Direct Theorems Research Articles

A Weighted Generalization of the Classical Kantorovich Operator. II: Saturation

In [1], we have introduced a new weighted type of modification of the classical Kantorovich operator. The advantage of this operator is that there is no restriction on the parameters of the weight, and the class of functions is wider than in the earlier version of the weighted operator (cf. the monograph of Ditzian and Totik [3]). Direct and converse theorems and a Voronovskaya-type relation were proved. Here we solve the saturation problem of the operator (Theorem 2.1). We follow the method developed in [3], but the details are much more involved. A surprising fact emerges in determining the trivial class of saturation (Theorem 3.1).

Approximation by trigonometric polynomials of functions having (α, ψ)- derivatives in weighted variable exponent Lebesgue spaces

We prove direct simultaneous and converse approximation theorems by trigonometric polynomials for functions f and (α, ψ)-derivatives of f in weighted Lebesgue spaces with variable exponent. Bibliography: 11 titles.

Improved direct product theorems for randomized query complexity

The “direct product problem” is a fundamental question in complexity theory which seeks to understand how the difficulty in computing a function on each of k independent inputs scales with k. We prove the following direct product theorem (DPT) for query complexity: if every T-query algorithm has success probability at most \({1 - \varepsilon}\) in computing the Boolean function f on input distribution μ, then for α ≤ 1, every \({\alpha \varepsilon Tk}\)-query algorithm has success probability at most \({(2^{\alpha \varepsilon}(1-\varepsilon))^k}\) in computing the k-fold direct product \({f^{\otimes k}}\) correctly on k independent inputs from μ. In light of examples due to Shaltiel, this statement gives an essentially optimal trade-off between the query bound and the error probability. Using this DPT, we show that for an absolute constant α > 0, the worst-case success probability of any α R 2(f) k-query randomized algorithm for \({f^{\otimes k}}\) falls exponentially with k. The best previous statement of this type, due to Klauck, Špalek, and de Wolf, required a query bound of O(bs(f) k). Our proof technique involves defining and analyzing a collection of martingales associated with an algorithm attempting to solve \({f^{\otimes k}}\). Our method is quite general and yields a new XOR lemma and threshold DPT for the query model, as well as DPTs for the query complexity of learning tasks, search problems, and tasks involving interaction with dynamic entities. We also give a version of our DPT in which decision tree size is the resource of interest.

Approximations in several variables with freud-type and Ap-weights

We present direct (Jackson-type) and converse (Bernstein-Stechkin-type) theorems for polynomial approximations with Freud-type weights and trigonometric approximations with Ap-weights, in the case of several variables.

On approximation in weighted Smirnov–Orlicz classes

In this work, we investigate the approximation problems in weighted Smirnov–Orlicz classes. We prove a direct theorem for polynomial approximation of functions in certain subclasses of weighted Smirnov–Orlicz classes. The direct theorem is proved in terms of the modulus of smoothness. Also, from the main theorem some results are obtained. In the proof of the main theorem, a Jackson-Dzjadyk polynomial is used.

Direct and strong converse theorems for a general sequence of positive linear operators

We present direct and strong converse theorems for a general sequence of positive linear operators satisfying some functional equations. The results can be applied to some extensions of Baskakov and Szasz–Mirakyan operators.

Elementary proof of constructive versions of the tangent direction theorem and the implicit function theorem

Elementary proof of constructive versions of the tangent direction theorem and the implicit function theorem

Approximating Polynomials for Functions of Weighted Smirnov-Orlicz Spaces

Let and be, respectively, bounded and unbounded components of a plane curve satisfying Dini's smoothness condition. In addition to partial sum of Faber series of belonging to weighted Smirnov-Orlicz space (), we prove that interpolating polynomials and Poisson polynomials are near best approximant for . Also considering a weighted fractional moduli of smoothness, we obtain direct and converse theorems of trigonometric polynomial approximation in Orlicz spaces with Muckenhoupt weights. On the bases of these approximation theorems, we prove direct and converse theorems of approximation, respectively, by algebraic polynomials and rational functions in weighted Smirnov-Orlicz spaces and .

