estimates for riesz kernels of eigenfunction expansions of elliptic differential operators on compact manifolds let 2 and be the eigenvalues and an orthonormal system of eigenvectors of a second order elliptic differential operator on a compact n dimensional manifold m the riesz means of order of an integrable function on m are defined by r x m 1 2 2 x y y d y in this paper we study the kernels and the operator norms of he operators r on lp m 1 p 2 n 1 n 3 we also prove that if 1 p 2 n 1 n 3 and np n 1 2 then the operators r are of weak type p p uniformly with respect to

divergence of eigenfunction expansions we shall use a result of rellich on the convergence of spectral projections to analyze mean and almost everywhere convergence of eigenfunction expansions the basic result is a transplantation theorem which reduces these questions on manifolds to questions on the boundedness of fourier multiplier operators on the cotangent space results such as these were long known in the context of fourier series where they were introduced by stein and weiss 121 various authors gave generalizations but the most far reaching was obtained by mitjagin 8 who applied his results to elliptic problems in this paper we shall give several extensions of mitjagin s results in the main part of the paper we shall employ these extensions and the results of kenig and tomas s 61 to analyze certain nonelliptic problems this paper is divided into four sections in the first we shall give mitjagin s original proof that one may transplant problems on the convergence of eigenfunction expansions in lp mean to fourier multiplier problems on i we shall also extend this to convergence of other summability methods and also to joint eigenexpansions of commuting selfadjoint operators

on almost everywhere divergence of bochner riesz means on compact lie groups let g be a connected simply connected compact semisimple lie group of dimension n it has been shown by clerc ann inst fourier grenoble 24 1 149 172 1974 that for any f in l 1 g the bochner riesz mean s r delta f converges almost everywhere to f provided delta n 1 2 in this paper we show that at the critical index delta n 1 2 there exists an f in l 1 g such that begin aligned limsup r rightarrow infty big s r n 1 2 f x big infty quad a e x in g end aligned this is an analogue of a well known result of kolmogoroff fund math 4 1 324 328 1923 for fourier series on the circle and a result of stein ann math 2 74 140 170 1961 for bochner riesz means on the tori mathbb t n n ge 2 we also study localization properties of the bochner riesz mean s r n 1 2 f for f in l 1 g

a spectral multiplier theorem for a sublaplacian on su 2 we prove a hormander type spectral multiplier theorem for a sublaplacian on su 2 with critical index determined by the euclidean dimension of the group this result is the analogue for su 2 of the result for the heisenberg group obtained by d muller and e m stein and by w hebisch

well posedness of feedback systems insights into exact robustness analysis and approximate computations this paper establishes a framework for robust stability analysis of linear time invariant uncertain systems the uncertainty is assumed to belong to an arbitrary subset of complex matrices the concept used here is well posedness of feedback systems leading to necessary and sufficient conditions for robust stability based on this concept some insights into exact robust stability conditions are given in particular frequency domain and state space conditions for well posedness are provided in terms of hermitian form inequalities it is shown that these inequalities can be interpreted as small gain conditions with a generalized class of scalings given by linear fractional transformations lft using the lft scaled small gain condition in the state space setting the duality is established between the h sub spl infin norm condition with frequency dependent scalings and the parameter dependent lyapunov condition connections to the existing results including the structured singular value and the integral quadratic constraints are also discussed finally we show that our well posedness conditions can be used to give a less conservative yet computable bound on the real structured singular value this result is illustrated by numerical examples

on the asymptotics of the bochner riesz kernel on compact lie groups suppose g is an n dimensional compact connected lie group and for 0 let dr r 0 be the bochner riesz kernel on g we prove that the lebesgue constants dr 1 are of the same order as r n 1 2 as r tends to infinity 0 n 1 2 while dr 1 log r if n 1 2 as a convergence we find best conditions for the norm convergence of the bochner riesz means on the spaces l1 g and c g

complete quadratic lyapunov functionals for distributed delay systems this paper is concerned with the stability analysis of distributed delay systems using complete lyapunov functionals numerous articles aim at approximating their parameters thanks to a discretization method or polynomial modeling the interest of such approximations is the design of tractable sufficient stability conditions in the present article we provide an alternative method based on polynomial approximation which takes advantages of the legendre polynomials and their properties the resulting stability conditions are scalable with respect to the maximum degree of the polynomials and are expressed in terms of tractable linear matrix inequalities several examples of delayed systems are tested to show the effectiveness of the method

