Vallee Poussin Research Articles

Estimates for the Best Approximations of Periodic Functions in Terms of Deviations

We estimate the best approximation of a function by trigonometric polynomials in terms of deviations. To illustrate general results, we consider estimates connected with the Vallee–Poussin integral and the generalized Weierstrass integral.

Weak type inequalities for the ℓ 1-summability of higher dimensional Fourier transforms

Under some weak conditions on θ, it was verified in [21, 17] that the maximal operator of the l1-θ-means of a tempered distribution is bounded from Hp(ℝd) to Lp(ℝd) for all d/(d + α) < p ≤ ∞, where 0 < α ≤ 1 depends only on θ. In this paper, we prove that the maximal operator is bounded from Hd/(d+α)(ℝd) to the weak Ld/(d+α)(ℝd) space. The analogous result is given for Fourier series, as well. Some special cases of the l1-θ-summation are considered, such as the Weierstrass, Picard, Bessel, Fejer, de La Vallee-Poussin, Rogosinski and Riesz summations.

Lebesgue-type inequalities for the de la Vallée-poussin sums on sets of entire functions

For functions from the sets C ψ β L s , 1 ≤ s ≤ ∞, where ψ(k) > 0 and $ {\lim_{{k\to \infty }}}\frac{{\psi \left( {k+1} \right)}}{{\psi (k)}} $ , we obtain asymptotically sharp estimates for the norms of deviations of the de la Vallee-Poussin sums in the uniform metric represented in terms of the best approximations of the (ψ, β) -derivatives of functions of this kind by trigonometric polynomials in the metrics of the spaces L s . It is shown that the obtained estimates are sharp on some important functional subsets.

Приближение гладких функций в L p(x) 2π средними Валле–Пуссена

Приближение гладких функций в L p(x) 2π средними Валле–Пуссена

Uniform approximation on [−1, 1] via discrete de la Vallée Poussin means

Starting from the function values on the roots of Jacobi polynomials, we construct a class of discrete de la Vallee Poussin means, by approximating the Fourier coefficients with a Gauss---Jacobi quadrature rule. Unlike the Lagrange interpolation polynomials, the resulting algebraic polynomials are uniformly convergent in suitable spaces of continuous functions, the order of convergence being comparable with the best polynomial approximation. Moreover, in the four Chebyshev cases the discrete de la Vallee Poussin means share the Lagrange interpolation property, which allows us to reduce the computational cost.

Triangular summability of two-dimensional Fourier transforms

We consider the triangular summability of two-dimensional Fourier transforms, and show that the maximal operator of the triangular-θ-means of a tempered distribution is bounded from Hp(ℝ2) to Lp(ℝ2) for all 2/(2 + α) < p ≤ ∞; consequently, it is of weak type (1,1), where 0 < α ≤ 1 is depending only on θ. As a consequence, we obtain that the triangular-θ-means of a function f ∈ L1(ℝ2) converge to f a.e. Norm convergence is also considered, and similar results are shown for the conjugate functions. Some special cases of the triangular-θ-summation are considered, such as the Weierstrass, Picar, Bessel, Fejer, de la Vallee-Poussin, Rogosinski, and Riesz summations.

Integral representation of functions of bounded second Φ-variation in the sense of Schramm

In this article we introduce the concept of second \(\Phi\)-variation in the sense of Schramm for normed-space valued functions defined on an interval \([a,b] \subset \mathbb{R}\). To that end we combine the notion of second variation due to de la Vallee Poussin and the concept of \(\varphi\)-variation in the sense of Schramm for real valued functions. In particular, when the normed space is complete we present a characterization of the functions of the introduced class by means of an integral representation. Indeed, we show that a function \(f \in \mathbb{X}^{[a,b]}\) (where \(\mathbb{X}\) is a reflexive Banach space) is of bounded second \(\Phi\)-variation in the sense of Schramm if and only if it can be expressed as the Bochner integral of a function of (first) bounded variation in the sense of Schramm.

On the rate of approximation of functions in Lipschitz norms

We show that the problem on the rate of approximation of functions in Lipschitz norms is reduced to estimating approximations in C. Approximations in Lipschitz norms by Vallee Poussin sums are considered as an example.

