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Home , Heaviside step function

Decomposition of a Nonlinear Multivariate Function using the Heaviside

Eisuke Chikayama1,2,3,* 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 * Author to whom correspondence should be addressed. Tel.: these two expressions, the left- and right-hand sides of (1.2), are +81-25-239-3706. Fax: +81-25-239-3690. E-mail: mathematically equivalent, the step-function expression would be [email protected]. expected to find broad applications in many branches of Abstract Whereas the introduced by P. A. M. Dirac in mathematical physics. However, that has not been noticed or 1930 in his famous quantum mechanics text has been well studied, developed so far compared with the application of the a not famous formula related to the delta function using the delta-function expression. Here we demonstrate a unified formula Heaviside step function in a single-variable form, also given in Dirac’s text, has been poorly studied. We demonstrate the that extends this step-function expression for single-variable decomposition of a nonlinear multivariate function into a sum of functions to multiple-variable functions. It can be interpreted as in which each integrand is composed of a of the the decomposition of any nonlinear multivariate function with function and a direct product of Heaviside step functions. It is an respect to the Heaviside step function. extension of Dirac’s single-variable form to that for multiple variables. Moreover, it remains mathematically equivalent to the definition of the Dirac delta function with multiple variables, and offers a 2. Decomposition of nonlinear multivariate functions using the mathematically unified expression. Heaviside step function Definition. Let 