Chikayama SpringerPlus 2014, 3:704 http://www.springerplus.com/content/3/1/704
a SpringerOpen Journal
RESEARCH
Open Access
Decomposition of multivariate function using the Heaviside step function Eisuke Chikayama1,2,3 Correspondence:
[email protected] 1 Department of Information Systems, Niigata University of International and Information Studies, 3-1-1 Mizukino, Nishi-ku, Niigata-shi, Niigata 950-2292, Japan 2 Environmental Metabolic Analysis Research Team, RIKEN, 1-7-22 Suehiro-cho, Tsurumi-ku, Yokohama-shi, Kanagawa 230-0045, Japan 3 Image Processing Research Team, RIKEN, 2-1 Hirosawa, Wako-shi, Saitama 351-0198, Japan
Abstract 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. Keywords: Heaviside step function; Dirac delta function; Transform
1. Introduction P. A. M. Dirac introduced in 1930 a function, now called the Dirac delta function, to develop his theory of quantum mechanics (Dirac 1958). It takes value zero at x ≠ 0 and its integral is unity. A fundamental property derivable from the definition of the Dirac delta function is that any multivariate real function can be expressed with delta functions δ and integrals as follows, Z∞ Z∞ ⋅⋅⋅ Rðμ1 ; ⋅⋅⋅; μN Þ δ ðμ1 −X 1 Þ⋅⋅⋅δ ðμN −X N Þdμ1 ⋅⋅⋅dμN : ð1:1Þ RðX 1 ; X 2 ; ⋅⋅⋅; X N Þ ¼ −∞
−∞
The importance of this property is analogous to the Fourier transform (Bracewell 1965) 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 (Lee 1992); quantum similarity integrals (Safouhi and Berlu 2006); pseudopotential (Derevianko 2003); 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 (Schwartz 1966; Kreyszig 2011). 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 (Schwartz 1945). Further development is the generalized delta impulse (Corinthios 2003), 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 (Corinthios 2005, 2007). © 2014 Chikayama; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
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Transforming the integral expression in one variable using the Dirac delta function δ into one using the Heaviside step function σ,
Z∞
Z∞ RðxÞδ ðxÞdx ¼ Rð0Þ ¼ Rð∞Þ−
−∞
−∞
Z∞ dRðxÞ dRðxÞ σ ð−xÞdx σ ðxÞdx ¼ Rð−∞Þ þ dx dx −∞
ð1:2Þ is essentially described in Dirac’s quantum mechanics text (Dirac 1958). 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 Rassias 2014). 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 singlevariable 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 2.1. Definition
Let R(X1, X2, ⋅ ⋅⋅, XN) be a continuous real function defined for 0 ≤ Xi < ∞ and satisfies: Whose derivatives
∂α R α α ∂X 1 1 ⋯∂X NN
exist and continuous where α = α1 + ⋯ + αN and αi ≥ 0
is a natural number. ∂α R can be integrated with respect to a given Xi while the other variables are α α ∂X 1 1 ⋯∂X NN held fixed. α−1 ∂α−1 R Sequences of functions h1 ð⋯; X i þ h; ⋯Þ− ∂ αi −1R ð⋯; X i ; ⋯Þ αi −1
uniformly converge to
⋯∂X i ⋯ ∂α R for Xj α α ∂X 1 1 ⋯∂X NN
⋯∂X i
⋯
and j ≠ i.
