In this article, the author presents some results of the isometric linear extension from some spheres in the finite dimensional space s（n）. Moreover, the author presents the representation for the onto isometric map...In this article, the author presents some results of the isometric linear extension from some spheres in the finite dimensional space s（n）. Moreover, the author presents the representation for the onto isometric mappings in the space s. It is obtained that if V is a surjective isometry from the space s onto s with V（0） = 0, then V must be real linear.展开更多
We present results on approximate solutions to the biadditive equationf（x＋y,z-w）＋f（x-y,z＋w）=2f（x,z）-2f（y,w）on a restricted domain. The proof is based on a quite recent fixed point theorem in some function s...We present results on approximate solutions to the biadditive equationf（x＋y,z-w）＋f（x-y,z＋w）=2f（x,z）-2f（y,w）on a restricted domain. The proof is based on a quite recent fixed point theorem in some function spaces. Our main results state that, under some weak natural assumptions, functions satisfying the equation approximately （in some sense） must be actually solutions to it. In this way we obtain inequalities characterizing biadditive mappings and inner product spaces. Our outcomes are connected with the well known issues of Ulam stability and hyperstability.展开更多
In this paper, Noether theory of Lagrange systems in discrete case are studied. First, we briefly overview the wellknown Noether theory of Lagrange system in the continuous case. Then, we introduce some definitions an...In this paper, Noether theory of Lagrange systems in discrete case are studied. First, we briefly overview the wellknown Noether theory of Lagrange system in the continuous case. Then, we introduce some definitions and notations, such as the operators of discrete translation to the right and the left and the operators of discrete differentiation to the right and the left, and give the conditions for the invariance of the difference functional on the uniform lattice and the non-uniform one, respectively. We also deduce the discrete analog of the Noether-type identity. Finally, the discrete analog of Noether＇s theorem is presented. An example was discussed to illustrate these results.展开更多
In this note, we discuss a class of so-called generalized sampling functions. These functions are defined to be the inverse Fourier transform of a family of piecewise constant functions that are either square integrab...In this note, we discuss a class of so-called generalized sampling functions. These functions are defined to be the inverse Fourier transform of a family of piecewise constant functions that are either square integrable or Lebegue integrable on the real number line. They are in fact the generalization of the classic sinc function. Two approaches of constructing the generalized sampling functions are reviewed. Their properties such as cardinality, orthogonality, and decaying properties are discussed. The interactions of those functions and Hilbert transformer are also discussed.展开更多
基金Supported by the National Science Foundation of China (10571090) The Research Fund for the Doctoral Program of Higher Education (20010055013)
文摘In this article, the author presents some results of the isometric linear extension from some spheres in the finite dimensional space s（n）. Moreover, the author presents the representation for the onto isometric mappings in the space s. It is obtained that if V is a surjective isometry from the space s onto s with V（0） = 0, then V must be real linear.
文摘We present results on approximate solutions to the biadditive equationf（x＋y,z-w）＋f（x-y,z＋w）=2f（x,z）-2f（y,w）on a restricted domain. The proof is based on a quite recent fixed point theorem in some function spaces. Our main results state that, under some weak natural assumptions, functions satisfying the equation approximately （in some sense） must be actually solutions to it. In this way we obtain inequalities characterizing biadditive mappings and inner product spaces. Our outcomes are connected with the well known issues of Ulam stability and hyperstability.
基金Project supported by the National Natural Science Foundation of China(Grant No.10872037)the Natural Science Foundationof Anhui Province,China(Grant No.070416226)
文摘In this paper, Noether theory of Lagrange systems in discrete case are studied. First, we briefly overview the wellknown Noether theory of Lagrange system in the continuous case. Then, we introduce some definitions and notations, such as the operators of discrete translation to the right and the left and the operators of discrete differentiation to the right and the left, and give the conditions for the invariance of the difference functional on the uniform lattice and the non-uniform one, respectively. We also deduce the discrete analog of the Noether-type identity. Finally, the discrete analog of Noether＇s theorem is presented. An example was discussed to illustrate these results.
文摘In this note, we discuss a class of so-called generalized sampling functions. These functions are defined to be the inverse Fourier transform of a family of piecewise constant functions that are either square integrable or Lebegue integrable on the real number line. They are in fact the generalization of the classic sinc function. Two approaches of constructing the generalized sampling functions are reviewed. Their properties such as cardinality, orthogonality, and decaying properties are discussed. The interactions of those functions and Hilbert transformer are also discussed.
