We study solutions to convolution equations for functions with discrete support in R^n, a special case being functions with support in the integer points. The Fourier transform of a solution can be extended to a holom...We study solutions to convolution equations for functions with discrete support in R^n, a special case being functions with support in the integer points. The Fourier transform of a solution can be extended to a holomorphic function in some domains in C^n, and we determine possible domains in terms of the properties of the convolution operator.展开更多
We study the class of functions called monodiffric of the second kind by Isaacs. They are discrete analogues of holomorphic functions of one or two complex variables. Discrete analogues of the Cauchy-Riemann operator,...We study the class of functions called monodiffric of the second kind by Isaacs. They are discrete analogues of holomorphic functions of one or two complex variables. Discrete analogues of the Cauchy-Riemann operator, of domains of holomorphy in one discrete variable, and of the Hartogs phenomenon in two discrete variables are investigated. Two fundamental solutions to the discrete Cauchy-Riemann equation are studied: one with support in a quadrant, the other with decay at infinity. The first is easy to construct by induction; the second is accessed via its Fourier transform.展开更多
For a tuple A = （A1, A2,..., An） of elements in a unital algebra/3 over C, its projective spectrum P（A） or p（A） is the collection of z ∈ Cn, or respectively z ∈ pn-1 such that A（z） = z1A1＋z2A2＋…＋znAn is ...For a tuple A = （A1, A2,..., An） of elements in a unital algebra/3 over C, its projective spectrum P（A） or p（A） is the collection of z ∈ Cn, or respectively z ∈ pn-1 such that A（z） = z1A1＋z2A2＋…＋znAn is not invertible in/3. The first half of this paper proves that if/3 is Banach then the resolvent set PC（A） consists of domains of holomorphy. The second half computes the projective spectrum for the generating vectors of a Clifford algebra. The Chern character of an associated kernel bundle is shown to be nontrivial.展开更多
文摘We study solutions to convolution equations for functions with discrete support in R^n, a special case being functions with support in the integer points. The Fourier transform of a solution can be extended to a holomorphic function in some domains in C^n, and we determine possible domains in terms of the properties of the convolution operator.
文摘We study the class of functions called monodiffric of the second kind by Isaacs. They are discrete analogues of holomorphic functions of one or two complex variables. Discrete analogues of the Cauchy-Riemann operator, of domains of holomorphy in one discrete variable, and of the Hartogs phenomenon in two discrete variables are investigated. Two fundamental solutions to the discrete Cauchy-Riemann equation are studied: one with support in a quadrant, the other with decay at infinity. The first is easy to construct by induction; the second is accessed via its Fourier transform.
基金supported by National Natural Science Foundation of China(Grant No.11101079)and China Scholarship Council
文摘For a tuple A = （A1, A2,..., An） of elements in a unital algebra/3 over C, its projective spectrum P（A） or p（A） is the collection of z ∈ Cn, or respectively z ∈ pn-1 such that A（z） = z1A1＋z2A2＋…＋znAn is not invertible in/3. The first half of this paper proves that if/3 is Banach then the resolvent set PC（A） consists of domains of holomorphy. The second half computes the projective spectrum for the generating vectors of a Clifford algebra. The Chern character of an associated kernel bundle is shown to be nontrivial.
