Fourier Transformation of L 2 loc -functions

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1 J. Math. Tokushima Univ. Vol. 49 (205), Fourier Transformation of L 2 -functions By Professor Emeritus, The University of Tokushima Home Address : Kamifukuman Hachiman-cho Tokushima , Japan address : yoshifumi@md.pikara.ne.jp (Received September 30, 205) Abstract In this paper, we study the Fourier transformation of L 2 -functions and L 2 c-functions in order to investigate the natural statistical phenomena by using the theory of natural statistical physics. Thereby we prove the structure theorems of the image spaces FL 2 and FL 2 c. We study the convolution f g of a L 2 c-function f and a L 2 -function g. Further, we characterize the al Sobolev spaces and the space of solutions of Schrödinger equations. Here assume d. These results are the English version of Ito [7], chapter Mathematics Subject Classification. Primary 42B0; Secondary 42A38, 42A85, 46E30, 46E35, 46F20. Introduction In this paper, we study the Fourier transformation of L 2 -functions and L 2 c-functions and some applications. In section, we define the Fourier transformation and the inverse Fourier transformation of L 2 -functions. We show some examples of Fourier transformation of L 2 -functions. We prove the inversion formulas of the Fourier transformation and the inverse Fourier transformation of L 2 -functions.

2 40 In section 2, using Paley-Wiener theorem for L 2 -functions, we prove the structure theorems of the function spaces L 2 and L2 c and the structure theorems of the Fourier images FL 2 and FL2 c. In section 3, we study the convolution f g of a function f in L 2 c = L 2 c(r d ) and a function g in L 2 = L2 (Rd ). In section 4, we define the al Sobolev space H s (Rd ), ( < s < ), and study its fundamental properties. In section 5, we determine the space of solutions of Schrödinger equations which describe the law of natural statistical phenomena in the space R d. This space is determined by virtue of the framework of my theory of natural statistical physics. Here I show my heartfelt gratitude to my wife Mutuko for her help of typesetting this manuscript. Fourier transformation of L 2 -functions In this section, at first we define the Fourier transformation of L 2 -functions and its fundamental properties. Let R d be the d-dimensional Euclidean space. Here assume d. Further we denote L 2 = L2 (Rd ) as usual. For the points in R d x = t (x, x 2,, x d ), p = t (p, p 2,, p d ), we define px = (p, x) = p x + p 2 x p d x d, x = x 2 + x x2 d, p = p 2 + p p2 d. Let D = D(R d ) be the space of all C -functions with compact support in R d. Here we define the Fourier transformation F by the relation (Fφ)(p) = ( φ(x)e ipx dx, (p R d ) for φ D. FD denotes the space of the Fourier image of D by the Fourier transformation F. Further, let D = D (R d ) be the space of Schwartz distributions on R d.

3 Fourier Transformation of L 2 -functions 4 Here, for the dual pair D and D of two TVS s, we denote the dual inner product of T D and φ D as < T, φ > and, for the dual pair (FD) and FD, we denote its dual inner product of S (FD) and φ FD as < S, φ >. Now assume T D. Then, since we have F φ D for φ FD, we can define a continuous linear functional S : φ < T, F φ >, (φ FD) and we have S (FD). Namely, we have the equality < S, φ >=< T, F φ >. Then we define that S is a Fourier transform of T and denote it as S = FT. This is the new definition of the Fourier transformation of D. Since a Schwartz distribution is a generalized concept of functions, we had better to define the Fourier transformation of Schwartz distributions as in the same direction as the Fourier transformation of classical functions. Thus we define the new type of Fourier transformation of Schwartz distributions. Therefore, for the Fourier transform FT FD of T D, we have the relation < FT, Fφ >=< T, φ >, (φ D). This is a generalization of Parseval s foumula for L 2 -functions. Then the Fourier transformation F is a topological isomorphism from D to FD. Thus we have the isomorphisms D = FD = (FD). Here we denote the dual mapping of the Fourier transformation F : D FD as F : (FD) D. Then we have the equality F F = the identity mapping of D. We define the Fourier transformation of f L 2 considering it as an element of D. We say that the limit in the sense of the topologies of D or FD is the limit in the sense of generalized functions. Then we give the following definition. Definition. We define the Fourier transform (Ff)(p) of f L 2 by the relation (Ff)(p) = lim R ( f(x)e ipx dx x R in the sense of generalized functions. Then we denote Ff(p) as (Ff)(p) = ( f(x)e ipx dx.

