- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 06:04:18
3
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=RMI&vol=9&iss=1&update_since=2024-03-29
Revista Matemática Iberoamericana
Rev. Mat. Iberoamericana
RMI
0213-2230
2235-0616
General
10.4171/RMI
http://www.ems-ph.org/doi/10.4171/RMI
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2012)
9
1993
1
Semigroup Commutators under differences, II
Pierre
Charollois
Université Paris 6 Pierre et Marie Curie, PARIS CEDEX 05, FRANCE
General
1
50
10.4171/RMI/132
http://www.ems-ph.org/doi/10.4171/RMI/132
Non-separable bidimensional wavelet bases
Albert
Cohen
Université Pierre et Marie Curie, PARIS, FRANCE
Ingrid
Daubechies
Duke University, DURHAM, UNITED STATES
We build orthonormal and biorthogonal wavelet bases of $L^2(\mathbb R^2)$ with dilation matrices of determinant 2. As for the one dimensional case, our construction uses a scaling function which solves a two-scale difference equation associated to a FIR filter. Our wavelets are generated from a single cornpactly supported mother function. However, the regularity of these functions cannot be derived by the same approach as in the one dimensional case. We review existing techniques to evaluate the regularity of wavelets, and we introduce new methods which allow to estimate the smoothness of non-separable wavelets and scaling functions in the most general situations. We illustrate these with several examples.
General
51
137
10.4171/RMI/133
http://www.ems-ph.org/doi/10.4171/RMI/133
Exceptional modular form of weight 4 on an exceptional domain contained in $\mathbb C^27$
Henry
Kim
University of Chicago, CHICAGO, UNITED STATES
Resnikoff [12] proved that weights of a non trivial singular modular form should be integral multiples of 1/2, 1, 2, 4 for the Siegel, Hermitian, quaternion and exceptional cases, respectively. The $\theta$-functions in the Siegel, Hermitian and quaternion cases provide examples of singular modular forms (Krieg [10]). Shimura [15] obtained a modular form of half-integral weight by analytically continuing an Eisenstein series. Bump and Baily suggested the possibility of applying an analogue of Shimura's method to obtain singular modular forms, i. e. modular forms of weight 4 and 8, on the exceptional domain of 3 x 3 hermitian matrices over Cayley numbers. The idea is to use Fourier expansion of a non-holomorphic Eisenstein series defined by using the factor of automorphy as in Karel [7]. The Fourier coefficients are the product of confluent hypergeometric functions as in Nagaoka [11] and certain singular series which we calculate by the method of Karel [6]. In this note we describe a modular form of weight 4 which may be viewed as an analogue of a $\theta$ zero-value and as an application, we consider its Mellin transform and prove a functional equation of the Eisenstein series which is a Nagaoka's conjecture (Nagaoka [11]).
General
139
200
10.4171/RMI/134
http://www.ems-ph.org/doi/10.4171/RMI/134