Dyadic derivative

In mathematical analysis, the dyadic derivative is a concept that extends the notion of classical differentiation to functions defined on the dyadic group or the dyadic field. Unlike classical differentiation, which is based on the limit of difference quotients, dyadic differentiation is defined using dyadic (binary) addition and reflects the discontinuous nature of Walsh functions.

Definition

Pointwise dyadic derivative

For a function $f$ defined on [0,1), the first pointwise dyadic derivative of $f$ at a point $x$ is defined as:

$f^{[1]}(x)=\lim _{m\to \infty }\sum _{j=0}^{m-1}2^{j-1}[f(x)-f(x\oplus 2^{-j-1})]$

if this limit exists. Here, $\oplus$ denotes the dyadic addition operation, which is defined using the dyadic (binary) representation of numbers. That is, if

$x=\sum _{j=0}^{\infty }x_{j}2^{-j-1}$ and $y=\sum _{j=0}^{\infty }y_{j}2^{-j-1}$ with $x_{j},y_{j}\in \{0,1\}$,

then

$x\oplus y=\sum _{j=0}^{\infty }(x_{j}\oplus y_{j})2^{-j-1}$,

where

$x_{j}\oplus y_{j}=(x_{j}+y_{j}){\pmod {2}}$.

Higher-order dyadic derivatives are defined recursively: $f^{[r]}(x)=(f^{[r-1]})^{[1]}(x)$ for $r\in \mathbb {N}$.

Strong dyadic derivative

The strong dyadic derivative is defined in the context of function spaces. Let $X(0,1)$ denote one of the function spaces $L^{p}(0,1)$ for $1\leq p\leq \infty$ (L^p space); $L^{\infty }(0,1)$ (L^∞ space); or $C^{\oplus }[0,1]$ (the space of dyadically continuous functions). If $f\in X(0,1)$ and there exists $g\in X(0,1)$ such that

$\lim _{m\to \infty }\left\|\sum _{j=0}^{m-1}2^{j-1}[f(\cdot )-f(\cdot \oplus 2^{-j-1})]-g(\cdot )\right\|_{X}=0$,

then $g$ is called the first strong dyadic derivative of $f$, denoted by $g=D^{[1]}f$. Higher-order derivatives can be defined recursively similar to pointwise dyadic derivatives.

Properties

Similar to the classic derivative in calculus, the dyadic derivative possesses several properties.

Linearity

The dyadic derivative is a linear operator. If functions $f$ and $g$ are dyadically differentiable and $\alpha ,\beta$ are constants, then $\alpha f+\beta g$ is dyadically differentiable:

$(\alpha f+\beta g)^{[1]}=\alpha f^{[1]}+\beta g^{[1]}$.

Closure

The dyadic differentiation operator is closed; that is, if $f$ is in the domain of the operator, then its dyadic derivative also belongs to the same function space.

Inverse operator

There exists a dyadic integration operator that serves as an inverse to the dyadic differentiation operator, analogous to the fundamental theorem of calculus.

Relationship to the Walsh-Fourier transform

For functions $f$ where $D^{[1]}f\in L_{1}(G)$ exists, the Walsh-Fourier transform satisfies:

$[D^{[1]}f]^{\wedge }(\chi )=|\chi |{\hat {f}}(\chi )$

for all characters $\chi$, where $|\chi |$ represents the norm of the character.

Eigenfunctions

The Walsh functions $\psi _{k}$ are eigenfunctions of the dyadic differentiation operator with corresponding eigenvalues related to their index:

$D^{[1]}\psi _{k}=\psi _{k}^{[1]}=k\psi _{k}$

and

$D^{[r]}\psi _{k}=\psi _{k}^{[r]}=k^{r}\psi _{k}$.

This eigenfunction property makes Walsh functions naturally suited for analysis involving dyadic derivatives, similar to how complex exponentials $e^{ikx}$ are eigenfunctions of classical differentiation.

Characterization of differentiable functions

Thanks to a generalization of a result of Butzer and Wagner,

Theorem (Skvorcov—Wade). Let $f$ be continuous on $[0,1)$, and let $f^{[1]}$ exist for all but countably many points $x\in (0,1)$. Then $f$ is constant.

This result implies that it is more interesting to consider functions that are not continuous over the entire interval. A generalization of the above result shows that:

Theorem. A bounded function defined on $[0,1)$ with a countable set of discontinuities (exclusively of jump discontinuities) that have at most a finite number of cluster points is pointwise dyadically differentiable except on a countable set if and only if it is a piecewise constant function.

Examples

Function type	Example	Dyadic derivative	Notes
Constant functions	$f(x)=C$ for all $x\in [0,1)$	$f^{[1]}(x)=0$	Similar to classical calculus
Step functions	$g(x)={\begin{cases}3,&x\in [0,1/4)\\-1,&x\in [1/4,1)\end{cases}}$	$g^{[1]}(x)={\begin{cases}6,&x\in [0,1/4)\\-4,&x\in [1/4,1/2)\\-2,&x\in [1/2,3/4)\\0,&x\in [3/4,1)\end{cases}}$	Creates additional discontinuities
Dirichlet function	$d(x)={\begin{cases}1,&x\in [0,1)\cap \mathbb {Q} \\0,&x\in [0,1)\setminus \mathbb {Q} \end{cases}}$	$d^{[1]}(x)=0$ for all $x\in [0,1)$	Differentiable despite dense discontinuities
Walsh functions	$\psi _{k}(x)$	$\psi _{k}^{[1]}(x)=k\psi _{k}(x)$	Eigenfunctions of the dyadic derivative
Linear functions	$f(x)=x$	Not dyadic differentiable	Contrasts with classical calculus

History

The dyadic derivative was introduced by mathematician James Edmund Gibbs in the context of Walsh functions and further developed by Paul Butzer and Heinz-Joseph Wagner.

Further contributions came from C. W. Onneweer, who extended the concept to fractional differentiation and p-adic fields. In 1979, Onneweer provided alternative definitions to the dyadic derivatives.

See also

• Walsh function
• Haar wavelet
• Harmonic analysis
• Walsh transform