푅(푋1, 푋2, ⋯ , 푋푁) be a continuous real function 1. Introduction defined for 0 ≤ 푋푖 < ∞ and satisfies: 휕훼푅 P. A. M. Dirac introduced in 1930 a function, now called the Dirac  Whose 훼 훼 exist and continuous where 휕푋 1⋯휕푋 푁 delta function, to develop his theory of quantum mechanics [1]. 1 푁 훼 = 훼1 + ⋯ + 훼푁 and 훼푖 ≥ 0 is a natural number. The delta function, a valuable function in the field of mathematical 휕훼푅  훼1 훼푁 can be integrated with respect to a given 푋푖 physics, takes value infinity at 푥 = 0 and zero at 푥 ≠ 0; its 휕푋1 ⋯휕푋푁 is unity. Its fundamental property derivable from its while the other variables are held fixed. 훼−1 definition is that any nonlinear multivariate real function can be 1 휕 푅  Sequences of functions ( { 훼 −1 (⋯ , 푋푖 + ℎ, ⋯ ) − ℎ ⋯휕푋 푖 ⋯ expressed with delta functions 훿 and integrals as follows, 푖 ∞ ∞ 휕훼−1푅 훼푖−1 (⋯ , 푋푖, ⋯ )}) uniformly converge to ⋯휕푋푖 ⋯ 푅(푋1, 푋2, ⋯ , 푋푁) = ∫ ⋯ ∫ 푅(휇1, ⋯ , 휇푁)훿(휇1 ℎ→0 휕훼푅 훼 훼 for 푋 and 푗 ≠ 푖. −∞ −∞ 휕푋 1⋯휕푋 푁 푗 − 푋 ) ⋯ 훿(휇 −푋 )푑휇 ⋯ 푑휇 . 1 푁 1 푁 푁 1 푁 (1.1) Theorem. 푅(푋 , 푋 , ⋯ , 푋 ) can be decomposed into: The importance of this property is analogous to the Fourier 1 2 푁 푅(푋1, 푋2, ⋯ , 푋푁) = 푅(0, 0, ⋯ ,0) transform [2] for its ability to yield an alternative representation of ∞ ( ) any nonlinear multivariate function. Moreover, the delta function 휕푅 휇1, 0, ⋯ ,0 + ∫ 휎(푋1 − 휇1)푑휇1 휕휇1 can be applied as an alternative to the from the 0 view point of continuity, and applied to numbers of formulae in ∞ 휕푅(0, 휇2, 0, ⋯ ,0) the Fourier and Laplace transforms, and differential equations [3]. + ∫ 휎(푋2 − 휇2)푑휇2 + ⋯ 휕휇2 0 A more rigorous mathematical theory for the delta function has ∞ also been developed and expanded as the theory of distributions 휕푅(0, ⋯ ,0, 휇푖, 0, ⋯ ,0) + ∫ 휎(푋푖 − 휇푖)푑휇푖 by L. Schwartz [4]. 