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\author{Jonathan Leto}
\title{An Introduction to Fractional Splines}
\begin{document}
\bibliographystyle{siam}
\maketitle
\section{Preliminaries}
\subsection{Gamma Function and Generalized Binomial}
The gamma function is defined as
\begin{equation}
\Gamma(u) = \int_{0}^{\infty} x^{n-1} e^{-x} d x
\end{equation}
Since it is related to the factioral by the identity $ \Gamma(n+1) = n ! $ we can generalize the binomial
coefficients:
\begin{equation} \left( \begin{array}{c}u \\ v\end{array} \right) = \frac{ \Gamma( u + 1) } { \Gamma(v+1) \Gamma(u - v + 1) }
\end{equation}
\subsection{Function Spaces}
Here we will outline common function spaces and their properties, a good reference is \cite{Yosida}. A normed vector space $\mathscr{E}$ is called complete if every Cauchy sequence
in $\mathscr{E}$ converges to an element of $\mathscr{E}$. These are commonly referred to as Banach spaces after Stefan Banach (1892-1945). Similarly, a complete
inner product space is called a Hilbert space after David Hilbert (1862-1943). All Hilbert spaces are therefore Banach spaces, because an
inner product induces a norm, but the converse is not true. It turns out that a Banach space $\mathscr{B}$ is a Hilbert space if and only if the parrallelogram
identity holds
\begin{equation}
||u+v||^2 + ||u-v||^2 = 2( ||u||^2 + ||v||^2 )
\end{equation}
for all $u,v \in B $. For example, since $\mathbb{C}^n$ is complete it is therefore a Hilbert space.
Some well-known examples of complete Hilbert spaces are $\mathbb{R}^n$,
$\mathbb{C}^n$ and $\mathscr{L}^2$, the space of Lebesque integrable functions. But, you
may ask, need we worry about incomplete vector spaces? Yes. It turns out that
the space of continously-differentiable functions $\mathscr{C}^1$ is not
complete on any interval! An example of this is the sequence of functions
\begin{equation}
f_n(x) = 1 + x + \frac{1}{2!}x^2 + \cdots \frac{1}{n!} x^n
\end{equation}
which are polynomials of degree $n$ and have continuous derivatives of all orders. But the limit of this sequence
\begin{equation}
\lim_{n \rightarrow \infty} f_n(x) = e^{x}
\end{equation}
is not a polynomial and hence in a different space than the original sequence.
There are also many examples of a sequence of continuous functions which develop a discontinuity
as $n \rightarrow \infty$. You may be surprised to learn that even the space of continuous functions on compact support is incomplete,
and hence is a Banach space. Because most of the theory of differential equations requires continuity the main objects of study
in functional analysis are Banach spaces.
We may now define the Sobolev space $H^m(\mathbb{R})=W^m_2(\mathbb{R})$, which if $m$ is a positive integer corresponds to all functions $u$ such
that $D^m u \in \mathscr{L}^2$, i.e. $u \in H^m$ means that $u$ has up to $m$ derivatives which are Lebesque integrable. There
is a famous theorem called the Sobolev embedding theorem which basically says that Sobolev spaces $H^n$ with $n \in \mathbb{N}$
are proper subspaces of the continuous functions. But unlike $\mathscr{C}^n$ Sobolev spaces are complete and therefore are Hilbert spaces,
which from the unique point of view of an analyst, makes them easier to work with.
%\subsection{Distributions}
%Not all distributions have fractional derivatives, such as $e^{-x}$. Why?
\subsection{Fractional Derivatives}
The notion of a fractional derivative is almost as old as the derivative itself, evidenced by the fact that in 1695 L'Hopital asked Leibniz what $D^{1/2}$could mean. He replied that "It will lead to a paradox, from which one day useful consequences will be drawn."
Liouvilles's generalization to fractional derivative operators \cite{Liouville}, \cite{OldSpan} is given by
\begin{equation}
D ^\alpha f = f \star g = \int_{a}^{x} \frac{ (x-t)^{-\alpha-1}_+ } { \Gamma(-\alpha) } f(t) dt
\end{equation}
where
\begin{equation}
g_\alpha(x) = \frac{ x^{\alpha-1}_+ } { \Gamma(\alpha) }
\end{equation}, and $^*$ denotes the convolution.
and
where
\begin{equation}
x^\alpha_+ = \left\{ \begin{array}{cc} x^\alpha & x \geq 0\\0 & x < 0 \end{array} \right.
\end{equation}
is the one-sided power function. These one-sided power functions are the building blocks of
fractional splines.
\section{Constructing Fractional Splines}
Suppose one has $N+1$ data points $x_0 , x_1, \cdots, x_N$, then we can write the
fractional spline $s^\alpha ( x) $ as \cite{UnserBlu1}
\begin{equation}
s^\alpha(x) = \sum_{n=0}^{N} a_n ( x - x_n )^\alpha_+
\end{equation}
For convenience we shall now only consider the case when our data points are the integers, i.e $x_n=n$.