4 42 Here, when the integration domain is equal to the entire space R d, we omit the symbol of the integration domain. Let C = C(R d ) be the function space of all continuous functions on R d. Then we have the inclusion relation C L 2. Therefore, we can define the Fourier transformation of continuous functions which are not necessarily L 2 -functions considering that they are L 2 -functions. Example. We have the following equality: (F( ix) α )(p) = ( ( ix) α e ipx dx = ( δ (α) (p). Here α = (α, α 2,, α d ) denotes a multi-index of natural numbers. Especially, for α = 0 = (0, 0,, 0), we have the equality (F)(p) = ( e ipx dx = ( δ(p). Therefore, the Fourier transform of the constant function ( is equal 2π) d to the Dirac measure δ. Thereby, in general, the Fourier transform Ff of a L 2 -function f is not necessarily a L2 -function. As for this fact, my classmate Dr Kôzô Yabuta gives me this advice. Remark. The L 2 -function which determines the natural statistical distribution of a certain physical system must be a solution of a certain Schrödinger equation. In general, there is no non-constant polynomial solution of a certain Schrödinger equation. Therefore, in order to determine a natural statistical distribution, we have not to consider the Fourier transformation of non-constant polynomial functions. Now we give some examples of Fourier transforms of continuous functions. Example.2 Assume < p, q <. Then we have the following () and (2): () π sin qxe ipx dx = (δ(p q) δ(p + q)). 2π 2 i (2) π cos qxe ipx dx = (δ(p q) + δ(p + q)). 2π 2

5 Fourier Transformation of L 2 -functions 43 In the following Example.3 Example.5, the convergence of series is considered to be the convergence in the sense of generalized functions. Example.3 The Fourier transform ˆf(p) of Riemann s function is equal to ˆf(p) = π 2 i f(x) = n= n= sin(n 2 x) n 2, ( < x < ) n 2 (δ(p n2 ) δ(p + n 2 )), ( < p < ). Example.4 We assume that two constants a, b satisfy the following conditions (i) (iii): (i) 0 < a <. (ii) b is a odd number. (iii) We have ab > π. Then the Fourier transform ˆf(p) of Weierstrass function is equal to ˆf(p) = f(x) = π 2 a n cos(b n πx), ( < x < ) n= (δ(p b n π) + δ(p + b n π)), ( < p < ). n= Example.5 Assume that a is an even number. transform ˆf(p) of Cellérier function is equal to ˆf(p) = π 2 i f(x) = n= n= sin(a n x) a n, ( < x < ) Then the Fourier a n (δ(p an ) δ(p + a n )), ( < p < ). Example.6 Assume d. The constant function belongs to L 2 = L 2 (Rd ). For R > 0, we put χ R (x) = χ x R (x). Then we have χ R L 2 and we have χ R, (R )

6 44 in the topology of L 2 -convergence. Thus we have χ R, (R ) in the topology of D. Then we have, for R, ˆχ R (p) = ( χ R (x)e ipx dx ( e ipx dx = ˆ(p) = ( δ(p) in the topology of FD. Example.7 For n, we put Then we have χ n L 2 and we have χ n (x) = χ [ n, n] (x), (x R). χ n, (n ) in the topology of L 2 -convergence. Thus we have χ n, (n ) in the topology of D. Then we have, for n, ˆχ n ((p) = χ n (x)e ipx dx e ipx dx = ˆ(p) = 2πδ(p) 2π 2π in the topology of FD. Example.8 We have in the topology of FD. Proof We have the equality n 2π n sin pn δ(p), (n ) π p e ipx dx = ip 2π (eipn e ipn ) = Thus we have the conclusion by virtue of Example.7.// 2 π sin pn. p Example.9 Assume d. Let n = (n, n 2,, n d ) be a multi-index of positive natural numbers. We denote n = n + n n d. By using the notation of Example.7, we denote χ n (x) = χ n (x )χ n2 (x 2 ) χ nd (x d ), (x R d ),

