中国科学院数学与系统科学研究院期刊网

15 January 1954, Volume 4 Issue 1
    

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  • Acta Mathematica Sinica, Chinese Series. 1954, 4(1): 1-20. https://doi.org/10.12386/A1954sxxb0001
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    By a normal kernel, we mean a kernel k(x, y) satisfying the condition kk [x, y] =kk [x, y], i.ie.,Evidently real symmetric kernels, real skew-symmetric kernels,Hermitian kernels, skew-Hermitian kernels etc. are normal kernels. In this paper, we discuss the properties and solutions of the integral equations with normal kernels, especially the properties of the singular values (E.Schmidt’s characteristic values), the characteristic values of such kernels and their relations.The main results are:1. If k(x, y) is a L~2 nonmal kernel of which the set of singular values is {λ_h}, then the set of singular values of the n-th iterated kernel k~n[x, y] is {λ_h~n}.2.Let λ_h be a singular value of rank h_ρ of the kernel k(x, y), i.e., in the complete orthonormal system of adioint singular function {Φ_h(x), ψ_h(y)} there are h_ρ and only h_ρ pairs of functions Φ_(hi)(x), ψ_(hi)(y) (i=1,2,…h_ρ) with the same singular value λ_h such that (φ_(hi),φ_(hj)=δ_(ij), (ψ_(hi),ψ_(hj)=δ_(ij)(i, j = 1, 2,…, h_ρ) Then a necessary and sufficient condition for k(x,y) to be normal is that for each h, we should have where the a_(h,ij) ′s are constants such that the matrix △ = (a_(h,ij)) (i,j=1,2,…h_ρ) is unitary.3.If k(x, y) is normal,then where α_(h,ij)~((1))=α_(h,ij),(α_(h,ij)~((n)))=(α_(h,ij)~n=△~n=(α_(h,ij)~(-n)=△~(1-n). 4.For any L~2 normal kernel k(x,y) there exists a one to one correspondence between the set of characteristic values {μ_h} ahd the set of singular values {λ_h} each arranged in order of non-decreasing absolute value, such that |μ_h|=λ_h (h=1, 2, 3,…).5.Every L~2 normal kernel K(x, y) is expressible in the following form: where ~ denotes convergence in mean square, each k_h(x,y) is an algebraic kernel of the form with an unitary matrix (α_(h,ij)) (i,j = 1, 2,…h_ρ) so that and i) the rank of each characteristic value of k_h(x,y) is equal to its multiplicity; ii) corresponding to the characteristic values μ_(ih)(i = 1, 2,…, h_ρ) (equal or distinct) of k_h(x,y) there exists h_ρ linearly independent characteristic functions u_(hi)(x)(i=1,2,…, h_ρ) and h_ρ linearly independent transpose characteristic functions ν_(hi)(x)(i=1, 2,…, h_ρ) of k(x, y) so that u_(hi)(x) =μ_(hi)ku_(hi)[x],ν_(hi)(x)=μ_(hi)kν_(hi)[x],(i=1,2,…,h_ρ); iii) each of the u_(hi) (x)'s and of the ν_(hi)(x)'s is a linear combination of the φ_(hi)(x)'s and therefore is also a linear combination of the ψ_(hi)(x)′s;iv)if (u_(hi). u_(hj)) = δ_(ij) = (ν_(hi),ν_(hj)) (i, j=1, 2,…, h_ρ),then u_(hi)(x)=v_(hi)(x) (i=1,2,…h_ρ).(It is only known previously that7.If k(x,y)~k_h(x,y) is a L~2 normal kernel,let u_(hi)(x)(i=1,2,…,h_ρ)denote the complete orthonormal characteristic functions of K_h(x, y), then K(x, y) and all its iterated kernels and the solutions of the integral equation q(x) = φ(x) - λ k φ [x] are expressible in terms of the functions u_(hi)(x) (i= 1, 2.…, h_ρ) such that when λ is not a characteristic value of k(x, y), but, if λ=μ_(ts) is a characteristic value of k(x, y), then a necessary and sufficient condition for the solvability of the above integral equation is that q(x) should be orthogonal to all transpose characteristic functions of k(x, y)belonging to μ_(ts), and when this condition is satisfied, the solution is given by where stands for those values of h and i which make μ_(hi)=μ_(ts); the coefficients of u_(hi) (x) may take arbitrary values.8. Suppose k(x, y) is an L~2 normal kernel, and p(x), q(x) ε L~2, then (It is a generalization of a formula of Hilbert for symmetric kernels).As applications we give new proofs to some classical theorems, for exampie, 1) the existence theorem of characteristic values of normal kernels, and 2) the singular points of the resolvent kernel of any L~2 normal kernel are all simple poles, etc.
