Decomposition of multivariate function using the Heaviside step function
© Chikayama; licensee Springer. 2014
Received: 24 September 2014
Accepted: 13 November 2014
Published: 29 November 2014
Whereas the Dirac delta function introduced by P. A. M. Dirac in 1930 to develop his theory of quantum mechanics has been well studied, a not famous formula related to the delta function using the Heaviside step function in a single-variable form, also given by Dirac, has been poorly studied. Following Dirac’s method, we demonstrate the decomposition of a multivariate function into a sum of integrals in which each integrand is composed of a derivative of the function and a direct product of Heaviside step functions. It is an extension of Dirac’s single-variable form to that for multiple variables.
The importance of this property is analogous to the Fourier transform (Bracewell1965) for its ability to yield an alternative representation of any multivariate function in which the variables of the function are changed. The delta function can be seen in applications from physics to engineering: such as quantum mechanical states (Lee1992); quantum similarity integrals (Safouhi and Berlu2006); pseudopotential (Derevianko2003); a spin system with a classical environment (Calvani et al.2013); and generally, numbers of formulae in the Fourier and Laplace transforms, and differential equations (Schwartz1966; Kreyszig2011). A more rigorous mathematical theory for the delta function has also been developed and expanded under the branch in pure mathematics called the theory of distributions by L. Schwartz (Schwartz1945). Further development is the generalized delta impulse (Corinthios2003), which is an extension of the Dirac delta function to that on the complex plane and is applied to theories of generalized Laplace, z, Hilbert, and Fourier-related transforms (Corinthios2005,2007).
is essentially described in Dirac’s quantum mechanics text (Dirac1958). It is derived from the relation between the Dirac delta function and the derivative of the Heaviside step function. As these two expressions, the left- and right-hand sides of (1.2), are mathematically equivalent, the step-function expression would be expected to find potent applications in physics and engineering. One general example must be approximation theory (Milovanovic and Rassias2014). The other example is replacement of the Dirac delta functions with the Heaviside step functions by which (divergent) delta functions can be hidden from integrand. It helps a rigorous formalism by using only bounded functions without advanced Schwartz distributions. Nevertheless, the step-function expression has not been extended so far compared with the application of the delta-function expression. Here we demonstrate a unified formula that extends this step-function expression for single-variable functions to multiple-variable functions. It can be interpreted as the decomposition of any multivariate function with respect to the Heaviside step function.
2. Decomposition of multivariate functions using the Heaviside step function
Let R(X1, X2, …, X N ) be a continuous real function defined for 0 ≤ X i < ∞ and satisfies:
Whose derivatives exist and continuous where α = α1 + ⋯ + α N and α i ≥ 0 is a natural number.
can be integrated with respect to a given X i while the other variables are held fixed.
Sequences of functions uniformly converge to for X j and j ≠ i.
We follow Dirac’s method:
The Dirac delta function is regarded as a function (not a Schwartz distribution)
The derivative of the Heaviside step function is regarded as the Dirac delta function
defines a set of Heaviside step functions for each μ i ≥ 0;; α = α1 + ⋯ + α N ; and α i = 0 or 1.
We use mathematical induction.
with X i ≥0.
holds by Lemma 2.1 given below. Note that, for single-variable functions, Dirac described (Dirac1958) the essentially equivalent expression (1.2).
where Lemma 2.2 was also used.
Thus, assuming expression (2.6) for N leads to the same expression for N +1. The formula for N =1 also holds as described above. Therefore, (2.1) holds for any natural number N.
where F is R or its derivatives.
where a = a1 + … + a N and a i ≥ 0 is a natural number.
We have demonstrated the decomposition of a multivariate function as a sum of integrals of which each integrand is composed of a derivative and a direct product of Heaviside step functions. The expression offers a rigorous formalism by using only bounded functions without the Dirac delta and Schwartz distributions; applications in approximation theory of functions using the Heaviside step or sigmoid functions with suitable parameters and dimensionality; and potent applications to mathematical methods in physics and engineering.
I thank K. Soda for discussion.
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