In this paper, some conclusions related to the prime number theorem, such as the Mertens formula are improved by the improved Abelian summation formula, and some problems such as “Dirichlet” function and “W(n)” fu...In this paper, some conclusions related to the prime number theorem, such as the Mertens formula are improved by the improved Abelian summation formula, and some problems such as “Dirichlet” function and “W(n)” function are studied.展开更多
Polynomial splines have played an important role in image processing, medical imaging and wavelet theory. Exponential splines which are of more general concept have been recently investigated.We focus on cardinal expo...Polynomial splines have played an important role in image processing, medical imaging and wavelet theory. Exponential splines which are of more general concept have been recently investigated.We focus on cardinal exponential splines and develop a method to implement the exponential B-splines which form a Riesz basis of the space of cardinal exponential splines with finite energy.展开更多
A transformation formula containing four independent bases is found by a special inversion formula and it is applied to derive a few summation formulas for basic hypergeometric series only by elementary method. The hy...A transformation formula containing four independent bases is found by a special inversion formula and it is applied to derive a few summation formulas for basic hypergeometric series only by elementary method. The hypergeometric series, the limits of those formulas are also obtained.展开更多
Let f be a fixed Maass form for SL_3(Z)with Fourier coefficients A_(f)(m,n).Let g be a Maass cusp form for SL_2(G)with Laplace eigenvalue(1/4)+k^(2) and Fourier coefficientλ_(g)(n),or a holomorphic cusp form of even ...Let f be a fixed Maass form for SL_3(Z)with Fourier coefficients A_(f)(m,n).Let g be a Maass cusp form for SL_2(G)with Laplace eigenvalue(1/4)+k^(2) and Fourier coefficientλ_(g)(n),or a holomorphic cusp form of even weight k.Denote by S_(X)(f×g,α,β)a smoothly weighted sum of A_(f)(1,n)λ_(g)(n)e(αn~β)for X 0 are fixed real numbers.The subject matter of the present paper is to prove non-trivial bounds for a sum of S_(X)(f×g,α,β)over g as k tends to∞with X.These bounds for average provide insight for the corresponding resonance barriers toward the Hypothesis S as proposed by Iwaniec,Luo,and Sarnak.展开更多
Let f be a full-level cusp form for GLm(Z) with Fourier coefficients Af(n1,..., nm-1). In this paper,an asymptotic expansion of Voronoi's summation formula for Af(n1,..., nm-1) is established. As applications of t...Let f be a full-level cusp form for GLm(Z) with Fourier coefficients Af(n1,..., nm-1). In this paper,an asymptotic expansion of Voronoi's summation formula for Af(n1,..., nm-1) is established. As applications of this formula, a smoothly weighted average of Af(n, 1,..., 1) against e(α|n|β) is proved to be rapidly decayed when 0 < β < 1/m. When β = 1/m and α equals or approaches ±mq1/mfor a positive integer q, this smooth average has a main term of the size of |Af(1,..., 1, q) + Af(1,..., 1,-q)|X1/(2m)+1/2, which is a manifestation of resonance of oscillation exhibited by the Fourier coefficients Af(n, 1,..., 1). Similar estimate is also proved for a sharp-cut sum.展开更多
文摘In this paper, some conclusions related to the prime number theorem, such as the Mertens formula are improved by the improved Abelian summation formula, and some problems such as “Dirichlet” function and “W(n)” function are studied.
文摘Polynomial splines have played an important role in image processing, medical imaging and wavelet theory. Exponential splines which are of more general concept have been recently investigated.We focus on cardinal exponential splines and develop a method to implement the exponential B-splines which form a Riesz basis of the space of cardinal exponential splines with finite energy.
文摘A transformation formula containing four independent bases is found by a special inversion formula and it is applied to derive a few summation formulas for basic hypergeometric series only by elementary method. The hypergeometric series, the limits of those formulas are also obtained.
文摘Let f be a fixed Maass form for SL_3(Z)with Fourier coefficients A_(f)(m,n).Let g be a Maass cusp form for SL_2(G)with Laplace eigenvalue(1/4)+k^(2) and Fourier coefficientλ_(g)(n),or a holomorphic cusp form of even weight k.Denote by S_(X)(f×g,α,β)a smoothly weighted sum of A_(f)(1,n)λ_(g)(n)e(αn~β)for X 0 are fixed real numbers.The subject matter of the present paper is to prove non-trivial bounds for a sum of S_(X)(f×g,α,β)over g as k tends to∞with X.These bounds for average provide insight for the corresponding resonance barriers toward the Hypothesis S as proposed by Iwaniec,Luo,and Sarnak.
基金supported by National Natural Science Foundation of China(Grant No.10971119)Program for Changjiang Scolars and Innovative Research Team in University(Grant No.1264)
文摘Let f be a full-level cusp form for GLm(Z) with Fourier coefficients Af(n1,..., nm-1). In this paper,an asymptotic expansion of Voronoi's summation formula for Af(n1,..., nm-1) is established. As applications of this formula, a smoothly weighted average of Af(n, 1,..., 1) against e(α|n|β) is proved to be rapidly decayed when 0 < β < 1/m. When β = 1/m and α equals or approaches ±mq1/mfor a positive integer q, this smooth average has a main term of the size of |Af(1,..., 1, q) + Af(1,..., 1,-q)|X1/(2m)+1/2, which is a manifestation of resonance of oscillation exhibited by the Fourier coefficients Af(n, 1,..., 1). Similar estimate is also proved for a sharp-cut sum.
