共查询到20条相似文献,搜索用时 703 毫秒
1.
We generalize the standard Poisson summation formula for lattices so that it operates on the level of theta series, allowing
us to introduce noninteger dimension parameters (using the dimensionally continued Fourier transform). When combined with
one of the proofs of the Jacobi imaginary transformation of theta functions that does not use the Poisson summation formula,
our proof of this generalized Poisson summation formula also provides a new proof of the standard Poisson summation formula
for dimensions greater than 2 (with appropriate hypotheses on the function being summed). In general, our methods work to
establish the (Voronoi) summation formulae associated with functions satisfying (modular) transformations of the Jacobi imaginary
type by means of a density argument (as opposed to the usual Mellin transform approach). In particular, we construct a family
of generalized theta series from Jacobi theta functions from which these summation formulae can be obtained. This family contains
several families of modular forms, but is significantly more general than any of them. Our result also relaxes several of
the hypotheses in the standard statements of these summation formulae. The density result we prove for Gaussians in the Schwartz
space may be of independent interest. 相似文献
2.
F. Štampach 《Linear algebra and its applications》2011,434(5):1336-1353
A function f with simple and nice algebraic properties is defined on a subset of the space of complex sequences. Some special functions are expressible in terms of f, first of all the Bessel functions of first kind. A compact formula in terms of the function f is given for the determinant of a Jacobi matrix. Further we focus on the particular class of Jacobi matrices of odd dimension whose parallels to the diagonal are constant and whose diagonal depends linearly on the index. A formula is derived for the characteristic function. Yet another formula is presented in which the characteristic function is expressed in terms of the function f in a simple and compact manner. A special basis is constructed in which the Jacobi matrix becomes a sum of a diagonal matrix and a rank-one matrix operator. A vector-valued function on the complex plain is constructed having the property that its values on spectral points of the Jacobi matrix are equal to corresponding eigenvectors. 相似文献
3.
The main theorem proved in this paper consists of a multiplicative distribution formula for the Jacobi forms in two variables associated to Klein forms. This gives stronger versions of distribution formulae appearing in the literature. Indeed, as a first consequence of the main theorem, we deduce an optional proof of the distribution formula true for any elliptic function first found by Kubert and as a second consequence, we prove an ameliorated distribution formula for a certain zeta function previously treated by Coates, Kubert and Robert. Moreover, our main theorem provides the exact root of unity appearing in the distribution formula of Jarvis and Wildeshaus, a fact which could be useful in the K-theory of elliptic curves or more precisely, in the investigation of the elliptic analogue of Zagier's conjecture linking regulators and polylogarithms. 相似文献
4.
Masatoshi Ikeuchi Hiroshi Kobayashi Hideo Sawami Hiroshi Niki 《Journal of Computational and Applied Mathematics》1979,5(4):247-258
The spectral radius of the Jacobi iteration matrix plays an important role to estimate the optimum relaxation factor, when the successive overrelaxation (SOR) method is used for solving a linear system. The specific systems are finite difference forms of the Laplace equation satisfied on a rectanglar region with two different media. Though the potential function for the inhomogeneous closed region is continuous, the first order derivative is not continuous. So this requires internal boundary conditions or interface conditions. In this paper, the spectral radius of the Jacobi iteration matrix for the inhomogeneous rectangular region is formulated and the approximation for the explicit formula, suitable for the computation of the spectral radius, is deduced. It is also found by the proposed formula that the spectral radius and the optimum relaxation factor rigorously depend on the inhomogeneity or the internal boundary conditions in the closed region, and especially vary with the position of the internal boundary. These findings are also confirmed by the numerical results of the power method.The stationary iterative method using the proposed formula for calculating estimates of the spectral radius of the Jacobi iteration matrix is compared with Carré's method, Kulstrud's method and the stationary iterative method using Frankel's theoretical formula, all for the case of some numerical models with two different media. According to the results our stationary iterative method gives the best results ffor the estimate of the spectral radius of the Jacobi iteration matrix, for the required number of iterations to calculate solutions, and for the accuracy of the solutions.As a numerical example the microstrip transmission line is taken, the propating mode of which can be approximated by a TEM mode. The cross section includes inhomogeneous media and a strip conductor. Upper and lower bounds of the spectral radius of the Jacobi iteration matrix are estimated. Our method using these estimates is also compared with the other methods. The upper bound of the spectral radius of the Jacobi iteration matrix for more general closed regions with two different media might be given by the proposed formula. 相似文献
5.
