## Absolutely Convergent

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### Werner E G Muller - One of the best experts on this subject based on the ideXlab platform.

• ##### on the spectral side of arthur s trace formula absolute convergence
Annals of Mathematics, 2011
Co-Authors: Tobias Finis, Erez Lapid, Werner E G Muller
Abstract:

We derive a renement of the spectral expansion of Arthur’s trace formula. The expression is Absolutely Convergent with respect to the trace norm.

• ##### On the spectral side of Arthur's trace formula | absolute convergence
Annals of Mathematics, 2011
Co-Authors: Tobias Finis, Erez Lapid, Werner E G Muller
Abstract:

We derive a renement of the spectral expansion of Arthur’s trace formula. The expression is Absolutely Convergent with respect to the trace norm.

### Francois Denis - One of the best experts on this subject based on the ideXlab platform.

• ##### absolute convergence of rational series is semi decidable
Information & Computation, 2011
Co-Authors: Raphael Bailly, Francois Denis
Abstract:

This paper deals with absolute convergence of real-valued rational series, i.e. mappings r:Σ∗→R computed by weighted automata. An algorithm is provided, that takes a weighted automaton A as input and halts if and only if the corresponding series rA is Absolutely Convergent: hence, absolute convergence of rational series is semi-decidable. A spectral radius-like parameter ρ|r| is introduced, which satisfies the following property: a rational series r is Absolutely Convergent iff ρ|r|<1. We show that if r is rational, then ρ|r| can be approximated by Convergent upper estimates. Then, it is shown that the sum Σw∈Σ∗|r(w)| can be estimated to any accuracy rate. This result can be extended to any sum of the form Σw∈Σ∗|r(w)|p, for any integer p.

• ##### absolute convergence of rational series is semi decidable
Language and Automata Theory and Applications, 2009
Co-Authors: Raphael Bailly, Francois Denis
Abstract:

### Tobias Finis - One of the best experts on this subject based on the ideXlab platform.

• ##### on the spectral side of arthur s trace formula absolute convergence
Annals of Mathematics, 2011
Co-Authors: Tobias Finis, Erez Lapid, Werner E G Muller
Abstract:

We derive a renement of the spectral expansion of Arthur’s trace formula. The expression is Absolutely Convergent with respect to the trace norm.

• ##### On the spectral side of Arthur's trace formula | absolute convergence
Annals of Mathematics, 2011
Co-Authors: Tobias Finis, Erez Lapid, Werner E G Muller
Abstract:

We derive a renement of the spectral expansion of Arthur’s trace formula. The expression is Absolutely Convergent with respect to the trace norm.

### Ferenc Moricz - One of the best experts on this subject based on the ideXlab platform.

• ##### Absolutely Convergent fourier series and generalized zygmund classes of functions
Mathematica Scandinavica, 2009
Co-Authors: Ferenc Moricz
Abstract:

We investigate the order of magnitude of the modulus of smoothness of a function $f$ with Absolutely Convergent Fourier series. We give sufficient conditions in terms of the Fourier coefficients in order that $f$ belongs to one of the generalized Zygmund classes $(\mathrm{Zyg}(\alpha, L)$ and $(\mathrm{Zyg} (\alpha, 1/L)$, where $0\le \alpha\le 2$ and $L= L(x)$ is a positive, nondecreasing, slowly varying function and such that $L(x) \to \infty$ as $x\to \infty$. A continuous periodic function $f$ with period $2\pi$ is said to belong to the class $(\mathrm{Zyg} (\alpha, L)$ if 26740 |f(x+h) - 2f(x) + f(x-h)| \le C h^\alpha L\left(\frac{1}{h}\right)\qquad \text{for all $x\in \mathsf T$ and $h>0$}, 26740 where the constant $C$ does not depend on $x$ and $h$; and the class $(\mathrm{Zyg} (\alpha, 1/L)$ is defined analogously. The above sufficient conditions are also necessary in case the Fourier coefficients of $f$ are all nonnegative.

• ##### Absolutely Convergent Fourier series, classical function classes and Paley’s theorem
Analysis Mathematica, 2008
Co-Authors: Ferenc Moricz
Abstract:

This is a survey paper on the recent progress in the study of the continuity and smoothness properties of a function f with Absolutely Convergent Fourier series. We give best possible sufficient conditions in terms of the Fourier coefficients of f which ensure the belonging of f either to one of the Lipschitz classes Lip(α) and lip(α) for some 0 < α ≤ 1, or to one of the Zygmund classes Zyg(α) and zyg(α) for some 0 < α ≤ 2. We also discuss the termwise differentiation of Fourier series. Our theorems generalize those by R. P. Boas Jr., J. Nemeth and R. E. A. C. Paley, and a number of them are first published in this paper or proved in a simpler way.

