Several cubature formulas on the cubic domains are derived using the discrete Fourier analysis associated with lattice tiling, as developed in [10]. The main results consist of a new derivation of the Gaussian type cu...Several cubature formulas on the cubic domains are derived using the discrete Fourier analysis associated with lattice tiling, as developed in [10]. The main results consist of a new derivation of the Gaussian type cubature for the product Chebyshev weight functions and associated interpolation polynomials on [-1,1]^2, as well as new results on [-1, 1]^3. In particular, compact formulas for the fundamental interpolation polynomials are derived, based on n3/4 ＋ O（n^2） nodes of a cubature formula on [-1,1]^3.展开更多
The new inversion formula of the Laplace transform is considered. In the formula we use only the positive values ofx SiCoLT（x） = c S（x）, L（S（x）） = T（x）, c = const., x 〉 O,from the real axis. Si is the sinus...The new inversion formula of the Laplace transform is considered. In the formula we use only the positive values ofx SiCoLT（x） = c S（x）, L（S（x）） = T（x）, c = const., x 〉 O,from the real axis. Si is the sinus transform, Co is the cosines transform of Fourier and L is the Laplace transform.展开更多
The present research will concentrate on the topic of Fourier analysis on fractals.It mainly deals with the problem of determining spectral self-affine measures on the typical fractals:the planar Sierpinski family.The...The present research will concentrate on the topic of Fourier analysis on fractals.It mainly deals with the problem of determining spectral self-affine measures on the typical fractals:the planar Sierpinski family.The previous researches on this subject have led to the problem within the possible fifteen cases.We shall show that among the fifteen cases,the nine cases correspond to the spectral measures,and reduce the remnant six cases to the three cases.Thus,for a large class of such measures,their spectrality and non-spectrality are clear.Moreover,an explicit formula for the existent spectrum of a spectral measure is obtained.We also give a concluding remark on the remnant three cases.展开更多
The classical Hardy theorem asserts that f and its Fourier transform f can not both be very rapidly decreasing. This theorem was generalized on Lie groups and also for the Fourier-Jacobi transform. However, on SU（1, ...The classical Hardy theorem asserts that f and its Fourier transform f can not both be very rapidly decreasing. This theorem was generalized on Lie groups and also for the Fourier-Jacobi transform. However, on SU（1, 1） there are infinitely many “good” functions in the sense that f and its spherical Fourier transform y both have good decay. In this paper, we shall characterize such functions on SU（1, 1）.展开更多
A program of proving the Riemann hypothesis by using the Fourier analysis on global fields is given by Connes(1999). The difficulty for realizing the program lies in proving the validity of Connes' global trace fo...A program of proving the Riemann hypothesis by using the Fourier analysis on global fields is given by Connes(1999). The difficulty for realizing the program lies in proving the validity of Connes' global trace formula on an L2-space. In this paper, a new global trace formula is obtained on a Fr′echet space which gives the Weil distribution △(h).展开更多
基金supported by NSFC Grants 10601056,10431050 and 60573023supported by National Basic Research Program grant 2005CB321702supported by NSF Grant DMS-0604056.
文摘Several cubature formulas on the cubic domains are derived using the discrete Fourier analysis associated with lattice tiling, as developed in [10]. The main results consist of a new derivation of the Gaussian type cubature for the product Chebyshev weight functions and associated interpolation polynomials on [-1,1]^2, as well as new results on [-1, 1]^3. In particular, compact formulas for the fundamental interpolation polynomials are derived, based on n3/4 ＋ O（n^2） nodes of a cubature formula on [-1,1]^3.
文摘The new inversion formula of the Laplace transform is considered. In the formula we use only the positive values ofx SiCoLT（x） = c S（x）, L（S（x）） = T（x）, c = const., x 〉 O,from the real axis. Si is the sinus transform, Co is the cosines transform of Fourier and L is the Laplace transform.
基金supported by National Natural Science Foundation of China (Grant No.11171201)
文摘The present research will concentrate on the topic of Fourier analysis on fractals.It mainly deals with the problem of determining spectral self-affine measures on the typical fractals:the planar Sierpinski family.The previous researches on this subject have led to the problem within the possible fifteen cases.We shall show that among the fifteen cases,the nine cases correspond to the spectral measures,and reduce the remnant six cases to the three cases.Thus,for a large class of such measures,their spectrality and non-spectrality are clear.Moreover,an explicit formula for the existent spectrum of a spectral measure is obtained.We also give a concluding remark on the remnant three cases.
基金Project supported by Grant-in-Aid for Scientific Research(C)of Japan(No.16540168)the National Natural Science Foundation of China(No.10371004).
文摘The classical Hardy theorem asserts that f and its Fourier transform f can not both be very rapidly decreasing. This theorem was generalized on Lie groups and also for the Fourier-Jacobi transform. However, on SU（1, 1） there are infinitely many “good” functions in the sense that f and its spherical Fourier transform y both have good decay. In this paper, we shall characterize such functions on SU（1, 1）.
文摘A program of proving the Riemann hypothesis by using the Fourier analysis on global fields is given by Connes(1999). The difficulty for realizing the program lies in proving the validity of Connes' global trace formula on an L2-space. In this paper, a new global trace formula is obtained on a Fr′echet space which gives the Weil distribution △(h).