Jürg Kramer 《manuscripta mathematica》1986,54(3):279-322
We give a characterisation of Jacobi forms by classical modular forms from which we obtain dimension formulas for the spaces of Jacobi forms in certain cases. Then we consider the ordinary theta series to the quaternary quadratic forms of discriminant q2 (q an odd prime) representing 2; these possess a natural continuation to Jacobi forms for which we give a sufficient condition of linear independence. If this condition is fulfilled and if there is no cusp form of weight 4 with respect to o(q) which vanishes at the cusp 0 with a certain order then the classical theta series are also linear independent. 相似文献
6.
Tuncay Aktosun Abdon E. Choque‐Rivero 《Mathematical Methods in the Applied Sciences》2017,40(6):1964-1972
The Jacobi system on a full‐line lattice is considered when it contains additional weight factors. A factorization formula is derived expressing the scattering from such a generalized Jacobi system in terms of the scattering from its fragments. This is performed by writing the transition matrix for the generalized Jacobi system as an ordered matrix product of the transition matrices corresponding to its fragments. The resulting factorization formula resembles the factorization formula for the Schrödinger equation on the full line. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
7.
YoungJu Choie Olav K. Richter 《Proceedings of the American Mathematical Society》2007,135(8):2309-2315
We determine a class of functions spanned by theta series of higher degree. We give two applications: A simple proof of the inversion formula of such theta series and a classification of skew-holomorphic Jacobi forms.
8.
Soumya Das 《Archiv der Mathematik》2010,95(5):423-437
We introduce a certain differential (heat) operator on the space of Hermitian Jacobi forms of degree 1, show its commutation
with certain Hecke operators and use it to construct a map from elliptic cusp forms to Hermitian Jacobi cusp forms. We construct
Hermitian Jacobi forms as the image of the tensor product of two copies of Jacobi forms and also from the differentiation
of the variables. We determine the number of Fourier coefficients that determine a Hermitian Jacobi form and use the differential
operator to embed a certain subspace of Hermitian Jacobi forms into a direct sum of modular forms for the full modular group. 相似文献
9.
H. Kim 《Archiv der Mathematik》2002,79(3):208-215
We study multilinear differential operators on a space of Hermitian Jacobi forms as well as on a space of Hermitian modular forms of degree 2. First we define a heat operator and construct multilinear differential operators on a space of Hermitian Jacobi forms of degree 2. As a special case of these operators, we also study Rankin-Cohen type differential operators on a space of Hermitian Jacobi forms. And we construct multilinear differential operators on a space of Hermitian modular forms of degree 2 as an application of multilinear differential operators on Hermitian Jacobi forms. 相似文献
10.
Christian Batut. 《Mathematics of Computation》2001,70(233):395-417
From the classical Voronoi algorithm, we derive an algorithm to classify quadratic positive definite forms by their minimal vectors; we define some new invariants for a class, for which several conjectures are proposed. Applying the algorithm to dimension 5 we obtain the table of the 136 classes in this dimension, we enumerate the 118 eutactic quintic forms, and we verify the Ash formula.
11.
Hang Xue 《Israel Journal of Mathematics》2016,212(2):547-633
In this paper, we propose a conjectural formula, relating the Fourier–Jacobi periods of automorphic forms on U(n)×U(n) and the central value of some Rankin–Selberg L-function. This can be viewed as a refinement of the Gan–Gross–Prasad conjecture for unitary groups. We then use the relative trace formula technique to prove this conjectural formula in some cases. We also have give applications to the conjecture of Ichino–Ikeda and N. Harris on the Bessel period of automorphic forms on unitary groups. 相似文献
12.
F. Cléry V. Gritsenko 《Abhandlungen aus dem Mathematischen Seminar der Universit?t Hamburg》2013,83(2):187-217
In this paper, we study Jacobi forms of half-integral index for any even integral positive definite lattice L (classical Jacobi forms from the book of Eichler and Zagier correspond to the lattice A 1=〈2〉). We construct Jacobi forms of singular (respectively, critical) weight in all dimensions n≥8 (respectively, n≥9). We give the Jacobi lifting for Jacobi forms of half-integral indices and we obtain an additive lifting construction of new reflective modular forms which are natural generalizations to O(2,n) (n=4, 5 and 6) of the Igusa modular form Δ 5. 相似文献
13.