weighted norm inequalities gaussian bounds and sharp spectral multipliers abstract let l be a non negative self adjoint operator acting on l 2 x where x is a space of homogeneous type assume that l generates a holomorphic semigroup e t l whose kernels p t x y have gaussian upper bounds but there is no assumption on the regularity in variables x and y in this article we study weighted l p norm inequalities for spectral multipliers of l we show that sharp weighted hormander type spectral multiplier theorems follow from gaussian heat kernel bounds and appropriate l 2 estimates of the kernels of the spectral multipliers these results are applicable to spectral multipliers for large classes of operators including laplace operators acting on lie groups of polynomial growth or irregular non doubling domains of euclidean spaces elliptic operators on compact manifolds and schrodinger operators with non negative potentials

a multistage reduction technique for feedback stabilizing distributed time lag systems abstract a direct pole relocation theory is advanced for linear time invariant systems with distributed delays in both state and control variables the principal tools of the theory include i the finite cardinality of the unstable spectrum ii a set of matrices each of which is a left zero of the system characteristic quasi polynomial matrix and iii a linear transformation which reduces the delay system to a sufficiently high order delay free system whose spectrum contains the delay system unstable spectrum it is shown that if the delay system is spectrally stabilizable then it shares a common feedback stabilizing control law with its delay free counterpart this point of contact with a delay free system permits the determination of the control law using well established ordinary system methods the workability of the approach hinges on the ability to partition the unstable spectrum augmented with additional poles from the stable spectrum if necessary into n symmetric sets when this partition is impossible a spectral controllability invariance theorem facilitates resolution of the problem

the bochner riesz means for fourier bessel expansions abstract the bochner riesz means for fourier bessel expansions are analyzed we prove a uniform two weight inequality with potential weights for these means the result provides necessary and sufficient conditions for boundedness moreover we obtain some corollaries regarding the convergence of these means and the boundedness of other operators related to the fourier bessel series

on the consequences of a mihlin hormander functional calculus maximal and square function estimates we prove that the existence of a mihlin hormander functional calculus for an operator l implies the boundedness on l p of both the maximal multiplier operators and the continuous square functions build on spectral multipliers of l the considered multiplier functions are finitely smooth and satisfy an integral condition at infinity in particular multipliers of compact support are admitted

discretized lyapunov functional for systems with distributed delay and piecewise constant coefficients the stability problem for systems with distributed delay is considered using discretized lyapunov functional the coefficients associated with the distributed delay are assumed to be piecewise constant and the discretization mesh may be non uniform the resulting stability criteria are written in the form of linear matrix inequality numerical examples are also provided to illustrate the effectiveness of the method the basic idea can be extended to a more general setting with more involved formulation

sharp estimates for lebesgue constants on compact lie groups suppose g is a n dimensional compact connected semisimple lie group and dr is the spherical dirichlet kernel on g we prove the existence of a positive constant k such that dr 1 kr n 1 2 this complements the known result dr 1 hr n 1 2 we also prove that for a polyhedral dirichlet kernel dn the above inequalities hold with np in place of r n 1 2 p is the number of positive roots of g

on the convergence of riesz means on compact manifolds the purpose of this paper is to study the lp mapping properties of the riesz means of eigenfunction expansions associated to elliptic differential operators on compact manifolds in order to be more specific let us first introduce some notation let m be a compact connected c manifold of dimension n 2 and p an elliptic differential operator on m with c00 coefficients we shall assume that p is self adjoint with respect to some positive c00 density dx kept fixed throughout that is pu v u pv where as usual

stabilization of uncertain linear distributed delay systems with dissipativity constraints abstract this paper examines the problem of stabilizing linear distributed delay systems with nonlinear distributed delay kernels and dissipativity constraints specifically the nonlinear distributed kernel includes functions such as polynomials trigonometric and exponential functions by constructing a liapunov krasovskii functional related to the distributed kernels sufficient conditions for the existence of a state feedback controller which stabilizes the uncertain distributed delay systems with dissipativity constraints are given in terms of linear matrix inequalities lmis in contrast to existing methods the proposed scenario is less conservative or requiring less number of decision variables based on the application of a new derived integral inequality finally numerical examples are presented to demonstrate the validity and effectiveness of the proposed methodology

