=f(x)\equiv 1$ with $D(f)=[0,1]$ and $D(g)=[0,1]\cap \mathbb{Q}$. Then we have two different limits.

I see what you mean.

The Zibigniew Grande paper is assuming each $f_n$ has domain $X$. This is clear from the proof of Theorem 1, which assumes $f_n(x)$ exists for each $x$ where $f(x)$ exists. So the paper isn't relevant to what I want.

The notions of "inner limit" and "outer limit" for sequences of sets ( https://en.wikipedia.org/wiki/Limit_superior_and_limit_inferior#General_set_convergence ) apply to the sequence $Gr(f_n)$ without the assumption that the domain of $f_n$ is all of $X$.

 

[fresh_42]

If you have a recursive system, there is another approach you could try: changing the system into differential equations. The standard here is the other way around, from diff systems to numerical difference and recursion systems, but IIRC it is also possible to go from discrete to smooth.

Here is another interesting list of possibilities:
https://en.wikipedia.org/wiki/Recurrence_relation