Singular measures and convolution operators

We show that in the study of certain convolution operators, functions can be replaced by measures without changing the size of the constants appearing in weak type (1,1) inequalities. As an application, we prove that the best constants for the centered Hardy-Littlewood maximal operator associated to parallelotopes do not decrease with the dimension.Comment: 8 page

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

AbstractHurwitz found the Fourier expansion of the Bernoulli polynomials over a century ago. In general, Fourier analysis can be fruitfully employed to obtain properties of the Bernoulli polynomials and related functions in a simple manner. In addition, applying the technique of Möbius inversion from analytic number theory to Fourier expansions, we derive identities involving Bernoulli polynomials, Bernoulli numbers, and the Möbius function; this includes formulas for the Bernoulli polynomials at rational arguments. Finally, we show some asymptotic properties concerning the Bernoulli and Euler polynomials

Asymptotic estimates for Apostol-Bernoulli and Apostol-Euler polynomials

We analyze the asymptotic behavior of the Apostol-Bernoulli polynomials $\mathcal{B}_{n}(x;\lambda)$ in detail. The starting point is their Fourier series on $[0,1]$ 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 $\mathcal{E}_{n}(x;\lambda)$ via a simple relation linking them to the Apostol-Bernoulli polynomials.Comment: 16 page

Existence and reduction of generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials

summary:One can find in the mathematical literature many recent papers studying the generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials, defined by means of generating functions. In this article we clarify the range of parameters in which these definitions are valid and when they provide essentially different families of polynomials. In particular, we show that, up to multiplicative constants, it is enough to take as the “main family” those given by $$\Big ( \frac{2}{\lambda e^t+1} \Big )^\alpha e^{xt} = \sum _{n=0}^{\infty } \mathcal{E}^{(\alpha )}_{n}(x;\lambda ) \frac{t^n}{n!}\,, \qquad \lambda \in \mathbb{C}\setminus \lbrace -1\rbrace \,,$$ and as an “exceptional family” $$\Big ( \frac{t}{e^t-1} \Big )^\alpha e^{xt} = \sum _{n=0}^{\infty } \mathcal{B}^{(\alpha )}_{n}(x) \frac{t^n}{n!}\,,$$ both of these for $\alpha \in \mathbb{C}$

Bernoulli–Dunkl and Apostol–Euler–Dunkl polynomials with applications to series involving zeros of Bessel functions

We introduce Bernoulli–Dunkl and Apostol–Euler–Dunkl polynomials as generalizations of Bernoulli and Apostol–Euler polynomials, where the role of the derivative is now played by the Dunkl operator on the real line. We use them to find the sum of many different series involving the zeros of Bessel functions

A connection between power series and Dirichlet series

[EN] We prove that for any convergent Laurent series f(z) = ∞n=−k anzn with k ≥ 0, there is a meromorphic function F(s) on C whose only possible poles are among the integers n = 1, 2, ..., k, having residues Res(F; n) = a−n/(n − 1)!, and satisfying F(−n) = (−1)nn! an for n = 0, 1, 2, .... Under certain conditions, F(s) is a Mellin transform. In particular, this happens when f(z) is of the form H(e−z)e−z with H(z) analytic on the open unit disk. In this case, if H(z) = ∞ n=0 hnzn, the analytic continuation of H(z) to z = 1 is related to the analytic continuation of the Dirichlet series ∞n=1 hn−1n−s to the complex plane

Endpoint weak boundedness of some polynomial expansions

AbstractLet w(x) = (1 −x)α(1 + x)β on [− 1, 1], α,β⩾ − 12, and for each function f let Snf be the nth expansion in the corresponding orthonormal polynomials. We show that the operators f → uSn(u−1f) are not of weak (p, p)-type, where u is another Jacobi weight and p is an endpoint of the interval of mean convergence. The same result is shown for expansions associated to measures of the form dv = w(x) dx + Σki=1Miδai

Unconditional and quasi-greedy bases in L-p with applications to Jacobi polynomials Fourier series

We show that the decreasing rearrangement of the Fourier series with respect to the Jacobi polynomials for functions in L-p does not converge unless p = 2. As a by-product of our work on quasi-greedy bases in L-p(µ), we show that no normalized unconditional basis in L-p, p not equal 2, can be semi-normalized in L-q for q not equal p, thus extending a classical theorem of Kadets and Pelczynski from 1968.The first two authors were partially supported by the Spanish Research Grant Analisis Vectorial, Multilineal y Aplicaciones, reference number MTM2014-53009-P, and the last two authors were partially supported by the Spanish Research Grant Ortogonalidad, Teoria de la Aproximacion y Aplicaciones en Fisica Matematica, reference number MTM2015-65888-C4-4-P. The first-named author also acknowledges the support of Spanish Research Grant Operators, lattices, and structure of Banach spaces, with reference MTM2016-76808-P