• ##### Absolutely Convergent fourier series classical function classes and paley s theorem
Analysis Mathematica, 2008
Co-Authors: Ferenc Moricz
Abstract:

This is a survey paper on the recent progress in the study of the continuity and smoothness properties of a function f with Absolutely Convergent Fourier series. We give best possible sufficient conditions in terms of the Fourier coefficients of f which ensure the belonging of f either to one of the Lipschitz classes Lip(α) and lip(α) for some 0 < α ≤ 1, or to one of the Zygmund classes Zyg(α) and zyg(α) for some 0 < α ≤ 2. We also discuss the termwise differentiation of Fourier series. Our theorems generalize those by R. P. Boas Jr., J. Nemeth and R. E. A. C. Paley, and a number of them are first published in this paper or proved in a simpler way.

• ##### Absolutely Convergent Fourier integrals and classical function spaces
Archiv der Mathematik, 2008
Co-Authors: Ferenc Moricz
Abstract:

We study the continuity and smoothness properties of functions $$f \in L^{1}({\mathbb{R}})$$ whose Fourier transforms. $$\hat {f}$$ belong to $$L^{1}({\mathbb{R}})$$, and give sufficient conditions in terms of $$\hat {f}$$ to ensure that f belongs either to one of the Lipschitz classes Lip(α) and lip(α) for some 0 < α ≤ 1, or to one of the Zygmund classes Zyg(α) and zyg(α) for some 0 < α ≤ 2. These sufficient conditions are also necessary under an additional positivity assumption. Our theorems extend known results from periodic to nonperiodic functions.

• ##### Absolutely Convergent multiple fourier series and multiplicative lipschitz classes of functions
Acta Mathematica Hungarica, 2008
Co-Authors: Ferenc Moricz
Abstract:

We consider N-multiple trigonometric series whose complex coefficients cj1,...,jN, (j1,...,jN) ∈ ℤN, form an Absolutely Convergent series. Then the series $$\sum\limits_{(j_1 , \ldots ,j_N ) \in \mathbb{Z}^N } {c_{j_1 , \ldots j_N } } e^{i(j_1 x_1 + \ldots + j_N x_N )} = :f(x_1 , \ldots ,x_N )$$ converges uniformly in Pringsheim’s sense, and consequently, it is the multiple Fourier series of its sum f, which is continuous on the N-dimensional torus $$\mathbb{T}$$N, $$\mathbb{T}$$ := [−π, π). We give sufficient conditions in terms of the coefficients in order that >f belong to one of the multiplicative Lipschitz classes Lip (α1,..., αN) and lip (α1,..., αN) for some α1,..., αN > 0. These multiplicative Lipschitz classes of functions are defined in terms of the multiple difference operator of first order in each variable. The conditions given by us are not only sufficient, but also necessary for a special subclass of coefficients. Our auxiliary results on the equivalence between the order of magnitude of the rectangular partial sums and that of the rectangular remaining sums of related N-multiple numerical series may be useful in other investigations, too.

### Raphael Bailly - One of the best experts on this subject based on the ideXlab platform.

• ##### absolute convergence of rational series is semi decidable
Information & Computation, 2011
Co-Authors: Raphael Bailly, Francois Denis
Abstract:

This paper deals with absolute convergence of real-valued rational series, i.e. mappings r:Σ∗→R computed by weighted automata. An algorithm is provided, that takes a weighted automaton A as input and halts if and only if the corresponding series rA is Absolutely Convergent: hence, absolute convergence of rational series is semi-decidable. A spectral radius-like parameter ρ|r| is introduced, which satisfies the following property: a rational series r is Absolutely Convergent iff ρ|r|<1. We show that if r is rational, then ρ|r| can be approximated by Convergent upper estimates. Then, it is shown that the sum Σw∈Σ∗|r(w)| can be estimated to any accuracy rate. This result can be extended to any sum of the form Σw∈Σ∗|r(w)|p, for any integer p.

• ##### absolute convergence of rational series is semi decidable
Language and Automata Theory and Applications, 2009
Co-Authors: Raphael Bailly, Francois Denis
Abstract: