Poussin Means Research Articles

On a conjecture of S.P. Robinson

In his thesis, S.P. Robinson made a conjecture concerning the polynomials P n β ( z ) = 1 + ∑ k = 1 n ∏ j = 0 k − 1 n − j β + n + j z k , n ∈ N , β ⩾ 1 , namely that z P n β is prestarlike of order ( 3 − β ) / 2 . These polynomials are closely related to the de la Vallée Poussin means (the case β = 1 ). We prove this conjecture in a more general form and show that these functions constitute a sort of two-dimensional subordination chain. These results are then compared with similar ones for Cesáro means of various orders.

Approximation error of the Whittaker cardinal series in terms of an averaged modulus of smoothness covering discontinuous signals

The Whittaker–Shannon–Kotel'nikov sampling theorem enables one to reconstruct signals f bandlimited to [ − π W , π W ] from its sampled values f ( k / W ) , k ∈ Z , in terms of ( S W f ) ( t ) ≡ ∑ k = − ∞ ∞ f ( k W ) sinc ( W t − k ) = f ( t ) ( t ∈ R ) . If f is continuous but not bandlimited, one normally considers lim W → ∞ ( S W f ) ( t ) in the supremum-norm, together with aliasing error estimates, expressed in terms of the modulus of continuity of f or its derivatives. Since in practice signals are however often discontinuous, this paper is concerned with the convergence of S W f to f in the L p ( R ) -norm for 1 < p < ∞ , the classical modulus of continuity being replaced by the averaged modulus of smoothness τ r ( f ; W −1 ; M ( R ) ) p . The major theorem enables one to sample any bounded signal f belonging to a certain subspace Λ p of L p ( R ) , the jump discontinuities of which may even form a set of measure zero on R . A corollary gives the counterpart of the approximate sampling theorem, now in the L p -norm. The averaged modulus, so far only studied for functions defined on a compact interval [ a , b ] , had first to be extended to functions defined on the whole real axis R . Basic tools are the de La Vallée Poussin means and a semi-discrete Hilbert transform.

Almost Everywhere Convergence of Orthogonal Expansions of Several Variables

For a weighted L1 space on the unit sphere of Rd+1, in which the weight functions are invariant under finite reflection groups, a maximal function is introduced and used to prove the almost everywhere convergence of orthogonal expansions in h-harmonics. The result applies to various methods of summability, including the de La Vallee Poussin means and the Cesaro means. Similar results are also established for weighted orthogonal expansions on the unit ball and on the simplex of Rd.

Approximation by Means of h-Harmonic Polynomials on the Unit Sphere

The h-harmonics are analogues of the ordinary harmonics, they are orthogonal homogeneous polynomials on the sphere with respect to a weight function that is invariant under a reflection group. Two means of associated orthogonal expansions, the de la Vallee Poussin means and an analog of spherical means, are defined and their approximation behaviors are studied. A weighted modulus of smoothness is defined using the modified spherical means and is proved to be equivalent to a weighted K-modulus defined using the differential-difference h-spherical Laplacian. A Bernstein type inequality for the h-spherical Laplacian is also established.

Approximate Properties of the de la Vallee Poussin Means for the Discrete Fourier-Jacobi Sums

We consider the system of the classical Jacobi polynomials of degree at most N which generate an orthogonal system on the discrete set of the zeros of the Jacobi polynomial of degree N. Given an arbitrary continuous function on the interval [-1,1], we construct the de la Vallee Poussin-type means for discrete Fourier–Jacobi sums over the orthonormal system introduced above. We prove that, under certain relations between N and the parameters in the definition of de la Vall'ee Poussin means, the latter approximate a continuous function with the best approximation rate in the space C[-1,1] of continuous functions.

Asymptotic approximation by de la Vallée Poussin means and their derivatives

The concern of this paper is the study of local approximation properties of the de la Vallée Poussin means V n . We derive the complete asymptotic expansion of the operators V n and their derivatives as n tends to infinity. It turns out that the appropriate representation is a series of reciprocal factorials. All coefficients are calculated explicitly.

A continuous extension of the de la Vallée Poussin means

Among the many interesting results of their 1958 paper, G. Polya and I. J. Schoenberg studied the de la Vallee Poussin means of analytic functions. These are polynomial approximations of a given analytic function on the unit disk obtained by taking Hadamard products of the functionf with certain polynomialsV n (z), wheren is the degree of the polynomial. The polynomial approximationsV n *f converge locally uniformly tof asn→∞. In this paper, we define a subordination chainV λ (z),γ>0, |z|<1, of convex mappings of the disk that for integer values is the same as the previously definedV n (z). Iff is a conformal mapping of the diskD onto a convex domain, thenV λ *f→f locally uniformly as λ→∞, and in fact $$V_{\lambda _1 } * f(\mathbb{D}) \subset V_{\lambda _2 } * f(\mathbb{D}) \subset f(\mathbb{D})$$ when λ2 > λ1. We also consider Hadamard products of theV λ with complex-valued harmonic mappings of the disk.

On the partial sums, Cesáro and de la Valleé Poussin means of convex and starlike functions of order 1/2

In this paper we study certain properties of partial sums, cesaro and de la vallee Poussin means of convex and starlike functions.

Christoffel Functions and Fourier Series for Multivariate Orthogonal Polynomials

The Christoffel functions Λ n ( dμ) associated with a general nonnegative measure μ on R d are studied. The asymptotics of Λ n ( dμ) are derived for μ supported on [− 1,1] d . The estimates of Λ n ( dμ) are used to study the summability of the multivariate orthogonal polynomials associated with μ. The pointwise convergence of the partial sums of the orthogonal expansion, and their ( C, 1) and de la Vallée Poussin means are considered.

Approximation by double Walsh polynomials

We study the rate of approximation by rectangular partial sums, Cesàro means, and de la Vallée Poussin means of double Walsh‐Fourier series of a function in a homogeneous Banach space X. In particular, X may be Lp(I2), where 1≦p &lt; ∞ and I2 = [0, 1) × [0, 1), or CW(I2), the latter being the collection of uniformly W‐continuous functions on I2. We extend the results by Watari, Fine, Yano, Jastrebova, Bljumin, Esfahanizadeh and Siddiqi from univariate to multivariate cases. As by‐products, we deduce sufficient conditions for convergence in Lp(I2)‐norm and uniform convergence on I2 as well as characterizations of Lipschitz classes of functions. At the end, we raise three problems.

On the Approximation by the Generalized de la Vallée Poussin Means of Fourier Series in the SpacesWrHα

Mathematische NachrichtenVolume 120, Issue 1 p. 63-71 Article On the Approximation by the Generalized de la Vallée Poussin Means of Fourier Series in the Spaces WrHα Ryszard Gajewski, Ryszard Gajewski A. Mickiewicz University, Institute of Mathematics, Matejki, 48/49, Poznań, PolandSearch for more papers by this authorWłodzimierz Łenski, Włodzimierz Łenski A. Mickiewicz University, Institute of Mathematics, Matejki, 48/49, Poznań, PolandSearch for more papers by this author Ryszard Gajewski, Ryszard Gajewski A. Mickiewicz University, Institute of Mathematics, Matejki, 48/49, Poznań, PolandSearch for more papers by this authorWłodzimierz Łenski, Włodzimierz Łenski A. Mickiewicz University, Institute of Mathematics, Matejki, 48/49, Poznań, PolandSearch for more papers by this author First published: 1985 https://doi.org/10.1002/mana.19851200107Citations: 2AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volume120, Issue11985Pages 63-71 RelatedInformation

On the divergence of de la Vallée Poussin means of Fourier series

Пусть {λ n 1 t8 — монотонн ая последовательнос ть натуральных чисел. Дл я каждой функции feL(0, 2π) с рядом Фурье строятся обобщенные средние Bалле Пуссена $$V_n^{(\lambda )} (f;x) = \frac{{a_0 }}{2} + \mathop \sum \limits_{k = 1}^n (a_k \cos kx + b_k \sin kx) + \mathop \sum \limits_{k = n + 1}^{n + \lambda _n } \left( {1 - \frac{{k - n}}{{\lambda _n + 1}}} \right)\left( {a_k \cos kx + b_k \sin kx} \right).$$ Доказываются следую щие теоремы. 1. Если λn=o(n), то существуе т функция feL(0, 2π), для кот орой последовательность {Vn (λ)(ƒ;x)} расходится почти вс юду. 2. Если λn=o(n), то существуе т функция feL(0, 2π), для кот орой последовательность $$\left\{ {\frac{1}{\pi }\mathop \smallint \limits_{ - \pi /\lambda _n }^{\pi /\lambda _n } f(x + t)\frac{{\sin (n + \tfrac{1}{2})t}}{{2\sin \tfrac{1}{2}t}}dt} \right\}$$ расходится почти всю ду .

Nikolskii-type inequalities in connection with regular spectral measures

There are counterparts for polynomials in several variables as well as for entire functions of exponential type, also connected with the names of D. Jackson, M. Plancherel, G. Polya, G. Szeg6, A. Zygmund, for all the details and comprehensive bibliographical comments see [8, 15, 16]. The aim of the present note is to construct a framework which enables one to consider inequalities of type (1.2) within a general class of orthogonal expansions in Banach spaces. To this end, Section 2 develops a multiplier concept for Banach spaces Xwhich are admissible with respect to a regular spectral measure E for some appropriate Hilbert space H. This continues and extends slightly our approach given in [4] which already turned out to be quite useful in connection with other generalizations (cf. [16] and the literature cited there). Let us mention that the classical situation (1.2) is covered for 1 <p=< c~ with H = L ~ and X=LP~, 1 < p < ~ , or X = L ~ =(L~)*, respectively. Section 3 defines polynomials in this general frame and sets up de la Vallre Poussin (or delayed) means (see Theorem 2), a basic tool in the treatment of problems (such as (1.2)) coneemed with functions having compact spectra. Section 4 first deals with interpolation of admissible Banach spaces. Here we only discuss the real method of Lions-Peetre but other constructions such as the complex method of Calderon may be considered as well (see [9, 18] for details). This is then used in Theorem 3 to derive general Nikolskii-type inequalities. Finally, Section 5 is concerned with some applications to illustrate the wide applicability of the general results obtained. While Section 5.1 considers trigonometric polynomials in several variables for different scales of admissible Banach spaces, Section 5.2 treats the (eontinuous)

De la Vall� Poussin means of Fourier-Gegenbauer series

A bound is given for the order of growth of de la Valle Poussin means of a Fourier-Gegenbauer series.

On a conjecture concerning Schlicht functions

is schlicht in the unit circle. This will be disproved in the following lines. Let D be the image of the unit circle by w— J^f avz We denote by the symbols 5, 2 and K the classes of such power series for which D is schlicht, schlicht and star-shaped, schlicht and convex, respectively. Evidently KCZCS. Observe now that X)* z(E.K. By a recent result concerning de la Vallee Poussin means [2, p. 298] we conclude that