휕휇푖 0 Transforming the integral expression in one variable ∞ 휕푅(0, ⋯ ,0, 휇푁) using the Dirac delta function 훿 into one using the Heaviside step + ⋯ + ∫ 휎(푋푁 − 휇푁)푑휇푁 휕휇푁 function 휎, 0 ∞ ∞ ∞ ∞ 2 푑푅(푥) 휕 푅(휇1, 휇2, 0, ⋯ ,0) ∫ 푅(푥)훿(푥)푑푥 = 푅(0) = 푅(∞) − ∫ 휎(푥)푑푥 + ∫ ∫ 휎(푋1 − 휇1)휎(푋2 − 휇2)푑휇1 푑휇2 + ⋯ 푑푥 휕휇1휕휇2 −∞ −∞ 0 0 ∞ ∞ ∞ 2 푑푅(푥) 휕 푅(0, ⋯ ,0, 휇푗, 0, ⋯ ,0, 휇푘, 0, ⋯ ,0) = 푅(−∞) + ∫ 휎(−푥)푑푥 + ∫ ∫ 휎(푋푗 − 휇푗)휎(푋푘 푑푥 휕휇푗휕휇푘 −∞ 0 0 (1.2) − 휇푘)푑휇푗 푑휇푘 + ⋯ is essentially described in Dirac’s quantum mechanics text [1]. It is derived from a well-known relation between the Dirac delta function and the derivative of the Heaviside step function. As ∞ ∞ ∞ 2 휕 푅(0, ⋯ , 0, 휇푁−1, 휇푁) 휕푅(0, ⋯ ,0, 휇푖, 0, ⋯ ,0) + ∫ ∫ 휎(푋푁−1 − 휇푁−1)휎(푋푁 + ∫ 휎(푋푖 − 휇푖)푑휇푖 + ⋯ 휕휇푁−1휕휇푁 휕휇푖 0 0 0 ∞ ∞ 2 − 휇푁)푑휇푁−1 푑휇푁 휕 푅(0, ⋯ ,0, 휇 , 0, ⋯ ,0, 휇 , 0, ⋯ ,0) ∞ ∞ ∞ 푗 푘 3 + ∫ ∫ 휎(푋푗 − 휇푗)휎(푋푘 휕 푅(휇1, 휇2, 휇3, 0, ⋯ ,0) 휕휇푗휕휇푘 + ∫ ∫ ∫ 휎(푋1 − 휇1)휎(푋2 − 휇2)휎(푋3 0 0 휕휇1휕휇2휕휇3 0 0 0 − 휇푘)푑휇푗 푑휇푘 + ⋯ ∞ ∞ − 휇3)푑휇1 푑휇2 푑휇3 + ⋯ 휕푁푅(휇 , ⋯ , 휇 ) ∞ ∞ ∞ 1 푁 3 + ∫ ⋯ ∫ 휎(푋1 휕 푅(0, ⋯ ,0, 휇푙, 0, ⋯ ,0, 휇푚, 0, ⋯ ,0, 휇푛, 0, ⋯ ,0) 휕휇1 ⋯ 휕휇푁 + ∫ ∫ ∫ 휎(푋푙 0 0 휕휇푙휕휇푚휕휇푛 0 0 0 − 휇1) ⋯ 휎(푋푁 − 휇푁)푑휇1 ⋯ 푑휇푁 − 휇푙)휎(푋푚 − 휇푚)휎(푋푛 − 휇푛)푑휇푙 푑휇푚 푑휇푛 + ⋯ (2.6) ∞ ∞ ∞ ∞ 4 휕 푅(0, ⋯ ,0, 휇표, 0, ⋯ ,0, 휇푝, 0, ⋯ ,0, 휇푞, 0, ⋯ , 0, 휇푟, 0, ⋯ ,0) holds for some N. Using (2.6), the following, + ∫ ∫ ∫ ∫ 휎(푋표 ∞ ∞ 휕휇표휕휇푝휕휇푞휕휇푟 0 0 0 0 ∫ ⋯ ∫ 푅(휇1, ⋯ , 휇푁, 휇푁+1)훿(휇1 − 푋1) ⋯ 훿(휇푁−푋푁)푑휇1 ⋯ 푑휇푁 − 휇표)휎(푋푝 − 휇푝)휎(푋푞 − 휇푞)휎(푋푟 − 휇푟) 푑휇표 푑휇푝 푑휇푞 푑휇푟 + ⋯ ∞ ∞ 0 0 푁 휕 푅(휇1, ⋯ , 휇푁) = 푅(푋 , ⋯ , 푋 , 휇 ) + ∫ ⋯ ∫ 휎(푋1 − 휇1) ⋯ 휎(푋푁 − 휇푁)푑휇1 ⋯ 푑휇푁 , 1 푁 푁+1 휕휇1 ⋯ 휕휇푁 0 0 = 푅(0, 0, ⋯ ,0, 휇푁+1) + ⋯ ∞ (2.1) 휕푅(0, ⋯ ,0, 휇푖, 0, ⋯ ,0, 휇푁+1) where + ∫ 휎(푋푖 − 휇푖)푑휇푖 + ⋯ 휕휇푖 1 (푋 > 휇 ) 0 푖 푖 ∞ ∞ 휕푁푅(휇 , ⋯ , 휇 , 휇 ) + ∫ ⋯ ∫ 1 푁 푁+1 휎(푋 1 휕휇 ⋯ 휕휇 1 휎(푋푖 − 휇푖) = (푋푖 = 휇푖) 