Haar Bases forL2(Rn) and Algebraic Number Theory

Haar Bases for L(R) and Algebraic Number Theory

Gro chenig and Madych showed that a Haar-type orthonormal wavelet basis of L(R) can be constructed from the characteristic function /Q of a set Q if and only if Q is an affine image of an integral self-affine tile T which tiles R using the integer lattice Z. An integral self-affine tile T=T(A, D) is the attractor of an iterated function system T= i=1 A (T+di) where A # Mn(Z) is an expanding n_n...

Corrigendum and Addendum to: Haar Bases for L2(Rn) and Algebraic Number Theory

We correct an error in the proof of Theorem 1.5 in 4]. We also give a strengthened necessary condition for the existence of a Haar basis of the speciied kind for every integer matrix A that has a given irreducible characteristic polynomial f(x) with jf(0)j = 2: A. Potiopa 7] found that the expanding polynomial g(x) = x 4 +x 2 +2 violates this necessary condition. Thus there exists some 4 4 expa...

Lemmermeyer Algebraic Number Theory

Algebraic Number Theory

These are the notes for a course taught at the University of Michigan in F92 as Math 676. They are available at www.math.lsa.umich.edu/∼jmilne/. Please send comments and corrections to me at [email protected]. v2.01 (August 14, 1996.) First version on the web. v2.10 (August 31, 1998.) Fixed many minor errors; added exercises and index.

Algebraic Number Theory

2 Arithmetic in an algebraic number field 30 2.1 Finitely generated abelian groups . . . . . . . . . . . . . . . . . . 30 2.2 The splitting field and the discriminant . . . . . . . . . . . . . . 31 2.3 Number fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.4 The ring of integers . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.5 Ideals . . . . . . . . . . . . . . ....