Some Separate Quasi-Asymptotics Properties of Multidimensional Distributions and Application

Quasi-asymptotic behavior of functions as a method has its application in observing many physical phenomena which are expressed by differential equations. The aim of the asymptotic method is to allow one to present the solution of a problem depending on the large (or small) parameter. One application of asymptotic methods in describing physical phenomena is the quasi-asymptotic approximation. The aim of this paper is to look at the quasi-asymptotic properties of multidimensional distributions by extracted variable. Distribution T(x0,x) from S'(Ṝ+ ×R) has the property of the separability of variables, if it can be represented in form T(x0,x)=∑φi(x0)ψi (x) where distributions, φi(x0) from S'(Ṝ ) and ψi from S(R ), x0 from Ṝ+ and x is element R n different values of do not depend on each other. Distribution T(x0,x) the element S'(Ṝ+ ×R) is homogeneous and of order α at variable x0 is element Ṝ 1 + and x=x1,x2,...,xn from R n if for k>0 it applies that T(kx0,kx)=k α T(x0,x). The method of separating variables is one of the most widespread methods for solving linear differential equations in mathematical physics. In this paper, the results by V. S Vladimirov are used to present the proof of the basic theorems, regarding the quasi-asymptotic behavior of multidimensional distributions by a singular variable, with the application of quasiasymptotics to the solution of differential equations. Keywords: Distribution Spaces, Asymptotics, Separate Quasi-Asymptotics, Multidimensional Distributions 1. Introduction We use to mark the standard space of the Schwartz's rapidly decreasing functions, and to mark the corresponding space of the slowly increasing distributions [1, 7]. If and is a positive and a continuous function for 0, distribution has quasi-asymptotics at infinity (at zero) with respect to positive function , if the following is valid , (1) ∞ in with the distribution being , [1, 2-4, 7]. If 0 distribution has a trivial quasiasymptotics at infinity (that is, at zero) with respect to positive function . If (1) is true, function occurs as an auto-modal function. If is a positive and continuous function, and ∞, then we say that is an auto-modal function, if for a real number 0 there exists lim ! " ! # (2) Where by it converges evenly along a on each compact semi-axies 0, ∞ . Distribution $ is asymptotically homogeneous with respect to function of order % if: & ' #$ in $ (3) where the nucleus of fractional differentiation and integration # is defined by # ( ) ' *+, # , . % 0 /0 / 0 #$1 , . % 2 0, % 3 4 0 (4) with Γ % being the gamma function, and 6 being the Heaviside function [1-3, 6, 7]. The fractional derivative, [3-5, 7], of order % and 65 Nenad Stojanovic: Some Separate Quasi-Asymptotics Properties of Multidimensional Distributions and Application distribution of is defined by the formula 7# # ∗ (5) Distribution 9 : , :) ∈ ;′(R$ × R ) has the property of the separability of variables, if it can be represented in form 9(: , :) = ∑ >?(: )@(:) ? where distributions >(: ) ∈ ;′(R$ ) and @ ∈ ;(R , : ∈ R$ and : ∈ R for different values of i do not depend on each other, [15]. Distribution 9(: , :) ∈ ;′(R$ AAAA × R ) is homogeneous and of order % at variable: ∈ R$ AAAAand : = : , :B, ... , : ∈ R if for > 0 it applies that 9( : , :) = #9(: , :), [1, 3, 7, 8]. In other words, distribution 9(: , :) ∈ ;′(R$ AAAA × R ) is homogeneous and of order % at variable : and : if for each test function D(: , :) ∈ ;(R$ AAAA × R ) the following is valid E9(: , :), D FG , F H = B#$ $ E9(: , :), D(: , :)H, > 0. Indeed, E9( : , :), D(: , :)H = = Ishift : = : ⟹ : = : , : = : ⇒ : = : P = 1 $ E9(: , : ), D R:′ , :′ SH = T:′ = : : = : U = VW, E9(: , :), D FG , F H (6) For example, let it be that 9(: , :) ∈ ;′(R$ AAAA × R ) in the form of 9(: , :) = (: ) × (:) with distributions (: ) ∈ ′(R $), (:) (R ) being homogenous and of order % . Then, there is a number of equations that are valid: E9( : , :), D(: , :)H = E ( : ) × ( :), D(: , :) H = E ( : ), E ( :), D(: , :)HH = T ( : ) = # (: ) ( :) = # (:) U = B#E (: ), E (:), D(: , :)HH = B#E9(: , :), D(: , :)H. (7) From (6) and (7) we can see that the following equation is valid 1 $ E9(: , :), D : , : H = B#E9(: , :), D(: , :)H and from here, there is E9(: , :), D : , : H = B#$ $ E9(: , :), D(: , :)H. For example, distribution 9(: , :) ∈ ;′(R$ AAAA × R ) in the form of 9(: , :) = (: ) × (:) with (: ) ∈ ′(R $), (:) (R ), and (: ) being homogenous and of order % , and test function D(: , :) in the fom of D(: , :) = ∑ >? (: )>?B(:) ? with >? (: ) ∈ ( R $), >?B(:) ∈ (R ) and > (: ) , is a homogenous distribution of order % and then, for ) ( i ∀ the following is true E9( : , :), D(: , :)H = E ( : ) × (:), D(: , :) H =E ( : ), E (:), D(: , :)H H =E ( : ). E (:), ∑ >? (: )>?B(:) ? H H = YE ( : ), >? (: ) H ? E (:), >?B(:) H = # YE ( : ), >? (: ) H ? E (:), >?B(:) H = #E (: ) (:), D(: , :)H = #E9( : , :), D(: , :)H since the set of functions ∑ >? (: )>?B(:) ? is dense in ;(R$ AAAA × R ). This is followed by the claim, because it is valid in a dense set, and with its continuity, it extends to entire set in ;(R$ AAAA × R ). Homogeneity by the second variable is similarly defined [1, 6]. The homogeneity of distribution 9(: , :) ∈ ;′(R$ AAAA × R ) separable at variable : assuming that distribution >? (: ) ∈ ( R $) is homogeneous and of order % for each i, then, form these relations, it follows that 9( : , :) = Y >?( : )@(:) ? = # ∑ >?(: )@(:) = #9(: , :). ? (8) Homogeneity at variable x is similarly observed. Let there be distribution 9(: , :) ∈ ;′(R$ AAAA × R ) . Distribution 9(: , :) with : ∈ R$ and : ∈ R has quasiasymptotic at infinity at variable : relative to auto-modal function , if there is distribution Z(: , :) ≠ 0 such that lim → !( ) 9( : , :) = Z(: , :) in ;′(R$ AAAA × R ). (9) Quasi-asymptotics by the separated variable at zero is similarly defined, [1, 7]. Let us suppose a distribution 9(: , :) ∈ ;′(R$ AAAA × R ). Distribution 9(: , :) , : ∈ R$ and : ∈ R has quasiasymptotics at zero at variable : with respect to auto-modal function , if, and only if there is distribution Z(: , :) ≠ 0 such that lim → !( ) 9 FG , : = Z FG , : ≠ 0 (10) in ; \R$ AAAA × R ]. For distributions from ^ \R$ AAAA × R ] (or ; \R$ AAAA × R ] we define the fractional (rational) differentiation at variable : as a convolution #(: ) with (: , :) at, : by the following formula (#) = 7#(: ) ⋇ (: , :) (11) Pure and Applied Mathematics Journal 2020; 9(3): 64-69 66 which belongs to ^ \ $ AAAA × ]if ^ \ $ AAAA × ] that is, ; \ $ AAAA × ] if ; \ $ AAAA × ], (more in [3-6, 8, 12, 13]). 2. Some Quasi-Asymptotics Properties of Multidimensional Distributions We provide proof of some of the basic theorems that apply to multidimensional distributions, and their formulaic presentation can be seen in [1]. Theorem 1. If distribution 9(: , :) ∈ ;′(R$ AAAA × R ) is asymptotically homogeneous with respect to positive function `( ) at variable : or if the following is true lim → !