radial weights and mixed norm inequalities for the disc multiplier in this paper we show that the l2 weighted inequalities with radial weights of the multiplier for the ball are controlled by the maximal function defined by averages on arbitrary rectangles this solves the case of radial weights of a problem proposed by e stein in 1978 it also gives an elegant and simple proof of the boundedness of the multiplier for the ball on the spaces of mixed norm lradp lang2 for 2n n 1 p 2n n 1 obtained by a cordoba and by g mockenhaupt additional work is presented in order to extend these mixed norm estimates to the end points 2n n 1 and 2n n 1 in the sharp form of restricted weak type inequalities

radial fourier multipliers in high dimensions given a fixed p 2 we prove a simple and effective characterization of all radial multipliers of mathcal f l p left mathbb r d right provided that the dimension d is sufficiently large the method also yields new l q space time regularity results for solutions of the wave equation in high dimensions

mean and weak convergence of fourier bessel series abstract we study the uniform boundedness on some weighted lp spaces of the partial sum operators associated to fourier bessel series obtaining necessary and sufficient conditions for this boundedness in terms of the weights on the other hand we study the weak and restricted weak type of the partial sum operators in the end points of the interval of strong boundedness

singular integral operators with rough convolution kernels in all dimensions again under the assumption q e l log l it is conceivable that a variant of the arguments in 3 for the maximal operator could also work for the singular integral operator in fact in unpublished work the authors of 3 obtained a weak type 1 1 inequality in dimension d 7 however their arguments if applied to the singular integral operator lead to substantial technical difficulties and no

concerning the lp norm of spectral clusters for second order elliptic operators on compact manifolds in this paper we study the lp l2 mapping properties of a spectral projection operator for riemannian manifolds this operator is a generalization of the harmonic projection operator for spherical harmonics on s see c d sogge duke math j 53 1986 43 65 among other things we generalize the l2 restriction theorems of c fefferman e m stein and p tomas bull amer math sot 81 1975 477478 for the fourier transform in r to the setting of riemannian manifolds we obtain these results for the spectral projection operator as a corollary of a certain sobolev inequality involving d r2 for large r this sobolev inequality generalizes certain results for r of c kenig a ruiz and the author duke math j 55 1987 329 347 the main tools in the proof of the sobolev inequalities for riemannian manifolds are the hadamard parametrix cf l hbrmander acta mad 88 1968 341 370 and the analysis of linear partial differential equations vol iii springer verlag new york 1985 and oscillatory integral theorems of l carleson and p sjolin studia math 44 19x 287 299 and stein arm math stud 112 1986 307 357 0 1988 academic pnss iw

fourier bessel heat kernel estimates we provide sharp two sided estimates of the fourier bessel heat kernel and we give sharp two sided estimates of the transition probability density for the bessel process in 0 1 killed at 1 and killed or reflected at 0

quadratic separation for feedback connection of an uncertain matrix and an implicit linear transformation topological separation is investigated in the case of an uncertain time invariant matrix interconnected with an implicit linear transformation a quadratic separator independent of the uncertainty is shown to prove losslessly the closed loop well posedness several applications for lti descriptor system analysis are then given first some known results for stability and pole location of descriptor systems are demonstrated in a new way second contributions to robust stability analysis and time delay systems stability analysis are exposed these prove to be new even when compared to results for usual lti systems not in descriptor form all results are formulated as linear matrix inequalities lmis

delay range stability of a class of distributed time delay systems this paper is dedicated to the stability analysis of a class of uncertain distributed delay systems the kernel of which can be modeled as a polynomial function of the delay the results are constructed by rewriting the system as an uncertain interconnected model appropriate robust control tools i e quadratic separation are then used to address the stability issue to this end some relations that highlight relevant characteristics of the delayed term are added to the interconnected model leading then to the conservatism reduction finally numerical examples show the effectiveness of the proposed method

two weight norm inequalities for the cesaro means of hermite expansions an accurate estimate is obtained of the cesaro kernel for hermite expansions this is used to prove two weight norm inequalities for cesaro means of hermite polynomial series and for the supremum of these means these extend known norm inequalities even in the single power weight and unweighted cases an almost everywhere convergence result is obtained as a corollary it is also shown that the conditions used to prove norm boundedness of the means and most of the conditions used to prove the boundedness of the cesaro supremum of the means are necessary

the multiplier for the ball and radial functions abstract it is shown that the multiplier for the ball is restricted weak type on radial functions in l p r n when p 2n n 1 interpolation then yields a theorem of herz

critical exponents of two dimensional potts and bond percolation models critical exponents of the two dimensional q state potts model are calculated by means of finite size scaling and transfer matrix techniques for continuous q results for the temperature exponent agree accurately with the conjecture of den nijs 1979 the magnetic exponent is found to behave in accordance with the conjecture of nienhuis et al 1980

