arXiv subjects

Walter Trebels

Publications and source records attributed to Walter Trebels.

12 recordsLinked to original sources

arXiv2025

Extremal polynomials for the Rogosinski--Szeg\H{o} estimates of the third coefficient of nonnegative sine polynomials

In the class of normalized sine-polynomials $S(t),$ non-negative on $[0,\pi],$ W.Rogosinski and G.Szeg\H{o} 1950 considered a number of extremal problems and proved, among other things, sharp upper and lower estimates for the coefficient $a_3.$ Their proof is based on the Luk\'acs representation of non-negative algebraic polynomials. This method does not lead to the construction of polynomials attaining the extreme values. We consider the corresponding problem in the framework of normalized typically real polynomials $P(z)$ on the unit disc in $\mathbb C.$ By L.Fej\'er's method with the additional use of the Chebyshev polynomials of the second kind and their derivatives, we regain the sharp upper and lower estimates for $a_3$ and identify the extremal polynomials. The corresponding statements for sine polynomials follow by the observation $S(t)=\text{Im}\{P(e^{it})\}$. For odd $N$ the extremizers are unique, for even $N$ there is a one-parameter family of extremizers.

math.CV↗

arXiv2024

Extremizers for the Rogosinski-Szeg\"o estimate of the second coefficient in nonnegative sine polynomials

For the class of sine polynomials $b_1\sin t+b_2\sin2t+...+b_N\sin Nt,\; (b_N\not= 0),$ which are nonnegative on $(0,\pi)$, W. Rogosinski and G. Szeg\"o derived, among other things, exact bounds for $|b_2|$ via the Luk\'acs presentation of nonnegative algebraic polynomials and a variational type argument for exact bounds, but they did not find the extremizers. Within this algebraic framework, we construct explicit polynomials which attain these bounds and prove their uniqueness. The proof uses the Fej\'er -Riesz representation of nonnegative trigonometric polynomials, a 7-band Toeplitz matrix of arbitrary finite dimension, and Chebyshev polynomials of the second kind and their derivatives.

math.CA↗

arXiv2023

A unified approach to inequalities for K-functionals and moduli of smoothness

The paper provides a detailed study of crucial inequalities for smoothness and interpolation characteristics in rearrangement invariant Banach function spaces. We present a unified approach based on Holmstedt formulas to obtain these estimates. As examples, we derive new inequalities for moduli of smoothness and K-functionals in various Lorentz spaces.

math.FA↗

arXiv2018

Embeddings for spaces of Lorentz-Sobolev type

The purpose of this paper is to characterize all embeddings for versions of Besov and Triebel-Lizorkin spaces where the underlying Lebesgue space metric is replaced by a Lorentz space metric. We include two appendices, one on the relation between classes of endpoint Mikhlin-Hörmander type Fourier multipliers, and one on the constant in the triangle inequality for the spaces $L^{p,r} $ when $p<1$.

math.FA↗

arXiv2007

Low regularity classes and entropy numbers

We note a sharp embedding of the Besov space $B^\infty_{0,q}(\bbT)$ into exponential classes and prove entropy estimates for the compact embedding of subclasses with logarithmic smoothness, considered by Kashin and Temlyakov.

math.CA↗

arXiv1996

Hankel Multipliers And Transplantation Operators

Connections between Hankel transforms of different order for $L^p$-functions are examined. Well known are the results of Guy [Guy] and Schindler [Sch]. Further relations result from projection formulae for Bessel functions of different order. Consequences for Hankel multipliers are exhibited and implications for radial Fourier multipliers on Euclidean spaces of different dimensions indicated.

math.CA↗

arXiv1995

A Riemann--Lebesgue lemma for Jacobi expansions

A Lemma of Riemann--Lebesgue type for Fourier--Jacobi coefficients is derived. Via integral representations of Dirichlet--Mehler type for Jacobi polynomials its proof directly reduces to the classical Riemann--Lebesgue Lemma for Fourier coefficients. Other proofs are sketched. Analogous results are also derived for Laguerre expansions and for Jacobi transforms.

math.CA↗

arXiv1995

Ultraspherical multipliers revisited

Sufficient ultraspherical multiplier criteria are refined in such a way that they are comparable with necessary multiplier conditions. Also new necessary conditions for Jacobi multipliers are deduced which, in particular, imply known Cohen type inequalities. Muckenhoupt's transplantation theorem is used in an essential way.

math.CA↗

arXiv1994

Fractional integration for Laguerre expansions

The aim of this note is to provide a fractional integration theorem in the framework of Laguerre expansions. The method of proof consists of establishing an asymptotic estimate for the involved kernel and then applying a method of Hedberg \cite{pro}. We combine this result with sufficient $(p,p)$ multiplier criteria of Stempak and Trebels \cite{ST}. The resulting sufficient $(p,q)$ multiplier criteria are comparable with necessary ones of Gasper and Trebels \cite{laguerre}.

math.CA↗

arXiv1994

On a restriction problem of de Leeuw type for Laguerre multipliers

In 1965 K. de Leeuw \cite{deleeuw} proved among other things in the Fourier transform setting: {\it If a continuous function $m(ξ_1, \ldots ,ξ_n)$ on ${\bf R}^n$ generates a bounded transformation on $L^p({\bf R}^n),\; 1\le p \le \infty ,$ then its trace $\tilde{m}(ξ_1, \ldots ,ξ_m)=m(ξ_1, \ldots ,ξ_m,0,\ldots ,0), \; m<n,$ generates a bounded transformation on $L^p({\bf R}^m)$. } In this paper, the analogous problem is discussed in the setting of Laguerre expansions of different orders.

math.CA↗

arXiv1993

On necessary multiplier conditions for Laguerre expansions

The necessary multiplier conditions for Laguerre expansions derived in Gasper and Trebels \cite{laguerre} are supplemented and modified. This allows us to place Markett's Cohen type inequality \cite{cohen} (up to the $\log $--case) in the general framework of necessary conditions.

math.CA↗

arXiv1993

On weighted transplantation and multipliers for Laguerre expansions

Using the standard square--function method (based on the Poisson semigroup), multiplier conditions of Hörmander type are derived for Laguerre expansions in $L^p$--spaces with power weights in the $A_p$-range; this result can be interpreted as an ``upper end point'' multiplier criterion which is fairly good for $p$ near $1$ or near $\infty $. A weighted generalization of Kanjin's \cite{kan} transplantation theorem allows to obtain a ``lower end point'' multiplier criterion whence by interpolation nearly ``optimal'' multiplier criteria (in dependance of $p$, the order of the Laguerre polynomial, the weight).

math.CA↗