15 January 1959, Volume 9 Issue 1

    

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  ON MIKUSINSKI THEOREM OF BOUNDED MOMENTS

  Acta Mathematica Sinica, Chinese Series. 1959, 9(1): 13-16. https://doi.org/10.12386/A1959sxxb0003

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  ON THE GIBBS PHENOMENON FOR RIEMANN SUMMATION(R,k)OF GENERALIZED FOURIER SERIES

  Acta Mathematica Sinica, Chinese Series. 1959, 9(1): 28-36. https://doi.org/10.12386/A1959sxxb0005

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  We shall say that a series(?)is summable(R,k)to S if the series(?)converges for all values of h(≠O)in some neighborhood of the origin,and(?) be the Fourier series of the function f(x),and let(?)In the present note we establish the following theorems:Theorem 1.For the Cauchy principal value Fowrier seris(?)the set of all limiting values of the ratio(?)is the infinite interval:(?)Theorem 2.For the Complex-integration real generalized Fourier series(?)of the first derivative of the function(?)the set of àll limiting values of the ratiois the infinite interval:(?).Theorem 3.For the Complex-integration real generalized Fourier series(?)of the (k-1)-th derivative of the function 1/2 ctg x/2,x(k=4m+2,4m,4m+1,4m-1,m=1,2…),the set of all limiting values of the ratiois the whole number axis:(?)some integers).These theorems show that {R_h~k(x)} presents Gibbs phenomenon for generalizedFourier series,although it does not present for Lebesgue-integration Fourier series.
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  FUNCTIONS TYPICALLY REAL AND MEROMORPHIC IN A CIRCULAR RING

  Acta Mathematica Sinica, Chinese Series. 1959, 9(1): 37-50. https://doi.org/10.12386/A1959sxxb0006

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  In the circular ring R_q:q<|z|<1 let the function f(z) be meromorphic andsatisfy the condition(?)The class of all such functions f(z)will be denoted by T_r(R_q).Let the poles p_i of f(z)∈Tr(R_q)be arranged so that(?)Let the Laurent expansion for f(z)in the ring p′<|z|1)is odd,where(?)when n is even;and(?)when n(>1)is odd;(?) when n is even;(?)when n(>1)is odd.All the above estimates are precise.Theorem 6.Corresponding to a function f(z)of T_r(R_q)there exists a functiong(z)regular and typically real in R_q,such that(?)where the p_j(j=O,±1,±2,...)are the poles of f(z)in R_q.From this result we can deduce an integral representation of f(z)∈T_r(R_q).
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  ON INTRINSIC INEQUALITIES ASSOCIATED WITH CERTAIN CONTINUOUS MAPPINGS AND THEIR APPLICATION TO FIBER SPACES

  Acta Mathematica Sinica, Chinese Series. 1959, 9(1): 51-68. https://doi.org/10.12386/A1959sxxb0007

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  Between two polyhedra X and Y if there exists a map,f:X→Y,whichinduces isomorphisms into of cohomology groups of Y with respect to any givencoefficient group into the corresponding cohomology groups of X,then there mustbe some“internal”properties between X and Y.Our purpose is to express theseproperties by means of algebraic relations.Of course we may also consider theproblem if f induces such homomorphisms f:H~(Y, G)→H~n(X,G)which arealways onto.Due to the development of theory of proper isomorphisms we havehad a number of easily computable homotopy invariants associated with a polyhedron.The simplest among those invariants are block invariants.In this paper we givecomplete exposition of inequalities between block invariants of X and Y if thereis a map f:X→Y such that f:H~n(Y,G)→H~n(X,G)is always isomorphism intoor homomorphism onto for all n and G.The results are applicable to fiber theorybecause the fiber map p:E→B induces homomorphisms p:H~n(B,G)→H~n(E,G)which are always isomorphisms into if there is a section.An outline of this paperhas appeared in Science Record,Academia Sinica,1958.
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  CHARACTERIZATIONS OF CERTAIN K-SPACES

  Acta Mathematica Sinica, Chinese Series. 1959, 9(1): 69-75. https://doi.org/10.12386/A1959sxxb0008

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  A K_n is a Riemannian n-space whose curvature tensor satisfiesand(?)or(?)where κ_1 is a vector field.The purpose of this paper is to give some characteriza-tions of certain K-spaees.We prove the following theorems:1.In all K-spaces other than a flat extension of V_2,the Rieei tensor R_(lj)is given bywhere p is a scalar and λ_i is a null parallel vector field.In such a space, R_(lj)isgiven by R_(lj)=ρ(?), if and only if κ_1 is a null vector.2. κ_i is the plane generated by R_(ij)if and only if the rank of R_(ij) is 1 and κ_iis a null vector.3.The necessary and sufficient condition that a V_n be a conformally flatK-spaee is that the Riemannian curvature is given by(?)where s is a scalar and κ_i is a null parallel vector field.4.A conformally flat space is a K-space if and only if it admits a nullparallel vector field.5.A K-space is a subprojective space if and only if it is conformally flat.6.A K-space which admits n-3 independent parallel vector field is a har-monic space if and only if it is an Einstein space which admits at least twonull parallel vector fields.7.A K_(2p_1)-space which is null extension of V_p(q>1)is harmonic if andonly if it is an Einstein space.8.Every K_(2p)-space which is null extension of V_p is harmonic.
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  SUR LE CYCLE DE MONTEL-MIRANDA DANS LA THEORIE DES FAMILLES NORMALES

  Acta Mathematica Sinica, Chinese Series. 1959, 9(1): 76-86. https://doi.org/10.12386/A1959sxxb0009

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  Miranda a on 1935 éfabli le critère de normalitéque Montel lui avait signaléet qui est connu sous son nora.Pour une fonction prise de la famille de fonctionsvérifiant los conditions qu'exige ce critère,Valiron a donnépar le principe qu'avaitutiliséMiranda dans sa démonstration une inégalitéqui permet do limiter sonmodule et qui est du type de cello de Schottky.Mais le facteur1/1-rqui s'y trouveest affectéd'un exposant trop grand(équivalant à5);il est évidemment im-portant de trouver un résultat plus précis et,si c'est possible,un résultat comparableà.celui do Schottky.lei,nous obtenons par nne méthode tout àfair différente decellos des deux auteurs précédents une inégalitéde précision désirée el nous par-venons auThéozème.Soit f(z)une fonction holomorphe dans le cercle unité;si elle nes'y annule pas et si sa dérivée d'un ordre quelconque k n'y prend pas la valour 1,alors on a pour 0≤r<1 l'inégalité(?)oùH et K sent des constantes ne dépendant que do k.Ce théorème fournit aisément uno démonstration pour le critère de Miranda.Pour compléter le cycle correspondant a ce critère,que nous appelons cycle deMontel-Miranda,nous démontrons encore un théorème d'impossibilitéet un théorè-me d'extension.Pnis nous étendons cos résultats au cas o ùf no s'annule pas,mils f_(k)prendla valour 1 un nombre fini de lois