Approximation of Poisson integrals by de la Valleé-Poussin sums in uniform and integral metrics

On the classes of Poisson integrals of functions belonging to the unit balls of the spaces L s , 1 ≤ s ≤ ∞, we establish asymptotic equalities for upper bounds of approximations by de la Vallee-Poussin sums in the uniform metric. Asymptotic equalities are also obtained for the case of approximation by de la Vallee-Poussin sums in the metrics of the spaces L s , 1 ≤ s ≤ ∞, on the classes of Poisson integrals of functions belonging to the unit ball of the space L 1.

Approximation by de la Vallée-Poussin operators on the classes of functions locally summable on the real axis

For the least upper bounds of deviations of the de la Vallee-Poussin operators on the classes $ \hat{L}_\beta^\psi $ of rapidly vanishing functions ψ in the metric of the spaces $ {\hat{L}_p} $ , 1 ≤ p ≤ ∞, we establish upper estimates that are exact on some subsets of functions from $ {\hat{L}_p} $ .

Approximation of poisson integrals by repeated de la Vallée Poussin sums

We obtain asymptotic equalities for upper bounds of deviations of the repeated de la Vallee Poussin sums on classes of Poisson integrals. Under certain conditions, these equalities guarantee the solvability of the Kolmogorov–Nikol’skii problem for the repeated de la Vallee Poussin sums and classes of Poisson integrals. We indicate conditions under which the repeated de la Vallee Poussin sums guarantee a better order of approximation than ordinary de la Vallee Poussin sums.

Generalized Gibbs phenomenon for Fourier partial sums and de la Vallée-Poussin sums

It is well-known that the Fourier partial sums of a function exhibit the Gibbs phenomenon at a jump discontinuity. We study the same question for de la Vallee-Poussin sums. Here we find a new Gibbs function and a new Gibbs constant. When the function is continuous, a behavior similar to the Gibbs phenomenon also occurs at a kink. We call it the “generalized Gibbs phenomenon”. Let \(F_{n}(x):=\frac{k_{n}(g,x)-g(x)}{k_{n}(g,x_{0})-g(x_{0})}\), where x0 is a kink and where kn(g,x) represents Fourier partial sums and de la Vallee-Poussin sums. We show that Fn(x) exhibits the “generalized Gibbs phenomenon”. New universal Gibbs functions for both sums are derived.

The Elementary Proof of the Prime Number Theorem

P rime numbers are the atoms of our mathematical universe. Euclid showed that there are infinitely many primes, but the subtleties of their distribution continue to fascinate mathematicians. Letting p(n) denote the number of primes p B n, Gauss conjectured in the early nineteenth century that pðnÞ n=lnðnÞ. In 1896, this conjecture was proven independently by Jacques Hadamard and Charles de la Vallee-Poussin. Their proofs both used complex analysis. The search was then on for an ‘‘elementary proof’’ of this result. G. H. Hardy was doubtful that such a proof could be found, saying if one was found ‘‘that it is time for the books to be cast aside and for the theory to be rewritten.’’ But in the Spring of 1948 such a proof was found. Almost immediately there was controversy. Was the proof attributable to Atle Selberg or was the proof attributable to Atle Selberg and Paul Erd} os? For decades there seemed to be two mathematical camps with wildly different viewpoints. In the twenty-first century the controversy has finally subsided. Among previous discussions of the controversy, we mention particularly Goldfeld [1] (from which the previously mentioned quotation by Hardy is taken) and the book [3] of Paul Hoffman. Ernst Straus was in a unique position to observe the beginnings of the controversy. He then held a position at the Institute for Advanced Study as a special assistant to Albert Einstein. (We believe Straus is the only person to have joint papers with Einstein and Erd} os.) Straus had already worked a great deal with Erd} os, and this work would continue throughout his life. Sometime in the early 1970s (we aren’t sure of the exact dates), Straus wrote the account we present here. He did not want the notes to be published while the participants were still alive. Ernst Straus was, for us and for many of his friends, a man of great wisdom. He certainly attempted in these notes to give as faithful an account of the events as he could. Whether he succeeded is a judgment for the reader to make. Certainly, he was far closer to Erd} os than to Selberg. Let us be clear that we two authors both have an Erd} os number of one, and our own associations with Erd} os were long and profound. We feel that the Straus recollections are an important historic contribution. We also believe that the controversy itself sheds considerable light on the changing nature of mathematical research. In November 2005, Atle Selberg was interviewed by Nils A. Baas and Christian F. Skau [4]. He recalled the events of 1948 with remarkable precision. We quote extensively from that account. Selberg had first shown that X