h→0
2.2. Premise
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 2.3. Theory
We demonstrate that the defined function R(X1, X2, ⋯, XN) can be decomposed into: ∞ X Z
RðX 1 ; X 2 ; ⋯; X N Þ ¼ →
α ∈f0;1gN
0
Z∞ α N ∂ Rðα1 μ1 ; ⋯; αN μN Þ Y ⋯ fσ ðX i −μi Þdμi gαi ; αN α1 ∂μ1 ⋯∂μN i¼1 0
ð2:1Þ
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8 1 > > > > < 1 σ ðX i −μi Þ ¼ >2 > > > : 0
where
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ðX i > μi Þ ðX i ¼ μi Þ ðX i < μi Þ →
defines a set of Heaviside step functions for each μi ≥ 0; α ¼ ðα1 ; ⋯; αN Þ; α = α1 + ⋯ + αN; and αi = 0 or 1. 2.4. Proof
The formula (2.1) may be expressed as:
∞ X Z
RðX 1 ; X 2 ; ⋯; X N Þ ¼ →
α ∈f0;1gN 0
Z∞ α N ∂ Rðα1 μ1 ; ⋯; αN μN Þ Y ⋯ fσ ðX s −μs Þdμs gαs αN α1 ∂μ1 ⋯∂μN s¼1
ð2:2Þ
0
Z∞ ∂Rðμ1 ; 0; ⋯; 0Þ ¼ Rð0; 0; ⋯; 0Þ þ σ ðX 1 −μ1 Þdμ1 ∂μ1 0
Z∞ ∂Rð0; μ2 ; 0; ⋯; 0Þ þ σ ðX 2 −μ2 Þdμ2 ∂μ2 0
Z∞ ∂Rð0; ⋯; 0; μi ; 0; ⋯; 0Þ þ⋯ þ σ ðX i −μi Þdμi ∂μi 0
Z∞ ∂Rð0; ⋯; 0; μN Þ þ⋯ þ σ ðX N −μN ÞdμN ∂μN 0
Z∞Z∞ 2 ∂ Rðμ1 ; μ2 ; 0; ⋯; 0Þ þ σ ðX 1 −μ1 Þσ ðX 2 −μ2 Þdμ1 dμ2 ∂μ1 ∂μ2 0 0 Z∞Z∞ ∂2 R 0; ⋯; 0; μ ; 0; ⋯; 0; μ ; 0; ⋯; 0 j k þ⋯ þ σ X j −μj σ X k −μk dμj dμk ∂μj ∂μk 0 0
Z∞Z∞ 2 ∂ Rð0; ⋯; 0; μN−1 ; μN Þ þ⋯ þ σ ðX N−1 −μN−1 Þσ ðX N −μN ÞdμN−1 dμN ∂μN−1 ∂μN Z∞Z∞Z∞
0 0
∂3 Rðμ1 ; μ2 ; μ3 ; 0; ⋯; 0Þ σ ðX 1 −μ1 Þσ ðX 2 −μ2 Þσ ðX 3 −μ3 Þdμ1 dμ2 dμ3 ∂μ1 ∂μ2 ∂μ3
þ
0 0 0 Z∞Z∞Z∞
þ⋯ þ 0 0 0
∂3 Rð0; ⋯; 0; μl ; 0; ⋯; 0; μm ; 0; ⋯; 0; μn ; 0; ⋯; 0Þ σ ðX l −μl Þ ∂μl ∂μm ∂μn
σ ðX m −μm ÞσðX n −μn Þdμl dμm dμn Z∞Z∞Z∞Z∞ ∂4 R 0; ⋯; 0; μ ; 0; ⋯; 0; μ ; 0; ⋯; 0; μ ; 0; ⋯; 0; μ ; 0; ⋯; 0 o p q r þ⋯ þ ∂μo ∂μp ∂μq ∂μr 0 0 0 0
σ ðX o −μo Þσ X p −μp σðX q −μq Þσ ðX r −μr Þdμo dμp dμq dμr
Z∞ þ⋯ þ 0
Z∞ N ∂ Rðμ1 ; ⋯; μN Þ ⋯ σ ðX 1 −μ1 Þ⋯σ ðX N −μN Þdμ1 ⋯dμN : ∂μ1 ⋯∂μN 0
We use mathematical induction.
