Wavelet bases on a manifold

Using the decomposition method, we present in this paper constructions of multiresolution analyses on a compact Riemannian manifold M of dimension n(n∈N). These analyses are generated by a finite number of basic functions and are adapted to the study of the Sobolev spaces H¹(M) and .     

Wavelet bases in

Some years ago, compactly supported divergence-free wavelets were constructed which also gave rise to a stable (biorthogonal) wavelet splitting of . These bases have successfully been used both in the analysis and numerical treatment of the Stokes and Navier-Stokes equations. In this paper, we construct stable wavelet bases for the stream function spaces . Moreover, -free vector wavelets are constructed and analysed. The relationship between and are expressed in terms of these wavelets. We obtain discrete (orthogonal) Hodge decompositions.

Our construction works independently of the space dimension, but in terms of general assumptions on the underlying wavelet systems in that are used as building blocks. We give concrete examples of such bases for tensor product and certain more general domains . As an application, we obtain wavelet multilevel preconditioners in and .

     

SAGBI bases in rings of multiplicative invariants

Let k be a field and G be a finite subgroup of . We show that the ring of multiplicative invariants has a finite SAGBI basis if and only if G is generated by reflections. Received: March 5, 2002     

Wavelet bases in generalized Besov spaces

In this paper we obtain a wavelet representation in (inhomogeneous) Besov spaces of generalized smoothness via interpolation techniques. As consequence, we show that compactly supported wavelets of Daubechies type provide an unconditional Schauder basis in these spaces when the integrability parameters are finite.     

Wavelet bases of Hermite cubic splines on the interval

In this paper a pair of wavelets are constructed on the basis of Hermite cubic splines. These wavelets are in C¹ and supported on [−1,1]. Moreover, one wavelet is symmetric, and the other is antisymmetric. These spline wavelets are then adapted to the interval [0,1]. The construction of boundary wavelets is remarkably simple. Furthermore, global stability of the wavelet basis is established. The wavelet basis is used to solve the Sturm–Liouville equation with the Dirichlet boundary condition. Numerical examples are provided. The computational results demonstrate the advantage of the wavelet basis. Dedicated to Dr. Charles A. Micchelli on the occasion of his 60th birthday Mathematics subject classifications (2000) 42C40, 41A15, 65L60. Research was supported in part by NSERC Canada under Grants # OGP 121336.     

On standard bases in rings of differential polynomials

We consider Ollivier’s standard bases (also known as differential Gröbner bases) in an ordinary ring of differential polynomials in one indeterminate. We establish a link between these bases and Levi’s reduction process. We prove that the ideal [x ^p ] has a finite standard basis (w.r.t. the so-called β-orderings) that contains only one element. Various properties of admissible orderings on differential monomials are studied. We bring up the question of whether there is a finitely generated differential ideal that does not admit a finite standard basis w.r.t. any ordering.     

Amalgamation bases for commutative rings without nilpotent elements

SCR=the class of commutative rings without nilpotent elements.Theorem,R is an amalgamation base for SCR iff rad (I=Ann²(I) forI ⊆R finitely generated. Supplement. IfR ε SCR thenR is contained in an amalgamation basis for SCR having no new idempotent elements. CR=the class of commutative rings. Theorem.R is an amalgamation base for CR iffR is a pureR-submodule of any commutative ring extendingR.     

Subfields of the adele ring

We establish a natural one-to-one correspondence between immediate extensions of a Dedekind ring and subfields of its ring of adeles containing its ring of fractions. As a consequence of the existence theorems for immediate extensions proved previously, we state that e-closed fields within the class of countable subfields of the classical adele ring are surprising extensions of the field of rationals.     

Wavelet bases in rearrangement invariant function spaces

We point out that the well known characterization of spaces () in terms of orthogonal wavelet bases extends to any separable rearrangement invariant Banach function space on (equipped with Lebesgue measure) with nontrivial Boyd's indices. Moreover we show that such bases are unconditional bases of .

     

Lazy bases: a minimalist constructive theory of Noetherian rings

We give a constructive treatment of the theory of Noetherian rings. We avoid the usual restriction to coherent rings; we can even deal with non‐discrete rings. We introduce the concept of rings with certifiable equality which covers discrete rings and much more. A ring R with certifiable equality can be fitted with a partial ideal membership test for ideals of R. Lazy bases of ideals of R [X ] are introduced in order to derive a partial ideal membership test for ideals of R [X ]. It is then proved that if R is Noetherian, then so is R [X ]. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Gröbner bases and Stanley decompositions of determinantal rings

Supported by the Austrian Science Foundation (FWF) Project no. P6763     

Wavelet bases for a set of commuting unitary operators

Let (U=U ₁, ...,U _d ) be an orderedd-tuple of distinct, pairwise commuting, unitary operators on a complex Hilbert space , and letX:={x ₁, ...,x _r } such that is a Riesz basis of the closed linear spanV ₀ of . Suppose there is unitary operatorD on such thatV ₀ D V ₀ =:V ₁ andU ⁿ D=DU ^An for alln ^d , whereA is ad ×d matrix with integer entries and := det(A) 0. Then there is a subset inV ₁, withr( – 1) vectors, such that is a Riesz basis ofW ₀, the orthogonal complement ofV ₀ inV ₁. The resulting multiscale and decomposition relations can be expressed in a Fourier representation by one single equation, in terms of which the duality principle follows easily. These results are a consequence of an extension, to a set of commuting unitary operators, of Robertson's Theorems on wandering subspace for a single unitary operator [24]. Conditions are given in order that is a Riesz basis ofW ₀. They are used in the construction of a class of linear spline wavelets on a four-direction mesh.     

Wavelet bases in the Lebesgue spaces on the field of p-adic numbers

In this paper we prove that p-adic wavelets form an unconditional basis in the space L ^r (? _p ⁿ ) and give the characterization of the space L ^r (? _p ⁿ ) in terms of Fourier coefficients of p-adic wavelets.Moreover, the Greedy bases in the Lebesgue spaces on the field of p-adic numbers are also established.     

On constructing bases for ideals in polynomial rings over the integers

The author defines canonical bases for ideals in polynomial rings over Z and develops an algorithm for constructing such a basis for a given ideal. Results of previous authors are discussed and a comparison between those and the results obtained here is included.     

Standard monomial bases and geometric consequences for certain rings of invariants

Consider the diagonal action ofSL _n (K) on the affine spaceX = V^⊕m ⊕ (V*)^⊕q whereV = K ⁿ ,K an algebraically closed field of arbitrary characteristic andm,q > n. We construct a ‘standard monomial’’ basis for the ring of invariantsK[X] ^SL _n ^(K). As a consequence, we deduce thatK[X] ^SL _n ^(K) is Cohen-Macaulay. We also present the first and second fundamental theorems forSL _n (K)- actions.     

Construction of normal bases of rings of integral elements of certain fields of algebraic functions

This article is devoted to the construction of normal bases of rings of integral elements of certain fields of algebraic functions. These fields are algebraic extensions of the initial fields, given by appropriate algebraic equations. The extensions are given only by the monodromy group. The device of the matrix Riemann boundary-value problem on a Riemann surface is used.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 7, pp. 1004–1007, July, 1990.