Fourier expansions of arithmetical functions

Fourier Expansions of Arithmetical Functions

Fourier Expansions of Functions with Bounded Variation of Several Variables

In the first part of the paper we establish the pointwise convergence as t → +∞ for convolution operators ∫ Rd tdK (ty)φ(x− y)dy under the assumptions that φ(y) has integrable derivatives up to an order α and that |K(y)| ≤ c (1 + |y|)−β with α+β > d. We also estimate the Hausdorff dimension of the set where divergence may occur. In particular, when the kernel is the Fourier transform of a bound...

Arithmetical Functions I: Multiplicative Functions

Truth be told, this definition is a bit embarrassing. It would mean that taking any function from calculus whose domain contains [1,+∞) and restricting it to positive integer values, we get an arithmetical function. For instance, e −3x cos2 x+(17 log(x+1)) is an arithmetical function according to this definition, although it is, at best, dubious whether this function holds any significance in n...

Multivariate modified Fourier expansions

In this paper, we review recent advances in the approximation of multivariate functions using eigenfunctions of the Laplace operator subject to homogeneous Neumann boundary conditions. Such eigenfunctions are known explicitly on a variety of domains, including the d-variate cube, equilateral triangle and numerous other higher dimensional simplices. Practical construction of truncated expansions...

Quasi-orthogonal expansions for functions in BMO

For {φ_n(x)}, x ε [0,1] an orthonormalsystem of uniformly bounded functions, ||φ_n||_{∞}≤ M