Obviously this poses constraints which may not practical in certain situations, but it allows us
to use the fractional forward finite difference operator, defined as
\begin{equation}
\fffd{\alpha} f(x) = \sum_{k \geq 0 } (-1)^k \binom{ \alpha}{k} f(x -k)
\end{equation}
\section{Fractional B-Splines}
We now define the causal fractional B-splines \cite{UnserBlu1} by taking the $\alpha+1$-th fractional difference of the one-sided power function
\begin{subequations}
\begin{eqnarray}
\beta_{+}^{\alpha}(x) &=& \frac{1}{\Gamma(\alpha+1)} \fffd{\alpha+1} \ospf{x}{\alpha} \\
&=& \frac{1}{\Gamma(\alpha+1)} \sum_{k\geq 0} (-1)^k \binom{\alpha+1}{k} \ospf{(x-k)}{\alpha}
\end{eqnarray}
\end{subequations}
The anti-causal fractional B-splines are definied as
\begin{equation}
\beta_{-}^{\alpha}(x) = \beta_{+}^{\alpha}(-x)
\end{equation}
%In this paper we shall only need to differentiate
%polynomials to create splines, so we may use the natural generalization of $D^n x^p $ where factorials are
%replaced with Gamma functions:
%\begin{equation}
%D^\alpha x^p = \frac{\Gamma(p+1)}{\Gamma(p-\alpha+1)} x^{p-\alpha}
%\end{equation}
%as is related to the factioral by the identity $ \Gamma(n+1) = n ! $. We therefore can generalize the binomial
%coefficients:
%\begin{equation} \left( \begin{array}{c}u \\ v\end{array} \right) = \frac{ \Gamma( u + 1) } { \Gamma(v+1) \Gamma(u - v + 1) }
%\end{equation}
We will show that $\ospf{\beta}{\alpha}(x) \in \mathscr{L}^1$ when $\alpha > -1$ and $\ospf{\beta}{\alpha}(x) \in \mathscr{L}^2$ when $\alpha > -\frac{1}{2}$. We will
also show that fractional B-splines decay proportional to $|x|^{-\alpha-2}$.
To symmetrize the B-splines and have the ability to compute inner products, one may define the symmetric B-splines of degree $\alpha$:
\begin{equation}
\beta_{\star}^{\alpha} = \ospf{\beta}{\frac{\alpha-1}{2}} \star \nospf{\beta}{ \frac{\alpha-1}{2} }
\end{equation}
In the Fourier domain the symmetric B-splines have the convenient form:
\begin{equation}
\hat{ \beta}^{\alpha}_*(w) = \left|\frac{ \sin{w/2} }{w/2}\right|^{\alpha+1}
\end{equation}
Using this we find that the explicit time domain formula for the symmetric B-splines is ( for $\alpha \neq 2n$ )
\begin{equation}
\beta^{\alpha}_*(x) = \frac{1}{ 2 \sin{ \pi \alpha / 2 }} \Gamma( \alpha + 1) \sum{ (-1)^{k+1} \left| \begin{array}{c}\alpha+1\\k\end{array}\right| \left|x-k\right|^\alpha }
\end{equation}
where $\left| \begin{array}{c}\alpha\\k\end{array}\right| = \binom{\alpha}{k+\frac{\alpha}{2} }$ are the modified binomial coefficients, which can only
vanish if $\alpha$ is even.
\section{Properties of Fractional Splines}
In the Fourier domain we have
\begin{equation}
\hat{ \beta}_+^{\alpha} (w) = \left( \frac{ 1 - e^{-i w}}{i w} \right)^{\alpha+1}
\end{equation}
Using this one can establish that
\begin{equation}
D^\gamma \beta_+^{\alpha} = \fffd{\gamma} \beta_+^{\alpha-\gamma}
\end{equation}
which is a generalization of the property of polynomial splines. This property is important because it states that one can differentiate by
taking finite differences. Usually there is error in approximating a derivative with finite differences but it is exact with B-splines, which
makes them very nice for numerical work.
\subsection{Reproduction of Polynomials}
One of the most interesting properties of fractional splines is that they reproduce the polynomials of degree $\lceil \alpha \rceil $, i.e. the
non-integer part of $\alpha$ gives us one additional degree of approximation. We say that $\phi(x)$ reproduces the polynomials of
degree $n$ when there exist sequences $c_m(k)$ such that
\begin{equation}
x^m = \sum_{k\in \mathbb{Z}} c_m(k) \phi(x-k)
\end{equation}
for $m=0,1,\cdots ,n$. This means that any polynomial of degree $n$ is expressible as a linear combination of integer shifts of $\phi(x)$.
Another more useful form of this condition is
\begin{equation}
\sum_{k\in \mathbb{Z}} (x-k)^m \phi(x-k) = C_m
\end{equation}
for $m=0,1,\cdots,n$. If $C_0=1$ then $\phi(x)$ is said to satisfy the partition of unity. In the Fourier domain this relation
is call the Strang-Fix condition (of order $n+1$) and it looks like
\begin{subequations}
\begin{eqnarray}\label{eq:strangfix}
\hat{ \phi}(0) &=& 1 \\
\hat{ \phi}^{(m)}(2 k \pi) &=& 0
\end{eqnarray}
\end{subequations}
for $m=0,1,\cdots, n$ and $\forall k \in \mathbb{Z} - \{0\} $.
It is easy to verify that $\beta^0(0) = 1$ with one applicatin of L'Hopitals rule. It can also be shown that $\beta^\alpha(0)=1$ for all $\alpha > -1 $ .
To see that fractional B-splines satisfy the second property, take the first term in a taylor series expansion for $ \omega \rightarrow 2 k \pi $ we find
\begin{equation}
\hat{ \phi} (\omega) = K ( \omega - 2 k \pi ) ^{\alpha +1 - m }
\end{equation}
This means that $\hat{\beta}^{\alpha}$ will satisfy \eqref{eq:strangfix} when $\alpha + 1 -m > 0$ or
\begin{equation}
\forall m \leq \lceil \alpha \rceil
\end{equation}
This leads to the strange fact that one can use shifts of $\sqrt{x}$ to approximate linear polynomials!
\subsection{ Decay Properties }
One can show that $ \beta^{\alpha} \in \mathbb{W}_2^r$ for all
$ r < \alpha + \frac{1}{2} $. This comes from the fact that as $ x \rightarrow \infty$ we have the asysmptotic form
\begin{equation}
\beta_+^\alpha(x) = \frac{ \Gamma(\alpha+2) \sin{\pi \alpha} }{ \pi x^{\alpha+2 }} \sum_{n=1}^{\infty} \frac{ e^{ 2 n i \pi x} }{ (2 n i \pi )^(\alpha+1) } + o(x^{-\alpha-2})
\end{equation}
From this we can surmise that $\beta_+^\alpha \in \mathscr{L}^1$ for $\alpha > -1$ and $\beta_+^\alpha \in \mathscr{L}^2$ for $\alpha > -1/2 $.
\subsection{Spline Spaces at scale $a$}
The fractional splines also have a fractional order of approximation, which require the idea of a spline space at a scale $a$.
The Spline space of scale $a$ is defined as
\begin{equation}
\mathbf{S}_a^{\alpha} = \left\{ s_a : \exists c \in \ell^2, s_a(x) = \sum_{ k \in \mathbb{Z} } c(k) \beta^\alpha(\frac{x}{a}-k) \right\}
\end{equation}
These can be thought of as stretching the basis functions by a factor of $a$ and respacing them. This scale can also be thought of as a sampling rate.
Now we can state more precisely the order of approximation of the fractional spline. If $f \in W^{\alpha+1}_2$ then
\begin{equation}\label{eq:fracorder}
|| f - \mathscr{P}_a|| \sim a^{\alpha+1}
\end{equation}
as $a \rightarrow 0$, where $ \mathscr{P}_a$ is the a projection operator defined in terms of elements from $\mathbf{S}_a^{\alpha}$ and it's dual space. For all the gory details, refer to \cite{UnserBlu1}. Note that \eqref{eq:fracorder} is of fractional order for general $\alpha$, which is not encountered with polynomial splines of any kind.
\subsection{Variational Properties}
Given the uniform samples ${f(k T)}$ of a smooth function $f \in \mathscr{L}^2$, find the interpolation function whose samples coincide with those of $f$ and whose derivative of order $\alpha$ has minimal $\mathscr{L}^2$ norm. The surprising result is that the solution $g$ is unique and lives in the vector space $\mathbf{S}_T^{2 \alpha-1} = span\{ \beta^{2 \alpha -1}_*(\frac{x}{T} -k ) \} $.
\section{Applications of Fractional Splines}
Why does one need fractional splines? When are these the right tool? These are not unimportant questions, but they are often anwered in
passing or not at all in the mathematical literature. The utility of fractional splines is that by varying $\alpha$ one controls the properties
of a family of functions, such as the order of approximation, decay rate, Hoelder continutiy, Sobolev regularity, etc. These can then be combined
in various ways to form bases for interpolation or wavelet bases which have all the required properties, such as satisfying a two-scale relation.
Fractional splines have been studied for their use in representing data in computer tomography applications. Fractional
differential equations arise in many physics situations (\cite{FractalOps}, \cite{Hilfer}), and fractional splines may
be an efficient way to represent the intermediate data in the algorithms needed to solve them.
Fractional splines will also need to be generalized to non-uniform grids, if they
are to be useful in adaptive mesh algorithms.
\section{conclusion}
The essential beginning of fractional anything is taking the factorial, a
discrete function, and asking: What happens when we make it continuous? Euler
answered this long ago with the Gamma function $\Gamma(x)$, which was then in
turn used to make the binomial coefficients $\binom{n}{k}$ continuous, which
leads to a watershed of new mathematical tools and results. Just as the fractional
derivative operators interpolate the integer derivatives, fractional splines
interpolate the polynomial splines and they all owe this ability to the interpolation
of the Gamma function. No doubt that there is still much to research in terms of
practical software to implement these ideas as well as how these new tools can be used
in sampling theory, signal/image processing \cite{Unser} , seismological instruments \cite{FJ} and many
other fields.
\bibliography{splines.bib}
\end{document}