Estimates of the norm of the Mercer kernel matrices with discrete orthogonal transforms

The paper is related to the lower and upper estimates of the norm for Mercer kernel matrices. We first give a presentation of the Lagrange interpolating operators from the view of reproducing kernel space. Then, we modify the Lagrange interpolating operators to make them bounded in the space of continuous function and be of the de la Vallee Poussin type. The order of approximation by the reproducing kernel spaces for the continuous functions is thus obtained, from which the lower and upper bounds of the Rayleigh entropy and the l2-norm for some general Mercer kernel matrices are provided. As an example, we give the l2-norm estimate for the Mercer kernel matrix presented by the Jacobi algebraic polynomials. The discussions indicate that the l2-norm of the Mercer kernel matrices may be estimated with discrete orthogonal transforms.

Tauberian conditions for almost convergence

In [Acta Math. 80(1948), 167–190], G. G. Lorentz characterized almost convergent sequences in \(\mathbb{R}\) (or in \(\mathbb{C}\)) in terms of the concept of uniform convergence of the de la Vallee-Poussin means. In this paper, we present Tauberian results which relate almost convergence to norm convergence or to the (C, 1) convergence. Our results generalize Kronecker lemma. As a consequence, we prove that almost convergence and norm convergence are equivalent for the sequence of the partial sums of the Fourier series of \(f \in L^p(T)\) (or \(f \in C(T)\)), where \(1\leq p \leq \infty\). We also show that our results can be used to derive Fatou’s theorem.

Some difference sequence spaces defined by a sequence of ϕ -functions

In this paper, we introduce new difference sequence spaces combining with de la Vallee-Poussin mean using by a sequence of modulus functions and ϕ -functions. We also studied connections between statistically convergence related with this space. Keywords Difference sequence, Modulus function, ϕ -function, De la Vallee-Poussin means, Statistical convergence Mathematics Subject Classification (2000) 46A45, 40F05, 46A80

Absolutely Continuous Functions with Values in a Metric Space

We present a general theory of absolutely continuous paths with values in metric spaces using the notion of metric derivatives. Among other results, we prove analogues of the Banach-Zarecki and Vallee Poussin theorems.

Subordinate Solutions of a Differential Equation

In 2003, Ruscheweyh and Suffridge settled a conjecture of Polya and Schoenberg on subordination of the de la Vallee Poussin means of an analytic function by defining a continuous extension of the de la Vallee Poussin means using a differential equation. We extend this differential equation to a more general setting and observe that a similar subordination result with convex functions holds. Through an integral operator of Bernardi, particular convex subordination chains are constructed with specified limiting functions. Finally, we show the importance of convexity by producing an example of a family of starlike solutions that fails to be a subordination chain.

Maximum principle for functional equations in the space of discontinuous functions of three variables

The paper is devoted to the maximum principles for functional equations in the space of measurable essentially bounded functions. The necessary and sufficient conditions for validity of corresponding maximum principles are obtained in a form of theorems about functional inequalities similar to the classical theorems about differential inequalities of the Vallee Poussin type. Assertions about the strong maximum principle are proposed. All results are also true for difference equations, which can be considered as a particular case of functional equations. The problems of validity of the maximum principles are reduced to nonoscillation properties and disconjugacy of functional equations. Note that zeros and nonoscillation of a solution in a space of discontinuous functions are defined in this paper. It is demonstrated that nonoscillation properties of functional equations are connected with the spectral radius of a corresponding operator acting in the space of essentially bounded functions. Simple sufficient conditions of nonoscillation, disconjugacy and validity of the maximum principles are proposed. The known nonoscillation results for equation in space of functions of one variable follow as a particular cases of these assertions. It should be noted that corresponding coefficient tests obtained on this basis cannot be improved. Various applications to nonoscillation, disconjugacy and the maximum principles for partial differential equations are proposed.

Approximation of Continuous Functions of Low Smoothness by de la Vallee-Poussin Operators

We study some problems of the approximation of continuous functions defined on the real axis. As approximating aggregates, the de la Vallee-Poussin operators are used. We establish asymptotic equalities for upper bounds of the deviations of the de la Vallee-Poussin operators from functions of low smoothness belonging to the classes \(\hat C^{\bar \psi } \mathfrak{N}\).