7 Fourier Transformation of L 2 -functions 45 Then we have in the topology of FD. ˆχ n (p) = ˆχ n (p )ˆχ n2 (p 2 ) ˆχ nd (p d ), (p R d ). ˆχ(p) ( δ(p), ( n ) Proof By virtue of Example,7, because we have ˆχ nj (p j ) 2πδ(p j ) for j d, we have the conclusion. // Theorem. We use the same notation as Example.9. Then, for we denote χ n (x) = χ n (x )χ n2 (x 2 ) χ nd (x d ), (x R d ), ˆχ n (p) = ˆχ n (p )ˆχ n2 (p 2 ) ˆχ nd (p d ), (p R d ). For f(x) L 2, we put f n(x) = χ n (x)f(x). Then we have f n (x) L 2. Now, when we consider that f n and f are elements of D, we denote their Fourier transformations as Ff n = ˆf n and Ff = ˆf. Then we have in the topology of FD. Proof When n, we have ˆf n ˆf, ( n ) f n (x) f(x), (x R d ) in the topology of L 2. Therefore, when n, we have f n f in the topology of D. Since we have f n = χ n f, we have the equality ˆf n = (χ n f) = ( ˆχ n f in FD. Here the symbol denotes the convolution. By virtue of Example.9, we have ˆχ n ( δ, ( n ). Thus, when n, we have fˆ n = ( 2π) χˆ d n ˆf δ ˆf = ˆf

8 46 in the topology of FD. // When we use the notation in Theorem., we have f ˆ n L 2 and ˆf n (p) = ( f n (x)e ipx dx. Therefore we have the equality lim n ( f n (x)e ipx dx = ˆf(p) in FD. In this sense, we use the notation ˆf(p) = ( f(x)e ipx dx for ˆf(p) FD. Here we consider this integral in the sense of convergence in the topology of FD. In this case, we say that this integral converges in the sense of generalized functions. Similarly, we define the Fourier inverse transformation as follows. Definition.2(Fourier inverse transformation) We define the Fourier inverse transformation of g(p) L 2 by the relation (F g)(x) = lim R ( g(p)e ipx dp p R in the sense of generalized functions. We denote (F g)(x) as (F g)(x) = ( g(p)e ipx dp. Theorem.2 Let α = (α, α 2,, α d ) be a multi-index of natural numbers. Assume that f(x) L 2 and Dα f(x) L 2 hold. Then we have the following () and (2): () F(( ix) α f)(p) = D α (Ff)(p). (2) F(D α f)(p) = (ip) α (Ff)(p). In Theorem.2, the symbols x α and D α etc. are the same as usually used. Namely D α f means a L 2 -derivatives, and Dα (Ff) means, in general, a partial derivative of Ff in FD and so on.

9 Fourier Transformation of L 2 -functions 47 Next we prove the Fourier inversion formula. Now we assume f L 2. Then, since we have we have f R (x) L 2, (0 < R < ), (Ff R )(p) L 2, (0 < R < ), Ff R = f R, (0 < R < ), F Ff R (x) = f R (x), (0 < R < ). Then, since we have f R (x) f(x), (R ) is the sense of generalized functions, we have the equality F Ff = f. Therefore we have the following inversion formula. Theorem.3(Inversion formula) For f(x) L 2, we have the following inversion formula f(x) = lim R ( (Ff R )(p)e ipx dp = ( e ipx dp f(y)e ipy dy. Here the integral converges in the sense of generalized functions. Namely we have F Ff = f. Similarly, for g(p) L 2, we denote the restriction of g to the closed ball p T as g T. Then we have F g T = g T, (0 < T < ), FF g T (p) = g T (p), (0 < T < ). Then, in the sense of generalized functions, we have Thus we have the equality g T (p) g(p), (T ). FF g(p) = g(p) in the sense of generalized functions. Therefore we have the following inversion formula. Theorem.4 (Inversion formula) For g L 2, we have the following inversion formula g(p) = ( (F g)(x)e ipx dx = ( e ipx dx g(q)e iqx dq.

10 48 Here the integral converges in the sense of generalized functions. Namely we have the equality FF g = g. Theorem.5 For f L 2, we have the equalities: F 2 f(x) = f( x), F 4 f(x) = f(x). 2 Structure theorems In this section, using Paley-Wiener theorem for L 2 -functions, we study the structure theorems of the function spaces L 2 and L2 c and the structure theorems of the Fourier images FL 2 and FL2 c. Now we choose an exhausting sequence {K j } of compact sets in R d which satisfies the following conditions (i) and (ii): (i) K K 2 R d, R d = K j. j= (ii) K j = cl(int(k j )), K j int(k j+ ), (j =, 2, 3, ). Then we denote the projective limit of projective system {L 2 (K j )} of Hilbert spaces as lim L2 (K j ). Then we have the isomorphism L 2 = lim L 2 (K j ) as TVS s. Here, since, for each j, the restriction mapping L 2 (K j+ ) L 2 (K j ) is a weakly compact mapping, L 2 is a FS -space. Further, because the system {L 2 (K j )} of Hilbert spaces can be considered as an inductive system, we denote the inductive limit as Then we have the isomorphism lim L2 (K j ). L 2 c = lim L 2 (K j ) as TVS s. Here L 2 c denotes the TVS of all L 2 -functions with compact support. Then, since, for each j, the inclusion mapping L 2 (K j ) L 2 (K j+ ) is a weakly compact mapping, L 2 c is a DFS -space.

