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A gradient-projective basis of compactly supported wavelets in dimension n > 1

Guy Battle (2013)

Open Mathematics

A given set W = W X of n-variable class C 1 functions is a gradient-projective basis if for every tempered distribution f whose gradient is square-integrable, the sum χ ( n f · W χ * ) W χ converges to f with respect to the norm ( · ) L 2 ( n ) . The set is not necessarily an orthonormal set; the orthonormal expansion formula is just an element of the convex set of valid expansions of the given function f over W. We construct a gradient-projective basis W = W x of compactly supported class C 2−ɛ functions on ℝn such that [...]...

A note on multilinear Muckenhoupt classes for multiple weights

Songqing Chen, Huoxiong Wu, Qingying Xue (2014)

Studia Mathematica

This paper is devoted to investigating the properties of multilinear A P conditions and A ( P , q ) conditions, which are suitable for the study of multilinear operators on Lebesgue spaces. Some monotonicity properties of A P and A ( P , q ) classes with respect to P⃗ and q are given, although these classes are not in general monotone with respect to the natural partial order. Equivalent characterizations of multilinear A ( P , q ) classes in terms of the linear A p classes are established. These results essentially improve and extend...

Carleson measures associated with families of multilinear operators

Loukas Grafakos, Lucas Oliveira (2012)

Studia Mathematica

We investigate the construction of Carleson measures from families of multilinear integral operators applied to tuples of L and BMO functions. We show that if the family R t of multilinear operators has cancellation in each variable, then for BMO functions b₁, ..., bₘ, the measure | R t ( b , . . . , b ) ( x ) | ² d x d t / t is Carleson. However, if the family of multilinear operators has cancellation in all variables combined, this result is still valid if b j are L functions, but it may fail if b j are unbounded BMO functions, as we indicate...

Convergence of greedy approximation II. The trigonometric system

S. V. Konyagin, V. N. Temlyakov (2003)

Studia Mathematica

We study the following nonlinear method of approximation by trigonometric polynomials. For a periodic function f we take as an approximant a trigonometric polynomial of the form G ( f ) : = k Λ f ̂ ( k ) e i ( k , x ) , where Λ d is a set of cardinality m containing the indices of the m largest (in absolute value) Fourier coefficients f̂(k) of the function f. Note that Gₘ(f) gives the best m-term approximant in the L₂-norm, and therefore, for each f ∈ L₂, ||f-Gₘ(f)||₂ → 0 as m → ∞. It is known from previous results that in the case of...

Convolution of radius functions on ℝ³

Konstanty Holly (1994)

Annales Polonici Mathematici

We reduce the convolution of radius functions to that of 1-variable functions. Then we present formulas for computing convolutions of an abstract radius function on ℝ³ with various integral kernels - given by elementary or discontinuous functions. We also prove a theorem on the asymptotic behaviour of a convolution at infinity. Lastly, we deduce some estimates which enable us to find the asymptotics of the velocity and pressure of a fluid (described by the Navier-Stokes equations) in the boundary...

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