harmonic analysis operators associated with multidimensional bessel operators we establish that the maximal operator and the littlewood paley g function associated with the heat semigroup defined by multidimensional bessel operators are of weak type 1 1 we also prove that riesz transforms in the multidimensional bessel setting are of strong type p p for every 1 p

pseudospectral differencing methods for characteristic roots of delay differential equations in d breda s maset and r vermiglio ima j numer anal 24 2004 pp 1 19 and d breda the infinitesimal generator approach for the computation of characteristic roots for delay differential equations using bdf methods research report udmi rr17 2002 dipartimento di matematica e informatica universita degli studi di udine udine italy 2002 the authors proposed to compute the characteristic roots of delay differential equations ddes with multiple discrete and distributed delays by approximating the derivative in the infinitesimal generator of the solution operator semigroup by runge kutta rk and linear multistep lms methods respectively in this work the same approach is proposed in a new version based on pseudospectral differencing techniques we prove the spectral accuracy convergence behavior typical of pseudospectral schemes as also illustrated by some numerical experiments

functional calculus for generators of symmetric contraction semigroups we prove that every generator of a symmetric contraction semigroup on a finite measure space admits for 1 p a hormander type holomorphic functional calculus on lp in the sector of angle p arcsin 1 2 p the obtained angle is optimal

bochner riesz profile of anharmonic oscillator l d2dx2 x we investigate spectral multipliers bochner riesz means and the convergence of eigenfunction expansion corresponding to the schrodinger operator with anharmonic potential l d2dx2 x we show that the bochner riesz profile of the operator l completely coincides with such profile of the harmonic oscillator h d2dx2 x2 it is especially surprising because the bochner riesz profile of the one dimensional standard laplace operator is known to be essentially different and the case of operators h and l resembles more the profile of multidimensional laplace operators another surprising element of the main obtained result is the fact that the proof is not based on restriction type estimates and instead an entirely new perspective has to be developed to obtain the critical exponent for bochner riesz means convergence

the weak type l1 convergence of eigenfunction expansions for pseudodifferential operators soit m une variete compacte connexe lisse sans bord de dimension n 2 soit p d un operateur pseudodifferentiel elliptique d ordre 1 sur m qui est auto adjoint par rapport a une densite positive lisse dx on demontre la convergence faible de type l 1 pour les developpements en fonctions propres

two weight norm inequalities for the cesaro means of generalized hermite expansions we prove two weight norm inequalities for cesaro means of generalized hermite polynomial series and for the supremum of these means a result about weak boundedness and an almost everywhere convergence result are also obtained

equiconvergence theorems for fourier bessel expansions with applications to the harmonic analysis of radial functions in euclidean and non euclidean spaces we shall prove an equiconvergence theorem between fourier bessel expansions of functions in certain weighted lebesgue spaces and the classical cosine fourier expansions of suitable related functions these weighted lebesgue spaces arise naturally in the harmonic analysis of radial functions on euclidean spaces and we shall use the equiconvergence result to deduce shag results for the pointwise almost everywhere convergence of fourier integrals of radial functions in the lorentz spaces l p q r n also we shall briefly apply the above approach to the study of the harmonic analysis of radial functions on noneuclidean hyperbolic spaces

weak type estimates for bochner riesz spherical summation multipliers we consider the bochner riesz multiplier tofw 1 1 12 tjw 8 0 where denotes the fourier transform it is shown that the multiplier operator to is weak type po po acting on lpo rn radial functions where po is the critical value 2n n 1 28

ising like critical phenomena in two dimensional percolation numerical evidence a recent analytic theory of two dimensional isotropic percolation indicates that the singular behaviour near pc is identical to that appropriate for the associated dilute ising critical point a specific prediction supported by recent series expansion studies is that the mean number of clusters k p presents a singularity of the form k p approximately mod p pc mod 2ln mod ln mod p pc rather than the accepted form k p approximately mod p pc mod 2 alpha with alpha 2 3 a novel numerical and renormalisation group finite size scaling analysis of the nature of the singularity in k p is presented in support of the new theory which implies the absence of a separate universality class for two dimensional percolation processes this study is consistent with the effective values of the exponents alpha 0 and nu 1

bounds for spectral multipliers on nilpotent groups a criterion is given for the lp boundedness of a class of spec tral multiplier operators associated to left invariant homogeneous subelliptic second order differential operators on nilpotent lie groups generalizing a theo rem of hormander for radial fourier multipliers on euclidean space the order of differentiability required is half the homogeneous dimension of the group improving previous results in the same direction