  • Acta Mathematica Sinica, Chinese Series. 1954, 4(1): 21-32. https://doi.org/10.12386/A1954sxxb0002
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    We say in this paper that a characteristic function (c.f.) belongs to the class (U), if it can equal to another c.f. in the neighborhood of zero without equalling it identically. A c.f. is said to belong to (U) if it does not belong to (U).Summing up the known examples of c.f.'s belonging to (U), we state the following result: If g(t) is a real-valued even function, with g(o) = 1, which is convex, non-negative and non-increasing on the positive t-axis, then g(t) is α c.f. and (unless g(t)=1) belongs to (U).It is found that the c.f.'s pertaining to a sub-class of stable laws; exp { - (1- ic sgn t) |t|~α} (A) belong to (U) The precise result is: For every 0<α<1, there exists a positive constant b, depending on α only, such that for all |c| ≤b the c.f. (A) belongs to (U).In an attempt to discover c.f.'s which belong to (U) and which are differentiable an unlimited number of times we find the following theorem helpful:Let q(x) be summable and Hermitian on the whole x-axis, and not equivalent to zero. Let the Fourier transform of q(x) vanish on an interval. Then the probability density function (p.d.f.) |q(x)| has its c.f. belonging to (U). We notice in passing that the p.d.f's |q(x)|=(1+x~2)~(-λ)(λ>1;q(x)=(1-ix)~(-λ)) and |q(x)| = |x|~(-n) | sin x|~n(n=2,3,4,…,q(x) = x~(-n) sin x)~n)are of this type.This theorem has the following consequence:Let θ(t) be a real and measurable function on (0,∞), with and let Let e~(-f(x)) be summable over (-∞,∞). Then the p.d.f, e~(-f(x)) has its c.f. belonging to (U).By taking θ(t) = α (1-α) t~(α-2) (0 < α < 1) we obtain the p.d.f. exp(-|x|~α). It can be shown that a proper choise of θ(t) will give the p.d.f.p(x)=c~(-|x|), 0≤|x|≤A;p(x)=e~(-|x|/ψ(|x|)),|x|>A, where ψ(x) is any member of the hierarchy (ln x)~λ, (ln x)(ln.ln x)~λ,…, (λ>1), and where A is determined by ψ(A)=1. It is an open question to discover a p.d.f. whose tails are even smaller than those of p(x) and whose c.f. belongs to (U).The rest of the paper is devoted to two theorems about the convergence of a sequence of c.f's in the neighborhood of zero. They are stated as follows:Theorem 6. Let {F_n(x)}be a sequence of distribution functions, with the corresponding sequence of c.f.'s {f_n(t)}. If f_n(t) tends to a limit h(t) for |t|<δ, and if the function h(t) is continuous at t=0, then the sequence { F_n(x) }is compact (i.e. every convergent sub-sequence of it converges to a distribution function).Theorem 7. Concerning any c.f. f(t) the following two statements are equivalent:1°f(t) belongs to (U);2°any sequence of c.f.'s which converges to f(t) in the neighborhood of zero must converge to it for all t.With the help of these two last theorems we are able to show that a recent theorem of Zygmund (Second Berkeley Symposium (1951, pp. 369-372) is a consequence of a. result of Marcinkiewicz (Fund. Math., 31 (1938), pp. 86-102) to the effect that a sufficient condition for a c.f. to belong to (U) is that the corresponding distribution function F(x) should satisfy the condition where r is certain positive constant.
  • Acta Mathematica Sinica, Chinese Series. 1954, 4(1): 33-79. https://doi.org/10.12386/A1954sxxb0003
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    The English translation of this article will appear in the Acta Scientia Sinica, vol. 3, no. 2, to be published in June, 1954.
  • Acta Mathematica Sinica, Chinese Series. 1954, 4(1): 81-86. https://doi.org/10.12386/A1954sxxb0004
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  • 龔昇
    Acta Mathematica Sinica, Chinese Series. 1954, 4(1): 87-103. https://doi.org/10.12386/A1954sxxb0005
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  • Acta Mathematica Sinica, Chinese Series. 1954, 4(1): 105-112. https://doi.org/10.12386/A1954sxxb0006
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    Let the k-symmetric function be regular and schlicht in the unit circle |z|<1. WriteSzego proves that all σ_n(z) are schlicht in the circle |z| <1/4.The present author establishes that all σ_n~((2))(z) are schlicht in the circle. The last result has been conjectured by, Joh. As for σ_n~((3)) (z), the problem has been studied by Ilief, and is solved completely in this note. We prove the following theorems:Theorem 1. All σ_n~((3))(z) are schticht in the circle The number can not be replaced by any large one.Comrbining the above results, we can stateTheorem 2. All σ_n~((k)) (z) are schlicht in the circle where k=1 2, 3. The number can not be replaced by any larger one.Let be regular and schlicht in the domain 1 < |ζ| < ∞. Denoting we can establish the followingTheorem 3. (i) If n>12,φ_n(ζ) is schlicht in the domain ∞>|ζ|≥(1-5lnn/n)~(-1/2).(ii) All φ_n(ζ) are schlicht in the domain ∞>|ζ|≥3/2.