We prove an addition formula for Jacobi functions \(\varphi _\lambda ^{\left( {\alpha ,\beta } \right)} \left( {\alpha \geqq \beta \geqq - \tfrac{1}{2}} \right)\) analogous to the known addition formula for Jacobi polynomials. We exploit the positivity of the coefficients in the addition formula by giving the following application. We prove that the product of two Jacobi functions of the same argument has a nonnegative Fourier-Jacobi transform. This implies that the convolution structure associated to the inverse Fourier-Jacobi transform is positive. 相似文献
14.
Kohnen introduced a limit process for Siegel modular forms that produces Jacobi forms. He asked if there is a space of real-analytic Siegel modular forms such that skew-holomorphic Jacobi forms arise via this limit process. In this paper, we initiate the study of harmonic skew-Maass–Jacobi forms and harmonic Siegel–Maass forms. We improve a result of Maass on the Fourier coefficients of harmonic Siegel–Maass forms, which allows us to establish a connection to harmonic skew-Maass–Jacobi forms. In particular, we answer Kohnen’s question in the affirmative. 相似文献
15.
Shayne Waldron 《Constructive Approximation》2011,33(3):405-424
Given a suitable weight on ℝ
d
, there exist many (recursive) three-term recurrence relations for the corresponding multivariate orthogonal polynomials.
In principle, these can be obtained by calculating pseudoinverses of a sequence of matrices. Here we give an explicit
recursive three-term recurrence for the multivariate Jacobi polynomials on a simplex. This formula was obtained by seeking the best
possible three-term recurrence. It defines corresponding linear maps, which have the same symmetries as the spaces of Jacobi
polynomials on which they are defined. The key idea behind this formula is that some Jacobi polynomials on a simplex can be
viewed as univariate Jacobi polynomials, and for these the recurrence reduces to the univariate three-term recurrence. 相似文献
16.
Zhang Peixuan 《分析论及其应用》1994,10(1):47-57
In this paper, we discuss the relation between the partial sums of Jacobi series on an elliptic region and the corresponding
partial sums of Fourier series. From this we derive a precise approximation formula by the partial sums of Jacobi series on
an elliptic region. 相似文献
17.
We prove an addition formula for Jacobi functions
analogous to the known addition formula for Jacobi polynomials. We exploit the positivity of the coefficients in the addition
formula by giving the following application. We prove that the product of two Jacobi functions of the same argument has a
nonnegative Fourier-Jacobi transform. This implies that the convolution structure associated to the inverse Fourier-Jacobi
transform is positive.
The first author was partially supported by the Danish Natural Science Research Council. 相似文献
18.
Eichler and Zagier developed a theory of Jacobi forms to understand and extend Maass' work on the Saito‐Kurokawa conjecture. Later Skoruppa introduced skew‐holomorphic Jacobi forms, which play an important role in understanding liftings of modular forms and Jacobi forms. In this paper, we explain a relation between Jacobi forms and skew‐holomorphic Jacobi forms in terms of a group cohomology. More precisely, we introduce an isomorphism from the direct sum of the space of Jacobi cusp forms on and the space of skew‐holomorphic Jacobi cusp forms on with the same half‐integral weight to the Eichler cohomology group of with a coefficient module coming from polynomials. 相似文献
19.
Mathieu moonshine attaches a weak Jacobi form of weight zero and index one to each conjugacy class of the largest sporadic simple group of Mathieu. We introduce a modification of this assignment, whereby weak Jacobi forms are replaced by semi-holomorphic Maass–Jacobi forms of weight one and index two. We prove the convergence of some Maass–Jacobi Poincaré series of weight one, and then use these to characterize the semi-holomorphic Maass–Jacobi forms arising from the largest Mathieu group. 相似文献
20.
M. Manickam B. Ramakrishnan 《Transactions of the American Mathematical Society》2000,352(6):2601-2617
In this paper, we set up Shimura and Shintani correspondences between Jacobi forms and modular forms of integral weight for arbitrary level and character, and generalize the Eichler-Zagier isomorphism between Jacobi forms and modular forms of half-integral weight to higher levels. Using this together with the known results, we get a strong multiplicity 1 theorem in certain cases for both Jacobi cusp newforms and half-integral weight cusp newforms. As a consequence, we get, among other results, the explicit Waldspurger theorem.