1 푁 2 0 0 − 휇1) ⋯ 휎(푋푁 − 휇푁)푑휇1 ⋯ 푑휇푁 , {0 (푋푖 < 휇푖) (2.7) holds because 푅(푋1, ⋯ , 푋푁, 휇푁+1) can be regarded as one of the (2.2) 푅(푋1, ⋯ , 푋푁) appending a parameter 휇푁+1 . Multiplying both defines a set of Heaviside step functions for each 휇푖 ≥ 0. sides of (2.7) by 훿(휇푁+1 − 푋푁+1) and then integrating each term with respect to 휇푁+1, one obtains on the left-hand side, ∞ ∞ Proof. This proof is demonstrated using mathematical induction. ( ) ( ) ( ) ( Using the definition of the Dirac delta function, then ∫ ⋯ ∫ 푅 휇1, ⋯ , 휇푁, 휇푁+1 훿 휇1 − 푋1 ⋯ 훿 휇푁−푋푁 훿 휇푁+1 0 0 for any real function 푅(푋1, 푋2, ⋯ , 푋푁) ∞ ∞ − 푋푁+1)푑휇1 ⋯ 푑휇푁푑휇푁+1 , (2.8) 푅(푋1, 푋2, ⋯ , 푋푁) = ∫ ⋯ ∫ 푅(휇1, ⋯ , 휇푁)훿(휇1 −∞ −∞ and on the right-hand side, ∞ − 푋1) ⋯ 훿(휇푁−푋푁)푑휇1 ⋯ 푑휇푁 . (2.3) ∫{푅(0, 0, ⋯ ,0, 휇푁+1)}훿(휇푁+1 − 푋푁+1)푑휇푁+1 0 Therefore, ∞ ∞ ∞ ∞ 휕푅(0, ⋯ ,0, 휇푖, 0, ⋯ ,0, 휇푁+1) + ⋯ + ∫ {∫ 휎(푋푖 − 휇푖)푑휇푖} 훿(휇푁+1 푅(푋1, 푋2, ⋯ , 푋푁) = ∫ ⋯ ∫ 푅(휇1, ⋯ , 휇푁)훿(휇1 휕휇푖 0 0 0 0 − 푋푁+1)푑휇푁+1 − 푋1) ⋯ 훿(휇푁−푋푁)푑휇1 ⋯ 푑휇푁 ∞ ∞ ∞ 푁 (2.4) 휕 푅(휇1, ⋯ , 휇푁, 휇푁+1) + ⋯ + ∫ {∫ ⋯ ∫ 휎(푋1 − 휇1) ⋯ 휎(푋푁 with 푋 ≥ 0. 휕휇1 ⋯ 휕휇푁 푖 0 0 0 For single-variable functions (푁 = 1), ∞ ∞ − 휇 )푑휇 ⋯ 푑휇 } 훿(휇 − 푋 )푑휇 . 푑 푁 1 푁 푁+1 푁+1 푁+1 ∫ 푅(휇1)훿(휇1 − 푋1)푑휇1 = 푅(0) + ∫ 푅(휇1)휎(푋1 − 휇1)푑휇1 푑휇1 0 0 (2.9) (2.5) The order of the integrations can be changed because each holds by Lemma 2.1 given below. Note that, for single-variable integrand can be integrated with respect to its corresponding 휇푖 functions, Dirac described the essentially equivalent expression while holding other variables fixed. Therefore, (2.9) can be (1.2) [1]. To initiate the mathematical induction procedure, transformed into ∞ suppose that ∞ ∞ {∫ 푅(0, 0, ⋯ ,0, 휇푁+1)훿(휇푁+1 − 푋푁+1)푑휇푁+1} ∫ ⋯ ∫ 푅(휇 , ⋯ , 휇 )훿(휇 − 푋 ) ⋯ 훿(휇 −푋 )푑휇 ⋯ 푑휇 0 1 푁 1 1 푁 푁 1 푁 ∞ ∞ 0 0 휕푅(0, ⋯ ,0, 휇푖, 0, ⋯ ,0, 휇푁+1) ∞ + ⋯ + ∫ {∫ 훿(휇푁+1 휕푅(휇1, 0, ⋯ ,0) 휕휇푖 = 푅(0, 0, ⋯ ,0) + ∫ 휎(푋1 − 휇1)푑휇1 + ⋯ 0 0 휕휇1 0 − 푋푁+1)푑휇푁+1} 휎(푋푖 − 휇푖)푑휇푖 ∞ ∞ ∞ 휕푁푅(휇 , ⋯ , 휇 , 휇 ) Lemma 2.1. 