Francisco J. Ruiz

Asymptotic estimates for Apostol-Bernoulli and Apostol-Euler polynomials

We analyze the asymptotic behavior of the Apostol-Bernoulli polynomials

Two notes on convergence and divergence a.e. of Fourier series with respect to some orthogonal systems

\mathcal{B}_{n}(x;\lambda)

Asymptotic behaviour of orthogonal polynomials relative to measures with mass points

in detail. The starting point is their Fourier series on

Commutators for the maximal and sharp functions

[0,1]

Rearrangement of Hardy-Littlewood maximal functions in Lorentz spaces

which, it is shown, remains valid as an asymptotic expansion over compact subsets of the complex plane. This is used to determine explicit estimates on the constants in the approximation, and also to analyze oscillatory phenomena which arise in certain cases. These results are transferred to the Apostol-Euler polynomials

The Möbius inversion formula for Fourier series applied to Bernoulli and Euler polynomials

\mathcal{E}_{n}(x;\lambda)

A Simple Computation of ζ (2 k )

via a simple relation linking them to the Apostol-Bernoulli polynomials.

Comparing Cognitive, Metacognitive, and Acceptance and Commitment Therapy Models of Depression: a Longitudinal Study Survey.

We study some problems related to convergence and divergence a.e. for Fourier series in systems {φk}, where {φk} is either a system of orthonormal polynomials with respect to a measure dμ on [−1, 1] or a Bessel system on [0, 1]. We obtain boundedness in weighted L spaces for the maximal operators associated to Fourier-Jacobi and Fourier-Bessel series. On the other hand, we find general results about divergence a.e. of the Fourier series associated to Bessel systems and systems of orthonormal polynomials on [−1, 1]. §0. Introduction. Let dμ be a positive measure on a finite interval [a, b] ⊂ R and

Psychological inflexibility mediates the effects of self-efficacy and anxiety sensitivity on worry.

Abstract. General expressions are found for the orthonormal polynomials and the kernels relative to measures on the real line of the form μ+Mδc, in terms of those of the measures dμ and (x−c)dμ. In particular, these relations allow us to obtain that Nevai’s class M(0, 1) is closed for adding a mass point, as well as several bounds for the polynomials and kernels relative to a generalized Jacobi weight with a finite number of mass points.

Elementary reverse Hölder type inequalities with application to operator interpolation theory

The class of functions for which the commutator with the HardyLittlewood maximal function or the maximal sharp function are bounded on Lq are characterized and proved to be the same. For the Hilbert transform H , and other classical singular integral operators, a well known and important result due to Coifman, Rochberg and Weiss (cf. [2]) states that a locally integrable function b in R is in BMO if and only if the commutator [H, b], defined by [H, b]f = H(bf)− bH(f), is bounded in L, for some (and for all) q ∈ (1,∞). The cancellation implied by the commutator operation and the properties of singular integrals are crucial for the validity of the result. Later in [3], using real interpolation techniques, Milman and Schonbeck proved a commutator result that applies to the Hardy-Littlewood maximal operatorM as well as the sharp maximal operator. In fact the commutator result is valid for a large class of nonlinear operators which we now describe. Let us say that T is a positive quasilinear operator if it is defined on a suitable class of locally integrable functions D(T ) and satisfies i) Tf ≥ 0, for f ∈ D(T ), ii) T (αf) = |α|Tf , for α ∈ R and f ∈ D(T ), iii) |Tf − Tg| ≤ T (f − g), for f, g ∈ D(T ). We have (cf. [3]) Proposition 1. Let b be a nonnegative BMO function and suppose that T is a positive quasilinear operator which is bounded on L(w), for some 1 ≤ q < ∞ and for all w weights belonging to the Muckenhoupt class Ar for some r ∈ [1,+∞). Then [T, b] is bounded on L. In particular the result applies to the maximal operator and the sharp maximal function. Note that since the Hardy-Littlewood maximal operator M is a positive Received by the editors January 6, 1999. 2000 Mathematics Subject Classification. Primary 42B25, 46E30.