I am studying the properties of a particular class of functions, and I'd appreciate some help in proving a property of that class. I started with a class of functions and made some modifications to show that it forms a vector space and to show that it is closed under convolution. I have given the entire description below, and require help in proving that the set is closed under the operation of convolution (circular). I have also mentioned what methods I have used to prove that it forms a vector space, so that i can get cleared of the difficulty i am facing in arriving at the result of closure under convolution. ( in case you want to avoid reading the entire post linearly, the property needs to be proved on the set $S_{pc}$ and it is described in the section "closure under addition").
PS : Please do let me know if you want to know the motivation for such a study. I do not have any concrete reasons which can be expressed in mathematically precise terms, behind the motivation, but I have an intuition behind it.
PS 2 : A construction method and proof of non-emptiness of this class of functions is given in this Q&A.
Definition
Let $f \colon \mathbb{R} \to \mathbb{R}$ be a function, consider a point $x \in \mathbb{R}$ where $f$ is continuous, then the following two statements are equivalent.
- $Cf(x) = k$, where $k \in \{0\}\bigcup\mathbb{N}\bigcup\{\infty\}$.
- The function is continuous at $x$ and the maximum number of times $f$ is differentiable at $x$ is $k$.
Clarification (Added)
$Cf(x)=k$, where we know that $f$ is a function and $x$ is the point in its domain, then it means that the maximum number of times $f$ is differentiable at $x$ is $k$. This is not a good notation as someone could think that $Cf$ itself is one function different from f and not associated with it, but here I intended that $C$ tells about a proprty of the function at a point in its domain.....not a convincing notation......but i'd like know any ideas to make a good notation for it.
Definition of a class of functions
The set $S$ consists of functions $f \colon (0,1) \to \mathbb{R}$, which satisfy the following properties.
Given any $f \in S$ there exist a countable dense subset $D \subset (0,1)$ and maps $k,ck$ defined as $k \colon D \to \mathbb{N}$ and $ck \colon (0,1) \to \mathbb{N}\bigcup\{\infty\}$ which satisfy the following properties.
- $\forall n \in \mathbb{N}$, the pre-image $k^{-1}(\{n\})$ is a finite set.
- $\forall x \in $D$, ck(x) \ge k(x)$ but finite and $\forall x \in (0,1)$\D$, ck(x) = \infty$.
- $\forall x \in (0,1), Cf(x) = ck(x)$
- Whenever $Cf(x) = k$ is finite, for the $(k+1)^{nth}$ derivative of $f$ at $x$, the left and right limits exist and are not equal. (the left and right limits do not diverge).
Let $S_p$ be the set of all functions $f_p$ which are periodic versions of the functions $f \in S$.
The periodic version $f_p$ of the function $f \in S$ is defined as $f_p(x) = f(x) \forall x \in (0,1)$, $f_p(0) = f(0+)$ and $f_p(1) = f_p(0)$ and $ \forall x \in \mathbb{R}, f_p(x) = f_p(x+1)$.
Closure property under addition
EDIT : (This argument is false and the set $S_{pc}$ is not closed under addition) (see comment by Andrew)
Let $f_1,f_2 \in S_p$ such that $f_1$ is not same as $-f_2$. I am able to prove that if $f_3 = f_1 + f_2$, then $f_3 \in S_p$, by making the following considerations.
Let $D_1,k_1,ck_1$ be the required set and maps respectively for the function $f_1$ as per the definitions given above. Let $D_2,k_2,ck_2$ be the required set and maps of $f_2$.
Let $D_3,k_3,ck_3$ be defined as below.
$D_3 = D_1 \bigcup D_2$.
Let $l_1 \colon (0,1) \to \mathbb{N}\bigcup\{\infty\}$ and $l_2 \colon (0,1) \to \mathbb{N}\bigcup\{\infty\}$ are defined as
$l_1(x) = k_1(x)$ if $x \in D_1$ otherwise $\infty$. $l_2(x) = k_2(x)$ if $x \in D_2$ otherwise $\infty$.
$k_3(x)$ is assigned as $k_3(x) = \min\{l_1(x),l_2(x)\}$
and
Let $m_1 \colon (0,1) \to \mathbb{N}\bigcup\{\infty\}$ and $m_2 \colon (0,1) \to \mathbb{N}\bigcup\{\infty\}$ are defined as
$m_1(x) = ck_1(x)$ if $x \in D_1$ otherwise $\infty$. $m_2(x) = ck_2(x)$ if $x \in D_2$ otherwise $\infty$.
$ck_3(x)$ is assigned as $ck_3(x) = \min\{m_1(x),m_2(x)\}$.
By assigning $D_3,k_3,ck_3$ as mentioned above I am able to prove that $f_3 \in S_p$.
To get the closure property under addition, we can add the set of all constant functions $K$ to the set $S_p$ to form a new set $S_{pc}$.
Hence I am able to prove that the set $S_{pc}$ as defined above is closed under the addition operation (and it easily follows that the set $S_{pc}$ is closed under the multiplication operation as well).
There by I am able to show that the set $S_{pc}$ is indeed a vector space. (as it cab be easily seen that the set $S_{pc}$ is closed under scalar multiplication).
The Question
Where I need some help is to show that the set $S_{pc}$ is closed under the binary operation of circular convolution. Here by circular convolution, i mean the convolution operation with the convolution integral summed only over one period i.e., on $[0,1]$.
Specifically How should i make the choice of $D_3,k_3,ck_3$, the required set and maps for the resultant function of a convolution, to show that it belongs to $S_{pc}$.
What I know
Let $f_1$ is periodic with period $1$ and is smooth (within one period) except at $x = a_1 \in [0,1]$, where it is only $n_1$ times differentiable and Let $f_2$ is periodic with period $1$ and is smooth (within one period) except at $x = a_2 \in [0,1]$, where it is only $n_2$ times differentiable.
Now the function $f_3 = f_1 \star f_2$ is smooth (within one period) except at $x = a_1 - a_2$ where it is only $n_1 + n_2$ times differentiable.
But I am confused as to how to use this fact to arrive at the desired result.