1 푁 푁+1 ∞ + ⋯ + ∫ ⋯ ∫ {∫ 훿(휇푁+1 휕휇1 ⋯ 휕휇푁 0 0 0 ∫ 퐹(휇1, ⋯ , 휇푖, ⋯ , 휇푁)훿(휇푖 − 푋푖)푑휇푖 0 − 푋푁+1)푑휇푁+1} 휎(푋1 − 휇1) ⋯ 휎(푋푁 = 퐹(휇1, ⋯ , 휇푖−1, 0, 휇푖+1, ⋯ , 휇푁) ∞ 휕퐹(휇 , ⋯ , 휇 , ⋯ , 휇 ) − 휇푁)푑휇1 ⋯ 푑휇푁 . 1 푖 푁 + ∫ 휎(푋푖 − 휇푖)푑휇푖 . (2.10) 휕휇푖 0 The terms enclosed in braces in (2.10) can be transformed using (2.13) Lemma 2.1 as follows, ∞ Proof. From Lemma 2.2, ∞ 휕푅(0, ⋯ ,0, 휇푁+1) {푅(0, 0, ⋯ , 0, 0) + ∫ 휎(푋푁+1 − 휇푁+1)푑휇푁+1} ∫ 퐹(휇 , ⋯ , 휇 , ⋯ , 휇 )훿(휇 − 푋 )푑휇 휕휇푁+1 1 푖 푁 푖 푖 푖 0 ∞ 0 ∞ 휕푅(0, ⋯ ,0, 휇푖, 0, ⋯ , 0, 0) 푑 + ⋯ + ∫ { = ∫ 퐹(휇 , ⋯ , 휇 , ⋯ , 휇 ) 휎(휇 − 푋 )푑휇 휕휇푖 1 푖 푁 푖 푖 푖 0 푑휇푖 ∞ 0 2 ∞ 휕 푅(0, ⋯ ,0, 휇푖, 0, ⋯ ,0, 휇푁+1) 푑 + ∫ 휎(푋푁+1 = − ∫ 퐹(휇 , ⋯ , 휇 , ⋯ , 휇 ) 휎(푋 휕휇푁+1휕휇푖 1 푖 푁 푖 0 푑휇푖 0 − 휇푖)푑휇푖 . − 휇푁+1)푑휇푁+1} 휎(푋푖 − 휇푖)푑휇푖 (2.14) ∞ ∞ 휕푁푅(휇 , ⋯ , 휇 , 0) If 푋푖 > 0, + ⋯ + ∫ ⋯ ∫ { 1 푁 ∞ 휕휇1 ⋯ 휕휇푁 푑 0 0 − ∫ 퐹(휇1, ⋯ , 휇푖, ⋯ , 휇푁) 휎(푋푖 − 휇푖)푑휇푖 ∞ 푑휇 푁+1 푖 휕 푅(휇1, ⋯ , 휇푁, 휇푁+1) 0 + ∫ 휎(푋 ∞ 푁+1 = −[퐹(휇1, ⋯ , 휇푖, ⋯ , 휇푁)휎(푋푖 − 휇푖)]0 휕휇푁+1휕휇1 ⋯ 휕휇푁 ∞ 0 휕 + ∫ 퐹(휇1, ⋯ , 휇푖, ⋯ , 휇푁)휎(푋푖 − 휇푖)푑휇푖 − 휇푁+1)푑휇푁+1} 휎(푋1 − 휇1) ⋯ 휎(푋푁 휕휇푖 0 = −퐹(휇1, ⋯ , ∞, ⋯ , 휇푁)휎(푋푖 − ∞) − 휇푁)푑휇1 ⋯ 푑휇푁 , + 퐹(휇1, ⋯ ,0, ⋯ , 휇푁)휎(푋푖 − 0) (2.11) ∞ where Lemma 2.4 was also used. 휕 + ∫ 퐹(휇1, ⋯ , 휇푖, ⋯ , 휇푁)휎(푋푖 − 휇푖)푑휇푖 . Finally (2.11) becomes 휕휇푖 0 푅(0, 0, ⋯ , 0, 0) ∞ (2.15) 휕푅(0, ⋯ ,0, 휇푁+1) With 휎(푋푖 − ∞) = 0 and 휎(푋푖 − 0) = 1, we obtain + ∫ 휎(푋푁+1 − 휇푁+1)푑휇푁+1 + ⋯ ∞ 휕휇푁+1 0 ∞ ∫ 퐹(휇1, ⋯ , 휇푖, ⋯ , 휇푁)훿(휇푖 − 푋푖)푑휇푖 휕푅(0, ⋯ ,0, 휇푖, 0, ⋯ , 0, 0) 0 + ∫ 휎(푋푖 − 휇푖)푑휇푖 휕휇푖 = 퐹(휇1, ⋯ ,0, ⋯ , 휇푁) 0 ∞ ∞ ∞ 휕퐹(휇 , ⋯ , 휇 , ⋯ , 휇 ) 휕2푅(0, ⋯ ,0, 휇 , 0, ⋯ ,0, 휇 ) 1 푖 푁 푖 푁+1 + ∫ 휎(푋푖 − 휇푖)푑휇푖 . + ⋯ + ∫ ∫ 휎(푋푖 − 휇푖)휎(푋푁+1 휕휇푖 휕휇푖휕휇푁+1 0 0 0 (2.16) − 휇푁+1)푑휇푖푑휇푁+1 ∞ ∞ 푁 If 푋푖 = 0, the left-hand side of (2.13) is 휕 푅(휇1, ⋯ , 휇푁, 0) ∞ + ⋯ + ∫ ⋯ ∫ 휎(푋1 − 휇1) ⋯ 휎(푋푁 휕휇1 ⋯ 휕휇푁 ∫ 퐹(휇 , ⋯ , 휇 , ⋯ , 휇 )훿(휇 − 푋 )푑휇 0 0 1 푖 푁 푖 푖 푖 0 − 휇푁)푑휇1 ⋯ 푑휇푁 + ⋯ ∞ ∞ ∞ 휕푁+1푅(휇 , ⋯ , 휇 , 휇 ) 1 푁 푁+1 = ∫ 퐹(휇1, ⋯ , 휇푖, ⋯ , 휇푁)훿(휇푖)푑휇푖 + ∫ ⋯ ∫ 휎(푋1 휕휇푁+1휕휇1 ⋯ 휕휇푁 0 0 0 = 퐹(휇1, ⋯ , 휇푖−1, 0, 휇푖+1, ⋯ , 휇푁) − 휇1) ⋯ 휎(푋푁 − 휇푁)휎(푋푁+1 (2.17) − 휇푁+1)푑휇1 ⋯ 푑휇푁푑휇푁+1 . whereas the right-hand side of (2.13) is (2.12) 퐹(휇1, ⋯ , 휇푖−1, 0, 휇푖+1, ⋯ , 휇푁) Thus, assuming expression (2.6) for 푁 leads to the same ∞ expression for 푁 + 1. The formula for 푁 = 1 also holds as 휕퐹(휇1, ⋯ , 휇푖, ⋯ , 휇푁) + ∫ 휎(푋푖 − 휇푖)푑휇푖 described above. Therefore, the theorem holds for any natural 휕휇푖 0 number 푁. □

훼−1 = 퐹(휇1, ⋯ , 휇푖−1, 0, 휇푖+1, ⋯ , 휇푁) 1 휕 푅 0 lim (푟ℎ→0) = lim lim { 훼 −1 (⋯ , 휇푖 + ℎ, ⋯ , 휇푗, ⋯ ) 휇푗→0 ℎ→0 휇푗→0 ℎ 푖 휕퐹(휇1, ⋯ , 휇푖, ⋯ , 휇푁) ⋯ 휕휇푖 ⋯ + ∫ 휎(푋푖 − 휇푖)푑휇푖 (푖≠푗) (푖≠푗) 휕휇푖 훼−1 0 휕 푅 ∞ − (⋯ , 휇푖, ⋯ , 휇푗, ⋯ )} 휕퐹(휇 , ⋯ , 휇 , ⋯ , 휇 ) ⋯ 휕휇훼푖−1 ⋯ + ∫ 1 푖 푁 휎(푋 − 휇 )푑휇 푖 푖 푖 푖 훼−1 휕휇푖 1 휕 푅 0< = lim { (⋯ , 휇푖 + ℎ, ⋯ , 휇푗 = 0, ⋯ ) ℎ→0 ℎ 훼푖−1 = 퐹(휇1, ⋯ , 휇푖−1, 0, 휇푖+1, ⋯ , 휇푁) . ⋯ 휕휇푖 ⋯ (2.18) 휕훼−1푅 − (⋯ , 휇푖, ⋯ , 휇푗 = 0, ⋯ )} □ 훼푖−1 ⋯ 휕휇푖 ⋯ 훼 휕 푅(⋯ , 휇푖, ⋯ , 휇푗−1, 0, 휇푗+1, ⋯ ) = . Lemma 2.2. 훼푖 푑휎(휇 − 푋) 푑휎(푋 − 휇) ⋯ 휕휇푖 ⋯ = − . (2.23) 푑휇 푑휇 □ (2.19)

Proof. From Lemma 2.3, 휎(휇 − 푋) = 1 − 휎(푋 − 휇) . Therefore, 푑휎(휇−푋) 푑휎(푋−휇) 3. Concluding remarks by differentiating both sides, = − . □ 푑휇 푑휇 We have demonstrated the decomposition of a nonlinear multivariate function as a sum of integrals of which each integrand Lemma 2.3. is composed of a derivative and a direct product of Heaviside step 휎(−푋) = 1 − 휎(푋) . functions. The expression offers a mathematically unified and (2.20) systematic expansion equivalent to that given in terms of Dirac Proof. If 푋 > 0, 