Walter Trebels

On localized potential spaces

Etude des proprietes des espaces de Riemann-Liouville localises RL(q,α) qui sont une variante des espaces de potentiels localises

Multiplier criteria of Marcinkiewicz type for Jacobi expansions

It is shown how an integral representation for the product of Jacobi polynomials can be used to derive a certain integral Lipschitz type condition for the Cesaro kernel for Jacobi expansions. This result is then used to give criteria of Marcinkiewicz type for a sequence to be multiplier of type (p,p), 1 < p < oo, for Jacobi expansions.

q-Moduli of Continuity in Hp(D), p>0, and an Inequality of Hardy and Littlewood

Some aspects of the interplay between approximation properties of analytic functions and the smoothness of its boundary values are discussed. One main result describes the equivalence of a special q-modulus of continuity and an intrinsic K-functional. Further, a generalization of a theorem due to G. H. Hardy and J. E. Littlewood (1932, Math. Z.34, 403?439) on the growth of fractional derivatives is deduced with the help of this K-functional.

Equivalence of a

Within the setting of abstract Cesaro-bounded Fourier series a K-functional is introduced and characterized by the convergence behavior of some linear means. Applications are given within the framework of Jacobi, Laguerre and Hermite expansions. In particular, Ditzians (1996) equivalence result in the setting of Legendre expansions is covered.

K

The necessary multiplier conditions for Laguerre expansions derived in Gasper and Trebels \cite{laguerre} are supplemented and modified. This allows us to place Marketts Cohen type inequality \cite{cohen} (up to the

-functional with the approximation behavior of some linear means for abstract Fourier series

\log

On necessary multiplier conditions for Laguerre expansions

--case) in the general framework of necessary conditions.

Ulyanov-type inequalities and generalized Liouville derivatives

We study ( p, q )-inequalities of Ulyanov type for moduli of smoothness of fractional order in the L p and the L p (ℝ n ) setting, p ≥ 1. In particular, we obtain estimates for the modulus of smoothness of a generalized Liouville derivative of a function via the modulus of smoothness of the function itself. We give examples showing the sharpness of these inequalities.

A lower estimate for the Lebesgue constants of linear means of Laguerre expansions

S. G. Kal’neǐ derived in [5], [6] a quite sharp necessary condition for the multiplier norm of a finite sequence in the setting of Fourier-Jacobi series on L1 with “natural weight” (which ensures a nice convolution structure). In this paper, Kalneǐ’s problem is considered in the setting of Laguerre series on weighted L1-spaces; the admitted scale of weights contains in particular the appropriate “natural weights” occurring in transplantation and convolution.

On the Rate of Almost Everywhere Convergence of Abel-Cartwright means on Lp(Rn)

The main purpose of this article is to establish results concerning the rate of almost everywhere convergence of the Abel-Cartwright means Wt,γf of the multidimensional Fourier integral. A typical result for these means is the following: Let% MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A!