Chikayama SpringerPlus 2014, 3:704 http://www.springerplus.com/content/3/1/704
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Using the definition of the Dirac delta-function δ, then for any real function R(X1, X2, ⋯, XN)
Z∞ RðX 1 ; X 2 ; ⋯; X N Þ ¼
Z∞ ⋯ Rðμ1 ; ⋯; μN Þδ ðμ1 −X 1 Þ⋯δ ðμN −X N Þdμ1 ⋯dμN :
−∞
ð2:3Þ
−∞
Therefore,
Z∞ Z∞ RðX 1 ; X 2 ; ⋯; X N Þ ¼ ⋯ Rðμ1 ; ⋯; μN Þδ ðμ1 −X 1 Þ⋯δ ðμN −X N Þdμ1 ⋯dμN 0
ð2:4Þ
0
with Xi ≥0. The expression (2.2) for single-variable functions (N =1),
Z∞ Z∞ Rðμ1 Þδ ðμ1 −X 1 Þdμ1 ¼ Rð0Þ þ RðX 1 Þ ¼ 0
0
d Rðμ1 Þσ ðX 1 −μ1 Þdμ1 dμ1
ð2:5Þ
holds by Lemma 2.1 given below. Note that, for single-variable functions, Dirac described (Dirac 1958) the essentially equivalent expression (1.2). To initiate the mathematical induction procedure, suppose that the expression (2.2),
Z∞ Z∞ ⋯ Rðμ1 ; ⋯; μN Þδ ðμ1 −X 1 Þ⋯δ ðμN −X N Þdμ1 ⋯dμN 0
0
Z∞ ∂Rðμ1 ; 0; ⋯; 0Þ ¼ Rð0; 0; ⋯; 0Þ þ σ ðX 1 −μ1 Þdμ1 ∂μ1 0
Z∞ ∂Rð0; ⋯; 0; μi ; 0; ⋯; 0Þ þ⋯ þ σ ðX i −μi Þdμi ∂μi 0
Z∞ Z∞ Z∞ 2 ∂Rð0; ⋯; 0; μN Þ ∂ Rðμ1 ; μ2 ; 0; ⋯; 0Þ þ⋯ þ σ ðX N −μN ÞdμN þ ∂μN ∂μ1 ∂μ2 0
0
0
σðX 1 −μ1 Þσ ðX 2 −μ2 Þdμ1 dμ2 Z∞ Z∞ ∂2 R 0; ⋯; 0; μ ; 0; ⋯; 0; μ ; 0; ⋯; 0 j k þ⋯ þ σ X j −μj σ X k −μk dμj dμk ∂μj ∂μk 0
0
0
0
Z∞ Z∞ Z∞ 3 ∂ Rð0; ⋯; 0; μl ; 0; ⋯; 0; μm ; 0; ⋯; 0; μn ; 0; ⋯; 0Þ þ⋯ þ σ ðX l −μl Þ ∂μl ∂μm ∂μn 0
Z∞ Z∞ N ∂ Rðμ1 ; ⋯; μN Þ σ ðX m −μm ÞσðX n −μn Þdμl dμm dμn þ ⋯ þ ⋯ ∂μ1 ⋯∂μN 0
0
σ ðX 1 −μ1 Þ⋯σ ðX N −μN Þdμ1 ⋯dμN ;
ð2:6Þ
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holds for some N. Using (2.6), the following,
Z∞ Z∞ R X 1 ; ⋯; X N ; μNþ1 ¼ ⋯ R μ1 ; ⋯; μN ; μNþ1 δ ðμ1 −X 1 Þ⋯δ ðμN −X N Þdμ1 ⋯dμN 0
0
Z∞ ∂R 0; ⋯; 0; μi ; 0; ⋯; 0; μNþ1 ¼ R 0; 0; ⋯; 0; μNþ1 þ ⋯ þ σ ðX i −μi Þdμi ∂μi
0 Z∞ Z∞ ∂2 R 0; ⋯; 0; μ ; 0; ⋯; 0; μ ; 0; ⋯; 0; μ j k Nþ1 þ⋯ þ σ X j −μj σ X k −μk dμj dμk ∂μj ∂μk 0
0
Z∞ Z∞ Z∞ þ⋯ þ 0
0
0
∂3 R 0; ⋯; 0; μl ; 0; ⋯; 0; μm ; 0; ⋯; 0; μn ; 0; ⋯; 0; μNþ1 ∂μl ∂μm ∂μn
σ ðX l −μl Þσ ðX m −μm ÞσðX n −μn Þdμl dμm dμn Z∞ Z∞ N ∂ R μ1 ; ⋯; μN ; μNþ1 þ⋯ þ ⋯ σ ðX 1 −μ1 Þ⋯σ ðX N −μN Þdμ1 ⋯dμN ; ∂μ1 ⋯∂μN 0
0
ð2:7Þ holds because R(X1, ⋯, XN, μN + 1) can be regarded as one of the R(X1, ⋯, XN) appending a parameter μN + 1. Multiplying both sides of (2.7) by δ(μN + 1 − XN + 1) and then integrating each term with respect to μN + 1, one obtains on the left-hand side,
Z∞ Z∞ RðX 1 ; ⋯; X N ; X Nþ1 Þ ¼ ⋯ R μ1 ; ⋯; μN ; μNþ1 δ ðμ1 −X 1 Þ⋯δ ðμN −X N Þδ μNþ1 −X Nþ1 0
0
dμ1 ⋯dμN dμNþ1 ;
ð2:8Þ and on the right-hand side,
Z∞ 0
R 0; 0; ⋯; 0; μNþ1 δ μNþ1 −X Nþ1 dμNþ1 8 9 Z∞