11 Fourier Transformation of L 2 -functions 49 Since L 2 (K j ) is a self-dual space, we have the isomorphism L 2 = (L 2 c) as TVS s. Here(L 2 c) denotes the dual space of L 2 c and we define the dual inner product of f L 2 and g L2 c by the equality < f, g >= f(x)g(x)dx. Here the dual inner product is a bilinear functional which defines the duality relation of the pair of two TVS s L 2 and L2 c. Then, because we have the inclusion relation L 2 c L 2, we define the Fourier transformation of a L 2 c-function g(x) by using the Fourier transformation of L 2 - functions Fg(p) = ( g(x)e ipx dx. Further we define the Fourier transformation of a L 2 -function f by the relation Ff(p) = lim j ( f(x)e ipx dx K j in the sense of generalized functions in D and FD. By virtue of the definition of the Fourier transformation of f L 2, we have the equality < Ff, Fg >=< f, g > for any g D. Since a L 2 c-function g has the compact support, there exists some K j such that supp(g) K j holds by the definition of {K j }. Therefore, for an arbitrary k j, we have the equalities < f Kk, g >= f Kk (x)g(x)dx = f(x)g(x)dx =< f, g >. K k K j Here f Kk (x) denotes the image of f(x) L 2 by the restriction mapping L 2 L2 (K k ). Since we have the equality Ff Kk (p)fg( p)dp = f Kk (x)g(x)dx by virtue of Parseval s formula, we have the equality lim Ff Kk (p)fg( p)dp = lim f Kk (x)g(x)dx k k

12 50 = f Kj (x)g(x)dx = Ff Kj (p)fg( p)dp. Especially, supposing that we have D Kj L 2 (K j ), g D Kj, we have the equality Ff(p)Fg( p)dp = f(x)g(x)dx. We can choose a compact set K j arbitrarily. Thus, if we consider that g D Kj holds for an arbitrary D Kj, we have the equality in the above for an arbitrary g D. Then, because the dual inner product < f, g >= f(x)g(x)dx is defined for an arbitrary f L 2 and g L2 c, we have the equality < Ff, Fg >= Ff(p)Fg( p)dp = f(x)g(x)dx =< f, g > for an arbitrary f L 2 and an arbitrary g L2 c. Now we choose one exhausting sequence {K j } of compact sets in R d as in the above. Then, for the sequence we have the isomorphisms Further we have the isomorphisms L 2 (K ) L 2 (K 2 ), L 2 c = lim L 2 (K j ), L 2 = lim L 2 (K j ). L 2 c = L 2 (K j ), L 2 = L 2 (K j ). j= Then we have the isomorphisms Further, for the sequence j= FL 2 (K j ) = L 2 (K j ), (j =, 2, 3, ). FL 2 (K ) FL 2 (K 2 ),

13 Fourier Transformation of L 2 -functions 5 we have the isomorphisms FL 2 c = lim FL 2 (K j ) = lim L 2 (K j ) = L 2 c, FL 2 = lim FL 2 (K j ) = lim L 2 (K j ) = L 2. Then we have the relations FL 2 FD, FL 2 L 2. Therefore we have the following theorem. Theorem 2. We use the notation in the above. following isomorphisms () (4): () L 2 c = lim L 2 (K j ) = L 2 (K j ). j= Then we have the (2) FL 2 c = lim FL 2 (K j ). (3) FL 2 (K j ) = L 2 (K j ), (j =, 2, 3, ). (4) FL 2 c = L 2 c, FL 2 c L 2, L 2 c L 2. Further we have the following theorem. Theorem 2.2 We use the notation in the above. following isomorphisms () (3) and the relation (4): () L 2 = lim L 2 (K j ) = L 2 (K j ) = (L 2 c). j= Then we have the (2) FL 2 = lim FL 2 (K j ). (3) FL 2 = L 2. (4) FL 2 FD, FL 2 L2, L2 D.