two weight norm inequalities for cesaro means of laguerre expansions two weight lp norm inequalities are proved for cesaro means of laguerre polynomial series and for the supremum of these means these extend known norm inequalities even in the single power weight and unweighted cases by including all values of p 1 for all positive orders of the cesaro summation and all values of the laguerre parameter a 1 almost everywhere convergence results are obtained as a corollary for the cesaro means the hypothesized conditions are shown to be necessary for the norm inequalities necessity results are also obtained for the norm inequalities with the supremum of the cesaro means in particular for the single power weight case the conditions are necessary and sufficient for summation of order greater than one sixth

an iterative method for suboptimal control of linear time delayed systems abstract this article presents a new approach for solving the optimal control problem ocp of linear time delay systems with a quadratic cost functional the proposed method can also be used for designing optimal control time delay systems with disturbance in this study the variational iteration method vim is employed to convert the original time delay optimal control problem tdocp into a sequence of nonhomogeneous linear two point boundary value problems tpbvps the optimal control law obtained consists of an accurate linear feedback term and a nonlinear compensation term which is the limit of an adjoint vector sequence the feedback term is determined by solving riccati matrix differential equation by using the finite step iteration of a nonlinear compensation sequence we can obtain a suboptimal control law finally illustrative examples are included to demonstrate the validity and applicability of the technique

characterizations of hankel multipliers we give characterizations of radial fourier multipliers as acting on radial l p functions 1 p 2d d 1 in terms of lebesgue space norms for fourier localized pieces of the convolution kernel this is a special case of corresponding results for general hankel multipliers besides l p l q bounds we also characte rize weak type inequalities and intermediate inequalities involving lorentz spaces applications include results on interpolation of multiplier spaces

plancherel type estimates and sharp spectral multipliers abstract we study general spectral multiplier theorems for self adjoint positive definite operators on l2 x where x is any open subset of a space of homogeneous type we show that the sharp hormander type spectral multiplier theorems follow from the appropriate estimates of the l2 norm of the kernel of spectral multipliers and the gaussian bounds for the corresponding heat kernel the sharp hormander type spectral multiplier theorems are motivated and connected with sharp estimates for the critical exponent for the riesz means summability which we also study here we discuss several examples which include sharp spectral multiplier theorems for a class of scattering operators on r3 and new spectral multiplier theorems for the laguerre and hermite expansions

some inequalities for singular convolution operators in spaces suppose that a bounded function m satisfies a localized multiplier condition supt o ii0m tp imp 0 the purpose of this paper is to find easily verified conditions that 1 implies me mp the condition llmll 0 then we ask under which conditions supt 0 qm tp 1mp oc implies m c mp throughout this paper x will always be chosen as in the following definition x c co rn satisfies a tauber condition with respect to the dilations tp if for every x 0 there is a tx such that q tpx 0 sometimes we need special bump functions of the following kind let p e c rn be a p homogeneous distance function this means that p tpx tp x received by the editors august 21 1986 1980 mathermatics subject lassijication 1985 revision primary 42b15 42b20 42b25

on the range of validity of the 6 e expansion for percolation in 6 epsilon dimensions the critical properties of percolation can be computed starting from a phi 3 theory having the same symmetry of the one state potts model the authors argue that this fixed point must be unstable with respect to a phi 4 interaction in two dimensions therefore a phi 4 perturbation is relevant this suggestion is confirmed by an explicit two loops computation

sharp spectral multipliers for operators satisfying generalized gaussian estimates abstract let l be a non negative self adjoint operator acting on l 2 x where x is a space of homogeneous type assume that l generates a holomorphic semigroup e t l which satisfies generalized m th order gaussian estimates in this article we study singular and dyadically supported spectral multipliers for abstract self adjoint operators we show that in this setting sharp spectral multiplier results follow from plancherel or stein tomas type estimates these results are applicable to spectral multipliers for a large class of operators including m th order elliptic differential operators with constant coefficients biharmonic operators with rough potentials and laplace type operators acting on fractals

stability and l_ 2 performance analysis of stochastic delayed neural networks this brief focuses on the robust mean square exponential stability and l2 performance analysis for a class of uncertain time delay neural networks perturbed by both additive and multiplicative stochastic noises new mean square exponential stability and l2 performance criteria are developed based on the delay partition lyapunov krasovskii functional method and generalized finsler lemma which is applicable to stochastic systems the analytical results are established without involving any model transformation estimation for cross terms additional free weighting matrices or tuning parameters numerical examples are presented to verify that the proposed approach is both less conservative and less computationally complex than the existing ones