( ) 9( : , :) = Z(: , :) in ;′(R$ AAAA × R ) (12) then `( ) is an auto-modal function. Proof: Let (12) be true and let D(: , :) ∈ (R$ AAAA × R ) test function such that EZ(: , :), D(: , :) H ≠ 0. Then let the test function be of the following form D(: , :) = ∑ >? (: )>?B(:), ? so that >? (: ) ∈ ( R $), >?B(:) ∈ (R ) ∀., are continuous functions with the following feature: bcdd >? ⊂ R$ AAAA, bcdd >?B ⊂ R$ , bcdd D = bcdd >? × bcdd >?B ⊂ \R$ AAAA × R ], (∀.), f ⊂ R$ compact set. For D(: , :) and ∈ f it applies that " D FG " , : = " ∑ >? FG " >?B(:) ? . Now, the following is valid for distribution 9(: , :) ∈ ;′(R$ AAAA × R ) and test function D(: , :) ∈ (R$ AAAA × R ); E9( : , :), 1 D : , : H → ,"∈g hiiiiij EZ(: , :), 1 D : , : H For ∈ f, and using (: = : ) the following is valid `( ) `( ) E9( : , :) `( ) , 1 D : , : H = `( ) `( ) E9( :′ , :) `( ) , >(:′ , :)H = `( ) `( ) E9( : , :) `( ) , D(: , :)H → ,"∈g hiiiiij EZ(: , :), " D FG " , : H (13) Further, if we replace with , ∈ f, the following is valid !(" ) E9( : , :), D(: , :)H → hij EZ(: , :), D(: , :)H. (14) Using relations (12) and (13), we get the following relation !(" ) !( ) → hij Ek(FG,F),, lm nG l ,F H Ek(FG,F),m(FG,F)H . (15) From here, by inserting ( ) 0 0 ax x ′ = we get the following `( ) `( ) → hij EZ( :′ , :), D(:′ , :) H EZ(: , :), D(: , :)H . From here, we get the required relation `( ) `( ) → hij EZ( : , :), D(: , :) H EZ(: , :), D(: , :)H = &( ). From the existence of lim → !(" ) !( ) = &( ) following &( ) = # and `( ) = #o( ), and Karamata L function [16], it follows that function `( ) is an auto-modal function, even in the case of multi-variable distributions. Theorem 2. Let distribution 9(: , :) ∈ ;′(R$ AAAA × R ) be asymptotically homogeneous with respect to positive function `( ) at variable x . In this case, if the order of auto-modal function `( ) is equal to % , then distribution Z(: , :) in the following equation lim → !( ) 9( : , :) = Z(: , :) is equal to Z(: , :) = & #(: ) × (:), with & being the constant. Proof: It has already been shown in the case of distributions of one variable [1],[7], that distribution Z(:) ∈ $′ has the form of Z(:) = & #(:) with & being the constant, and #(:) being the nucleus of fractional differentiation. For Z(: , :) ∈ ;′(R$ AAAA × R ) let us suppose that (: ) ∈ ;′(R$ ) AAAAA, and (:) = (R ), and that distribution (: ) is homogeneous and of order %, and Z(: , :) = (: ) × (:) . Since for the function in the form of >(: )@(:) ∈ ;′(R$ × R ) the following applies lim → EE 1 `( ) 9(: , :), >(: )H , @(:) H =E& (:) #$ (: ), @(:)H = & #$ (: )E (:), @(:)H so distribution (:) = ′(R ), Z(: , :) is in the form of Z(: , :) = & #$ (: ) × (:). Theorem 3. If distribution 9(: , :)is separated at variable forms 0 x , then it has the following form: 9(: , :) = 9 (:) + 9B(: ) B(: ) and distribution 9(: , :) has the quasi-asymptotics of order % in relation to function #`(k) at a variable 0 x , if 9 and 9B have the same quasi-asimptotics in relation to function `(k). The reverse of the theorem is not valid. Proof. Let us show that distribution 9(: , :)is has quasiasymptotics of order % with respect to `(k) if 9 and 9B have the same quasiasymptotics. Let the test function D(: , :) be in the form of D(: , :) = ∑ >?(: )@?(:) ? . By the definition of quasi-asymptotics, the following applies: 67 Nenad Stojanovic: Some Separate Quasi-Asymptotics Properties of Multidimensional Distributions and Application 1 ` E9 : , :), D(: , :)H = E9( : , :) `( ) , Y >?(: )@?(:) ? H =Eq,( FG)r,(F)$qs( FG)rs(F) *!( ) , ∑ >?(: )@?(:) ? H = Eq,( FG)r,(F) !( ) , ∑ >?(: )@?(:) ? H + Eqs( FG)rs(F) !( ) , ∑ >?(: )@?(:) ? H. Since `( ) = #o( ) and 9 ( : ) = # 9 (: ) and 9B( : ) = # 9B(: ) therefore = E9 ( : ) (:) #o( ) , Y >?(: )@?(:) ? H + E9B( : ) B(:) #o( ) , Y >?(: )@?(:) ? H = 1 #o( ) E9 ( : ), E (:), Y >?(: )@?(:) ? HH + 1 #o( ) E9B( : ), E B(:), Y >?(: )@?(:) ? HH = 1 #o( ) YE9 ( : ), >?(: )H ? E (:), @?(:)H + 1 #o( ) YE9B( : ), >?(: )H ? E B(:), @?(:)H = # #o( ) YE9 ( : ), >?(: )H ? E (:), @?(:)H + # #o( ) YE9B( : ), >?(: )H ? E B(:), @?(:)H = t( ) ∑ E9 ( : ), >?