휎(−푋) = 0 = 1 − 휎(푋) . If 푋 < 0, 휎(−푋) = delta functions. It can further be approximated using, e.g., sigmoid 1 1 = 1 − 휎(푋) . If 푋 = 0, 휎(−푋) = = 1 − 휎(푋) . □ functions with suitable parameters, yielding a convenient form for 2 broad applications to other fields exploiting both analytical and Lemma 2.4. numerical methods. 훼 훼 휕 푅(⋯ , 휇푖, ⋯ , 휇푗, ⋯ ) 휕 푅(⋯ , 휇푖, ⋯ , 휇푗−1, 0, 휇푗+1, ⋯ ) References: 훼 | = 훼 , 푖 휇 =0 푖 [1] P.A.M. Dirac, The principles of quantum mechanics, 4th ed., Clarendon ⋯ 휕휇푖 ⋯ 푗 ⋯ 휕휇푖 ⋯ (푖≠푗) Press, Oxford, 1958, pp. 58-62. (2.21) [2] R.N. Bracewell, The and its applications, McGraw-Hill, New York, 1965. where 훼 = 훼1 + ⋯ + 훼푁 and 훼푖 ≥ 0 is a natural number. [3] L. Schwartz, Mathematics for the physical sciences, Hermann, Paris, Proof. Since derivatives of R is continuous, 1966. [4] L. Schwartz, Généralisation de la notion de fonction, de dérivation, de 훼 훼 휕 푅(⋯ , 휇푖, ⋯ , 휇푗, ⋯ ) 휕푅 (⋯ , 휇푖, ⋯ , 휇푗, ⋯ ) transformation de Fourier et applications mathématiques et physiques, Ann. 훼 | = lim 훼 Univ. Grenoble, 21 (1945) 57-74. 푖 휇 =0 휇푗→0 푖 ⋯ 휕휇푖 ⋯ 푗 ⋯ 휕휇푖 ⋯ (푖≠푗) (푖≠푗) 1 휕훼−1푅 Acknowledgements: I thank K. Soda for discussion. = lim lim { 훼 −1 (⋯ , 휇푖 + ℎ, ⋯ ) 휇푗→0 ℎ→0 ℎ ⋯ 휕휇 푖 ⋯ (푖≠푗) 푖 휕훼−1푅 − (⋯ , 휇 , ⋯ )} 훼푖−1 푖 ⋯ 휕휇푖 ⋯ = lim (푟ℎ→0), 휇푗→0 (푖≠푗) (2.22) where (푟ℎ→0) is a sequence of functions. The sequence of fu nction (푟ℎ→0) converges uniformly for 휇푗 and 푗 ≠ 푖 to 훼 휕 푅(⋯,휇푖,⋯,휇푗,⋯ ) 훼푖 . Similarly ⋯휕휇푖 ⋯ 1 휕훼−1푅 lim { (⋯ , 휇 + ℎ, ⋯ , 휇 , ⋯ ) − 휇 →0 ℎ 훼푖−1 푖 푗 푗 ⋯휕휇푖 ⋯ (푖≠푗) 휕훼−1푅 훼푖−1 (⋯ , 휇푖, ⋯ , 휇푗, ⋯ )} ⋯휕휇푖 ⋯ converges pointwise for h to 1 휕훼−1푅 { (⋯ , 휇 + ℎ, ⋯ , 휇 = 0, ⋯ ) − ℎ 훼푖−1 푖 푗 ⋯휕휇푖 ⋯ 휕훼−1푅 훼푖−1 (⋯ , 휇푖, ⋯ , 휇푗 = 0, ⋯ )} since it is continuous. Therefore, ⋯휕휇푖 ⋯ the order of the limits can be interchanged. Finally (2.22) be