on the weak behaviour of partial sums of legendre series sn j x denotes the nth partial sum of the legendre expansion of f x it was proved by h pollard in 4 that if f e lp 1 1 where 4 3 4 however j newman and w rudin in 3 showed that the inequality iisnv j ilp cp iifiip is false atp 4 orp 4 3 the aim of this paper is to examine the weak behaviour atp 4 much in the spirit of the hilbert transform at l 0 2wt we now need some notations to state our main results the lebesgue measure of any set e c 1 1 will be denoted as m e xe x will as usual denote the characteristic function of a set e we then have

ising like critical phenomena in two dimensional percolation series expansion evidence a recent analytic theory of two dimensional isotropic percolation indicates that the critical behaviour near pc is determined by the same renormalisation group fixed point describing the behaviour of the associated dilute ising model a specific prediction is that the mean number of clusters k p contains a singularity of the form k p approximately mod p pc mod 2ln mod ln mod p pc rather than the currently accepted form k p approximately mod p pc mod 2 alpha with alpha 2 3 novel series expansion studies for the site and bond percolation problems on the triangular and simple quadratic lattices respectively are presented in support of the new finding which implies the absence of a separate universality class for two dimensional percolation processes

improved bounds for stein s square functions supported in part by nrf grant 2009 0072531 korea micinn grant mtm2010 16518 spain erc grant 277778 europe and nsf grant 0652890 usa

weak type 1 1 bounds for rough operators ii in this paper we consider four operators arising in harmonic analysis in r n 1 they share one feature each is known to be bounded on lp for all 1 p o but for each it has been an open question whether a weak type 1 1 bound also holds in two cases we show that the weak 1 1 bound does hold and for the others we establish weaker estimates which are still sharper than the lp boundedness although the four operators are quite distinct they all involve convolution kernels which are related to the singular integrals of calderon and zygmund but which lack the smoothness required for the usual analysis in each case this defect is mitigated by the presence of some form of curvature consider the bochner riesz means

q state potts models in d dimensions migdal kadanoff approximation the first and second order phase transitions of the q state potts models are obtained in arbitrary dimension d critical and tricritical behaviours merge and annihilate at qc d clearing the way to first order transitions at q qc d by the condensation of effective vacancies the value of qc d decreases with increasing d from diverging as exp 2 d 1 at d to 1 to qc 2 3 81 cf exact value of 4 to lower values at d 2 for given d a changeover in critical behaviour occurs at q1 d as the critical fixed points merge from the potts lattice gas region to the undiluted potts limit it is suggested that the power law singularities of the percolation problem q to 1 have logarithmic corrections

l p norms of certain kernels on the n dimensional torus in this paper we study a class of kernels fr which generalize the bochner riesz kernels on the n dimensional torus our main result consists in upper estimates for the lp norms of fr as r tends to infinity as a consequence we prove a convergence theorem for means of functions belonging to suitable besov spaces 1 throughout this paper we identify the n dimensional torus tn n 2 with the n dimensional cube qn x e r n 1 2 0 we form the means fr g t f r m g m exp 2 7imt 1 m where t e tn m ranges on the integer lattice zn and denotes the convolution in tn in several cases one is able to estimate the l 1 tn norm of the kernel fr t e f r 1m exp 2 srimt 2 m and if 0 e s and f o 1 deduce convergence results for the means 1 as r x oc when g belongs to certain classes of periodic functions see e g 1 3 7 8 12 for instance if f x 1 i x12 8 when lxl 1 then fr k a is the familiar bochner riesz kernel in this case k i babenko cf 1 has shown that iikriil is of the same order as r n 1 2 as r tends to infinity n 2 0 8 n 1 2 while stein proved the estimate iikr il log r if 8 n 1 2 9 recently yudin 12 obtained the estimate lf o r n1 2 if f is the characteristic function of a closed balanced set c whose boundary has finite upper minkowski measure the method of yudin which also yields estimates for the lp tn norms of fr uses jackson approximation theorem which in tum involves estimates for the l2 r n modulus of continuity of f such a method can be adapted to a more general situation as we show in this paper in 2 4 we consider kernels of the type 2 associated with functionsf whose derivatives of certain orders are controlled by not necessarily positive received by the editors may 13 1980 1980 mathematics subject classification primary 42b99 secondary 46e35