14 52 3 Convolution In this section, we study the convolution f g of a function f in L 2 c = L 2 c(r d ) and a function g in L 2 = L2 (Rd ). Here assume d. We define the convolution f g of f L 2 c and g L by the relation (f g)(x) = f(x y)g(y)dy. Then we have the equality f(x y)g(y)dy = g(x y)f(y)dy. Therefore we have the following theorem. Theorem 3. For f L 2 c and g L 2, we have f g L2. Further we have the relation f g = g f. Theorem 3.2 Let α = (α, α 2,, α d ) be a multi-index of natural numbers. Then, for f L 2 c and g L 2, we have the equality D α (f g) = (D α f) g = f (D α g). Here the partial derivatives are considered in the sense of topologies of L 2 c and L 2. Corollary 3. Assume f L 2 c. Then the linear transformation of L 2 defined by the convolution is continuous in L 2. T f : g f g, (g L 2 ) Now assume that {g n } is a sequence of L 2 -functions and it converges to g L 2 in the topology of L2. Namely, assume that g n g, (n ) in the topology of L 2. Then we have T f (g n ) T f (g), (n ). Corollary 3.2 Assume g L 2. Then the linear mapping T g = f g, (f L 2 c) defined by the convolution is a continuous linear mapping from L 2 c into L 2.

15 Fourier Transformation of L 2 -functions 53 Therefore, if a sequence {f n } of functions in L 2 c convergences to f L 2 c in the topology of L 2 c, we have T g (f n ) T g (f), (n ). Here the convolution of a function f in L 2 c and a function g in L 2 is a separately continuous bilinear mapping L 2 c L 2 L2. Theorem 3.3 Assume f L 2 c and g L 2. Then we have F(f g) = ( F(f)F(g). 4 Characterization of the al Sobolev spaces In this section, we define the al Sobolev space H s (Rd ) and study its fundamental properties. As for the precise concerning these results, we refer to Ito [], [5], [6], [7]. This problem is the characterization of the al Sobolev space by using the Fourier transformation. For a real number s, we define L 2, s = L 2, s (R d ) to be the Hilbert space of all complex valued measurable functions f which satisfies the condition ( + x 2 ) s f(x) 2 dx <. Assume that s is a real number and F is the Fourier transformation of L 2 = L 2 (R d ). Then we define the Solobev space H s = H s (R d ) to be the Hilbert space H s (R d ) = {f L 2 (R d ); Ff L 2, s (R d )}. Especially when m is a natural number, the Solobev space H m = H m (R d ) is equal to the Sobolev space W m, 2 (R d ) = {f L 2 (R d ); D α f L 2, α m}. Here, for a multi-index α = (α, α 2,, α d ) of natural numbers, the simbol ( ) D α α ( ) αd f = f x x d denotes the L 2 -derivative. Further we put α = α + α α d. Then, for a real number s, the al Sobolev space H s = Hs (Rd ) is assumed to be the TVS of all complex valued measurable functions f(x) on

16 54 R d such that, for an arbitrary compact set K in R d, the function f K (x) = f(x)χ K (x) belongs to H s. Here χ K (x) denotes the characteristic function of the set K. Now, for a real number s, we define the vector space L 2, s L 2, s L 2, s = L2, s (Rd ) = { f L 2 ; ( + x 2 ) s f(x) L 2 }. by the condition is equal to the vector space of all complex valued measurable functions f on R d which satisfy the condition ( + x 2 ) s f(x) 2 dx < K for an arbitrary compact set K in R d. We define the seminorm f 2, s, K by the relation f 2, s, K = { K( + x 2 ) s f(x) 2 dx } /2. Here K denotes a compact set K in R d. Then the topology of L 2, s is defined by the system of seminorms { 2, s, K ; K is a compact set in R d }. Thereby L 2, s is a Fréchet space. For f L 2, s, we have the inequalities B K f(x) 2 dx ( + x 2 ) s f(x) 2 dx C K K K K f(x) 2 dx. Here B K and C K are two positive constants depending on K. Therefore, for an arbitrary real number s, we have the equality L 2, s = L2 as the sets of functions. Then L 2, s is equal to the LCV L2 endowed with the topology defined by the system of seminorms { 2, s, K ; K is a compact set in R d }. Now, for f L 2 and a compact set K in Rd, we define the seminorm f K of L 2 by the relation f K = ( f(x) 2 dx ) /2. K Thereby L 2 s is a Fréchet space. Here, because the topologies of L2, are equivalent, L 2, s and L2 are topologically isomorphic. Then we have the following theorem. Theorem 4. For a real number s, we have the equality H s = H(R s d ) = { f L 2 ; Ff L 2, s }. and L2