(: )H ? E (:), @?(:)H + t( ) ∑ E9B( : ), >?(: )H ? E B(:), @?(:)H = 1 o( ) E9 (: ) (:), Y >?(: )@?(:) ? H + 1 o( ) E9B(: ) B(:), Y >?(: )@?(:) ? H = 1 o( ) E9 (: ) (:), D(: , :)H + 1 o( ) E9B(: ) B(:), D(: , :)H = t( ) E9 (: ) (:) + 9B(: ) B(:), D(: , :)H. This shows that distribution 9(: , :) = 9 (:) + 9B(: ) B(: ) has quasi-asymptotics of order % with respect to function #`( ) at variable : if distributions 9 and 9B have the same quasi-asymptotics. The reverse of the theorem is not valid. To show this, it is enough to show that, for example, the following is not valid for distribution 9 (: ) = : #$ + : # and 9B(: ) = −: #$ + : # with respect to function #`( ). Indeed 1 `( ) E9( : , :), D(: , :)H = E9( : , :) `( ) , Y >?(: )@?(:) ? H = E #: #v : ( (:) − B(:)) + (:) + B(:) w `( ) , Y >?(: )@?(:) ? H = E #: #x : \ (:) − B(:)] + (:) + B(:)y #o( ) , Y >?(: )@?(:) ? H = E: #v : ( (:) − B(:)) + (:) + B(:) w o( ) , Y >?(: )@?(:) ? H = E: #v( : + 1 ) (:) + ( : − 1 ) B(:) w o( ) , Y >?(: )@?(:) ? H. From here, it can be seen that 9(: , :) has no quasiasymptotics when → ∞. Theorem 4. In order for distribution 9(: , :) ∈ ;′(R$ AAAA × R ) to be asymptotically homogeneous at infinity, with respect to auto-modal function `( ) at variable : , it is necessary, and it is also sufficient, that for each z ∈ R its Pure and Applied Mathematics Journal 2020; 9(3): 64-69 68 fractional derivative 9 7{ : , :) is asymptotically homogeneous with respect to {`( ). Proof: We define fractional differentiation in ;′(R$ AAAA × R ) with distribution 9(: , :) at : as convolution of distribution {(: ) ∈ ;′(R$ ) AAAAA and distribution 9(: , :) ∈ ;′(R$ AAAA × R i.e. 9(7{)(: , :) = 9(: , :) ∗ {(: ). Using the property of distribution {(: ) to be homogeneous and of order z − 1 , that is, using the validity of the following {( : ) = {7 {(: ), we get the following: lim → |!( ) E9(7{)( : , :), D(: , :)H, from here, if we put that , : = :′ , we get = lim → 1 {$ `( ) E9(7{)(:′ , :), D R:′ , :SH = lim → |W,!( ) E9(7{)(: , :), D FG , : H. By using the definition of convolution 9(: , :) ∗ {(: ) = 1 Γ(z) Θ(: ): {7 ∗ 9(: , :) = -({) ~ (:, : ){7 9( , :)  = 9(7{)(: , :). we can see that the last equation is precisely the z primitive integral for 9(: , :). Based on this, we have that 9(: , :) ∈ ;′(R$ AAAA × R , {(: ) ∈ ;′(R$ ) AAAAA, E9(: , :) ∗ {(: ), D(: , :)H = lim → E9(: , :) ∗ {(),  (: , )D(: + , :)H, with   being unit sequence. If there is a limes on the righthand side for each series  , → ∞ then the function from (RB) which converges to number one in RB and this limit does not depend on the choice of series  , → ∞ then we have that 9(: , :) ∗ {(: ) ∈ ′(R $ ) . Based on this, the last equation transforms into lim → 1 {$ `( ) E9(: , :) ∗ {(: ), D : , : H = lim → |W,!( ) E9(: , :) × {(),  (: ; )D FG$ , : H. Now, if we put that   = ′ : +  = : +  = : + ′ the last equation transforms into the following form: lim → 1 {`( ) E9(: , :), E {( ′), D : + ′, : HH lim → 1 {`( ) E9(: , :), E {( ), D : + , : HH (since {( ) = {7 {()) lim → {7 {`( ) E9(: , :), E {(), D : + , : HH. From the last equation, using the shift (: = : we get the following lim → 1 `( ) E9( :′ , :), E {(), D(:′ + , :)HH lim → 1 `( ) E9( : , :), E {(), D(: + , :)HH lim → !( ) E9( : , :), @(: , :)H,where function D(: , :) = @(: , :) = E {(), D(: + , :)H creates the auto-morphism of space (R$ AAAA × R ) → (R$ AAAA × R ). Theorem 5. Let it be that  ∈ N and that 9(: , :) ∈ ;′(R$ AAAA × R ) has quasi-asymptotics (: , :) at variable : with respect to auto-modal function `( ), → ∞ and let it be that :  ∈ M(,) , with M(,) being the space of the multiplier of distributions, then distribution :  ⋅ 9(: , :) also has quasi-asymptotics Z(: , :) = :  ⋅ (: , :) at : with respect to auto-modal function `( ). Proof. There is lim → E( : ) ∙ 9( : , :)  ∙ `( ) , D(: , :)H = = lim → E  ⋅ :  ⋅ 9( : , :) || ∙ `( ) , D(: , :)H = lim → E9( : , :) `( ) , : D(: , :)H = lim → E9( : , :) `( ) , : D(: , :)H = E (: , :), : D(: , :)H = E:  (: , :), D(: , :)H = EZ(: , :), D(: , :)H. From here we find that Z(: , :) = :  ⋅ (: , :). 