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 ,1), the first ''pointwise dyadic derivative'' of $f$ at a point $x$ is defined as:
:$f^(x) = \lim_ \sum_^ 2^[f(x) - f(x \oplus 2^)$/math>

if this limit exists. Here, $\oplus$ denotes the dyadic addition operation, which is defined using the dyadic (binary) representation of numbers. That is, if
:$x = \sum_^ x_j 2^$ and $y = \sum_^ y_j 2^$ with $x_j, y_j \in \$,
then
:$x \oplus y = \sum_^ (x_j \oplus y_j) 2^$,
where
:$x_j \oplus y_j = (x_j + y_j) \pmod 2$.

Higher-order dyadic derivatives are defined recursion (mathematics)">recursively: $f^(x) = (f^)^(x)$ 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 $1 \le p \le \infty$ (Lp_space, ''L''^''p'' space); $L^\infty(0, 1)$ (L-infinity, ''L''^''∞'' space); or $C^\oplus$, 1/math> (the space of dyadically continuous functions). If $f \in X(0, 1)$ and there exists $g \in X(0, 1)$ such that
:\lim_ \left\, \sum_^ 2^ (\cdot) - f(\cdot \oplus 2^)- g(\cdot) \right\, _X = 0,
then $g$ is called the first strong dyadic derivative of $f$, denoted by $g = D^f$.Engels, W. (1985). "On the characterization of the dyadic derivative". ''Acta Mathematica Hungarica'', 46(1-2): 47-56. 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:
:$(\alpha f + \beta g)^ = \alpha f^ + \beta g^$.Butzer, P. L., and Wagner, H. J. (1973). "Walsh-Fourier series and the concept of a derivative". ''Applicable Analysis'', 3(1): 29-46.

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.Onneweer, C.W. (1979). "On the definition of dyadic differentiation". ''Applicable Analysis'', 9(4): 267-278.

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 $D^f \in L_1(G)$ exists, the Walsh-Fourier transform satisfies:
: ^f(\chi) = , \chi, \hat(\chi)
for all characters $\chi$, where $, \chi,$ represents the norm of the character.Onneweer, C. W. (1977). "Fractional differentiation on the group of integers of a p-adic or p-series field". ''Analysis Mathematica'', 3(2): 119-130.

Eigenfunctions

The Walsh functions $\psi_k$ are eigenfunctions of the dyadic differentiation operator with corresponding eigenvalues related to their index:
:$D^\psi_k = \psi_k^ = k\psi_k$
and
:$D^\psi_k = \psi_k^ = k^r \psi_k$.
This eigenfunction property makes Walsh functions naturally suited for analysis involving dyadic derivatives, similar to how complex exponentials $e^$ 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 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 discontinuity (mathematics), 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

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.Gibbs, J. E., and Ireland, B. (1974). "Walsh functions and differentiation". ''Proceedings of the Symposium and Workshop on Applications of Walsh-Functions''. Naval Research Laboratory, Washington, D.C., pp. 1-29.Butzer, P. L., and Wagner, H. J. (1973). "Walsh-Fourier series and the concept of a derivative". ''Applicable Analysis'', 3(1): 29-46.

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

References

{{reflist

Mathematical analysis
Differential calculus
Harmonic analysis
Functional analysis