A note on absolute summability factors

Summary We obtain sufficient conditions for the series <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>\sum a_{n}\lambda_{n}$ to be absolutely summable of order $k$ by a<span lang=FR style='font-size:10.0pt; mso-ansi-language:FR'>triangular matrix.     

A note on two absolute summability methods

In this paper we have established a relation between |R,p _n ;δ|_k and |R,q _n ;δ|_k summability methods which generalizes the results of Bor [1] and Bosanquet [2].     

A localization theorem for Laguerre expansions

Regularity properties of Laguerre means are studied in terms of certain Sobolev spaces defined using Laguerre functions. As an application we prove a localization theorem for Laguerre expansions.     

Summability of Product Jacobi Expansions

Orthogonal expansions in product Jacobi polynomials with respect to the weight function W_α, β(x)=∏^d_j=1 (1−x_j)^α_j (1+x_j)^β_j on [−1, 1]^d are studied. For α_j, β_j>−1 and α_j+β_j−1, the Cesàro (C, δ) means of the product Jacobi expansion converge in the norm of L^p(W_{α, β}, [−1, 1]^d), 1p<∞, and C([−1, 1]^d) if

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Moreover, for α_j, β_j−1/2, the (C, δ) means define a positive linear operator if and only if δ∑^d_i=1 (α_i+β_i)+3d−1.     

A note on absolute summability factors

In this paper, by using an almost increasing and δ-quasi-monotone sequence, a general theorem on φ- | C, α | _k summability factors, which generalizes a result of Bor [3] on φ |C, 1| _k summability factors, has been proved under weaker and more general conditions.     

Two-weight norm inequalities for Cesàro means of Laguerre expansions

Two-weight norm inequalities are proved for Cesàro 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 for all positive orders of the Cesàro summation and all values of the Laguerre parameter -1$">. Almost everywhere convergence results are obtained as a corollary. For the Cesàro 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 Cesàro means; in particular, for the single power weight case the conditions are necessary and sufficient for summation of order greater than one sixth.

     

A note on absolute weighted mean summability factors

In this paper we have proved a main theorem concerning the | $\bar N$ , p _n; δ |_k summability methods, which generalizes a result of Bor and Özarslan [3].     

Monotonicity of zeros of Laguerre polynomials

Denote by x_nk(α), k=1,…,n, the zeros of the Laguerre polynomial . We establish monotonicity with respect to the parameter α of certain functions involving x_nk(α). As a consequence we obtain sharp upper bounds for the largest zero of .     

级数的常规可和，Cesàro可和与Abel可和的几点讨论

讨论级数常规可和、Cesaro可和与Abel可和的关系．利用数学分析级数理论,证明Abel可和适用范围最广,Cesaro可和其次,级数常规可和适用范围最小．这个结论丰富了经典级数理论,为实际应用中选用合适可和提供依据．     

A Cohen type inequality for Laguerre-Sobolev expansions

Let introduce the Sobolev type inner product

     

Divergent Laguerre series

We prove failure of a.e. convergence of partial sums of Laguerre expansions of functions for 4$">. The idea which is used goes back to Stanton and Tomas. We follow Meaney's paper (1983), where divergence results were proved in the Jacobi polynomial case.

     

A sharp weighted transplantation theorem for Laguerre function expansions

We find the sharp range of boundedness for transplantation operators associated with Laguerre function expansions in Lp spaces with power weights. Namely, the operators interchanging and are bounded in Lp(yδp) if and only if , where ρ=min{α,β}. This improves a previous partial result by Stempak and Trebels, which was only sharp for ρ?0. Our approach is based on new multiplier estimates for Hermite expansions, weighted inequalities for local singular integrals and a careful analysis of Kanjin's original proof of the unweighted case. As a consequence we obtain new results on multipliers, Riesz transforms and g-functions for Laguerre expansions in Lp(yδp).     

On a certain family of generalized Laguerre polynomials

Following the work of Schur and Coleman, we prove the generalized Laguerre polynomial is irreducible over the rationals for all n?1 and has Galois group A_n if n+1 is an odd square, and S_n otherwise. We also show that for certain negative integer values of α and certain congruence classes of n modulo 8, the splitting field of L_n^(α)(x) can be embedded in a double cover.     

On differential equations for Sobolev-type Laguerre polynomials

The Sobolev-type Laguerre polynomials are orthogonal with respect to the inner product

where , and . In 1990 the first and second author showed that in the case and the polynomials are eigenfunctions of a unique differential operator of the form

where are independent of . This differential operator is of order if is a nonnegative integer, and of infinite order otherwise. In this paper we construct all differential equations of the form

where the coefficients , and are independent of and the coefficients , and are independent of , satisfied by the Sobolev-type Laguerre polynomials . Further, we show that in the case and the polynomials are eigenfunctions of a linear differential operator, which is of order if is a nonnegative integer and of infinite order otherwise. Finally, we show that in the case and the polynomials are eigenfunctions of a linear differential operator, which is of order if is a nonnegative integer and of infinite order otherwise.

     

On the rate of pointwise strong summability of orthogonal expansions

There is introduced a modificated local modulus of continuity as a measure of pointwise strong summability of expansion with respect to polynomial-like orthonormal system. The approximation versions and generalizations of the some special cases of the known results of K. Tandori [6], [7] and G. Freud [4] are obtained.     

On |A|k summability factors

Summary The paper deals with absolute summability factors for infinite series. The main result obtained in this paper generalizes a recent paper of Mazhar.     

Hardy's inequalities for Hermite and Laguerre expansions

Hardy's inequalities are proved for higher-dimensional Hermite and special Hermite expansions of functions in Hardy spaces. Inequalities for multiple Laguerre expansions are also deduced.

     

Differential operators having Sobolev type Laguerre polynomials as eigenfunctions

We consider the polynomials orthogonal with respect to the Sobolev type inner product

where and is a nonnegative integer. It is the purpose of this paper to show that these polynomials are eigenfunctions of a class of linear differential operators containing one that is of finite order if is a nonnegative integer and

     

基于Laguerre函数的Landau型不等式

基于经典正交的广义 Laguerre多项式 Lsn( x)建立了 Landau型不等式 ,并且指出其不等式的系数在某种意义上是最好可能的 ,加强并推广了 Varma和 Bojanov的结果 .     

Summability of Fourier orthogonal series for Jacobi weight on a ball in

Fourier orthogonal series with respect to the weight function
on the unit ball in are studied. Compact formulae for the sum of the product of orthonormal polynomials in several variables and for the reproducing kernel are derived and used to study the summability of the Fourier orthogonal series. The main result states that the expansion of a continuous function in the Fourier orthogonal series with respect to is uniformly summable on the ball if and only if .