3. Example of the Use of Quasi-Asymptitics to the Solutions of Differential Equations Let o be a differential operator with constant coefficients {(:) = { and let ∈  , be such a distribution that convolution E ∗ exists in ' where E ∈ ′ is the fundamental solution of equation o(^)E = (:), [3, 6, 9, 11]. Then the solution c =  ∗ of differential equation o(^)c = (:), ∈ ′ has quasi-asymptotics of order % with respect to `( ) = #o( ) (with o( ) being the Karamata slow-varying function), if distribution ∈ ′ has such quasi-asymptotics, ′-distribution space. Proof: Let have the quasi-asymptotics with respect to `( ) = #o( ). Then the following is valid 1 `( ) E ( :), D(:)H = 1 `( ) E (:), D : H 69 Nenad Stojanovic: Some Separate Quasi-Asymptotics Properties of Multidimensional Distributions and Application 1 ` E (:) ∗ (:), > : H = 1 `( ) E o(^)E ∗ (:), > : H = 1 `( ) E RY #^#E(:) |#| S ∗ (:), > : H = 1 `( ) E Y #^# (E |#| ∗ )(:), > : H = 1 `( ) E o(^)(E ∗ )(:), > : H = 1 `( ) E o(^)c(:), > : H = 1 `( ) EY #^#c(:) |#| , > : H = 1 `( ) Y E^#c(:), #D : H |#| = 1 `( ) Y (−1)|#| Ec(:), ^# R #D : S H |#| = 1 `( ) E c(:), o∗(^) D : H = 1 `( ) E c( :), o(−^) D(:)H = E c( :) `( ) , o(−^) D(:)H. Therefore, we have the following: !( ) E ( :), D(:)H = E ( F) !( ) , o(−^) D(:)H, and, as per assumption, has the quasi-asymptotics, thus, distribution u has one also.


Introduction
We use to mark the standard space of the Schwartz's rapidly decreasing functions, and to mark the corresponding space of the slowly increasing distributions [1,7].
If and is a positive and a continuous function for 0, distribution has quasi-asymptotics at infinity (at zero) with respect to positive function , if the following is valid , ∞ in with the distribution being , [1,[2][3][4]7].
If 0 distribution has a trivial quasiasymptotics at infinity (that is, at zero) with respect to positive function . If (1) is true, function occurs as an auto-modal function. If is a positive and continuous function, and ∞, then we say that is an auto-modal function, if for a real number 0 there exists Where by it converges evenly along a on each compact semi-axies 0, ∞ . Distribution $ is asymptotically homogeneous with respect to function of order % if: where the nucleus of fractional differentiation and integration with Γ % being the gamma function, and 6 being the Heaviside function [1-3, 6, 7].

Some Quasi-Asymptotics Properties of Multidimensional Distributions
We provide proof of some of the basic theorems that apply to multidimensional distributions, and their formulaic presentation can be seen in [1].
Further, if we replace with , ∈ f, the following is valid Using relations (12) and (13), we get the following relation . By the definition of quasi-asymptotics, the following applies: From here, it can be seen that 9(: , :) has no quasiasymptotics when → ∞.
Proof: Let have the quasi-asymptotics with respect to `( ) = # o( ). Then the following is valid

Conclusion
Most of the theorems proved in this paper on quasiasymptotics of distributions at a separable variable have their analog in the case of one-dimensional distributions. In [1], Vladimirov showed a theorem that does not have a onedimensional analog, the consequence of which is very important, and on the basis of which the application of separated quasi-asymptotics in to the solutions of differential equations.