17 Fourier Transformation of L 2 -functions 55 Here Ff L 2, s is the Fourier transform of f L2. Since, in general, we happen to have Ff L 2 for f L2 s, Ff L2, is one restriction condition. In fact, though we have L 2, we have Hs. Especially, for a natural number m, we have the equalities H m = H m (R d ) = W m, 2 (R d ) = { f L 2 ; D α f L 2, α m }. Here, for a multi-index α = (α, α 2,, α d ) of natural numbers, D α f means the L 2 -derivatives as same as in the case of L2. Further we put α = α + α α d. Then, for f H m and an arbitrary compact set K in Rd, we define the seminorm f m, K of H m by the relation f m, K = ( D α f 2 ) /2 K. α m Thereby, the topology of H m is defined by the system of seminorms { m, K ; K is a compact set in R d }. Therefore H m is a Fréchet space. Especially we remark that H 0 W 0, 2 = L 2 holds. In the sequel, we denote H 0 as H. Then H is the closed subspace of L 2 Ṡince, for all real number s, we have L 2, s = L2 as sets of functions, we have, for all real number s H s = H similarly. Therefore, for the Fourier transform Ff(p) L 2 of f(x) H, we have the equality Ff(p) = lim R ( f(x)e ipx dx x R in the topology of L 2. Further, since we have Ff(p) L2 for f(x) Hs, we have the Fourier inversion formula f(x) = ( e ipx dp f(y)e ipy dy in the topology of L 2.

18 56 Remark 4. Since the conditions of definitions H s and H s are given by the integral estimates of the classical functions, we remark that H s and H s are some classes of classical functions and are characterized without using the theory of distributions. Further, the fact that L 2, s and Hs are different TVS s for some different real number s means that the definitions of the topologies of those TVS s are different 5 Characterization of solutions of Schrödinger equations In this section, we determine the space of solutions of Schrödinger equations which describe the law of natural statistical phenomena in the space R d. Here assume d. Now assume that, for p R d, ψ p (x) L 2 satisfies the condition ˆψ p (q) = δ p (q). Then we have ˆδ p (x) = ψ p (x). When we denote the subspace of D spanned by {ψ p, δ p ; p R d } as V {ψ p, δ p ; p R d }, we define the subspace N of D by the relation N = H V {ψ p, δ p ; p R d }. Then we have the inclusion relation L 2 H 2, L 2 N. The function space C 0 = C 0 (R d ) is the TVS of all continuous functions with compact support in R d. We say that a continuous linear functional µ on C 0 as a Radon measure on R d. The TVS of all Radon measures on R d is equal to the dual space (C 0 ). Then we have the inclusion relation N (C 0 ) D. When f N and f δ p, (p R d ), we define µ (C 0 ) by the relation µ(φ) = f(x)φ(x)dx, (φ C 0 ). Then, if we denote µ = µ f, the correspondence f µ f is one to one correspondence. Thereby, when f N and f δ p, (p R d ), we can identify f and µ = µ f (C 0 ).

19 Fourier Transformation of L 2 -functions 57 When f N and f H, we have f L 2 or f L 2. Therefore, we have Ff L 2 or Ff L 2 respectively. Further, because we have Fψ p = δ p, Fδ p = ψ p, (p R d ), the Fourier transformation F is the isomorphism Hence we have F : N FN. FN = N, FN L 2 + V {ψ p, δ p ; p R d }. Thus we have the following theorem. Theorem 5. We use the notations in the above. We define the subspace N of D by the relation N = H V {ψ p, δ p ; p R d }. Denoting the Fourier transformation of D as F, we have the following () and (2): () We have the isomorphism (2) We have the inclusion relation FN = N. FN L 2 + V {ψ p, δ p ; p R d }. Then, we consider that, in the space R d, (d ), the function in N which is a solution of a Schrödinger equation determines the natural statistical distribution state for the natural statistical phenomenon of some physical system. This fact is a restriction condition for a solution of a Schrödinger equation in R d. This is a restriction condition in order that a solution of a Schrödinger equation satisfies the condition postulated for the law of natural statistical physics. As for the laws of natural statistical physics, we refer to Ito [8]. References [] Y.Ito, Linear Algebra, Kyôritu, 987, (in Japanese). [2], Analysis, Vol., Science House, 99, (in Japanese).

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