Strong Direct Product Theorems for Quantum Communication and Query Complexity

A strong direct product theorem (SDPT) states that solving $n$ instances of a problem requires $\Omega(n)$ times the resources for a single instance, even to achieve success probability $2^{-\epsilon n}$ for a small enough constant $\epsilon>0.$ We prove that quantum communication complexity obeys an SDPT whenever the communication lower bound for a single instance is proved by the generalized discrepancy method, the strongest technique in that model. We prove that quantum query complexity obeys an SDPT whenever the query lower bound for a single instance is proved by the polynomial method, one of the two main techniques in that model. In both models, we prove the corresponding XOR lemmas and threshold direct product theorems.

Duality for Multitime Multiobjective Ratio Variational Problems on First Order Jet Bundle

We consider a new class of multitime multiobjective variational problems of minimizing a vector of quotients of functionals of curvilinear integral type. Based on the efficiency conditions for multitime multiobjective ratio variational problems, we introduce a ratio dual of generalized Mond‐Weir‐Zalmai type, and under some assumptions of generalized convexity, duality theorems are stated. We prove our weak duality theorem for efficient solutions, showing that the value of the objective function of the primal cannot exceed the value of the dual. Direct and converse duality theorems are stated, underlying the connections between the values of the objective functions of the primal and dual programs. As special cases, duality results of Mond‐Weir‐Zalmai type for a multitime multiobjective variational problem are obtained. This work further develops our studies in (Pitea and Postolache (2011)).

Approximation by polynomials in rearrangement invariant quasi Banach function spaces

In the present work we deal with the approximation properties of certain linear polynomial operators in rearrangement invariant quasi Ba- nach function spaces. We obtain some Jackson type direct theorem and sharp converse theorem of trigonometric approximation with respect to fractional positive order moduli of smoothness in these spaces.

Approximation of Functions by Rational Functions on Closed Curves of the Complex Plane

Let Γ be a quasi-smooth curve in the complex plane \({\mathbb{C}}\) . In this study, a direct theorem of approximation theory in certain subclasses of the functions which have continuous derivatives through order r on a closed curve Γ is proved.

Direct Ramsey theorem for structures involving relations and functions

We prove the direct structural Ramsey theorem for structures with relations as well as functions. The result extends the theorem of Abramson and Harrington and of Nešetřil and Rödl.

Direct and inverse theorems of rational approximation in the Bergman space

For positive numbers and let denote the Bergman space of analytic functions in the half-plane . For let be the best approximation by rational functions of degree at most . Also let and be numbers such that and . Then the main result of the paper claims that the set of functions such that is precisely the Besov space of analytic functions in .Bibliography: 23 titles.

The refined direct and converse inequalities of trigonometric approximation in weighted variable exponent Lebesgue spaces

Abstract Refined direct and converse theorems of trigonometric approximation are proved in the variable exponent Lebesgue spaces with weights satisfying some Muckenhoupt Ap -condition. As a consequence, the refined versions of Marchaud and its converse inequalities are obtained.

Weighted approximation by Baskakov-type operators

The uniform weighted approximation errors of Baskakov-type operators are characterized for weights of the form \(\left(\dfrac{x}{1+x}\right)^{\gamma_{0}}{(1+x)}^{\gamma_{\infty}}\) for γ0,γ∞∈[−1,0]. Direct and strong converse theorems are proved in terms of the weighted K-functional.

Approximation in weighted Orlicz spaces

Abstract We prove some direct and converse theorems of trigonometric approximation in weighted Orlicz spaces with weights satisfying so called Muckenhoupt’s A p condition.

Duality theorems for a new class of multitime multiobjective variational problems

In this work, we consider a new class of multitime multiobjective variational problems of minimizing a vector of functionals of curvilinear integral type. Based on the normal efficiency conditions for multitime multiobjective variational problems, we study duals of Mond-Weir type, generalized Mond-Weir-Zalmai type and under some assumptions of (?, b)-quasiinvexity, duality theorems are stated. We give weak duality theorems, proving that the value of the objective function of the primal cannot exceed the value of the dual. Moreover, we study the connection between values of the objective functions of the primal and dual programs, in direct and converse duality theorems. While the results in §1 and §2 are introductory in nature, to the best of our knowledge, the results in §3 are new and they have not been reported in literature.

Direct and inverse theorems in approximation theory for periodic functions in S.B. Stechkin’s papers and the development of these theorems

S.B. Stechkin made a considerable contribution to the important aspect of approximation theory for periodic functions concerned with direct and inverse theorems. We present the history of developing this research area and pose new problems.

Trigonometric approximation of functions in generalized Lebesgue spaces with variable exponent

We investigate the approximation properties of the trigonometric system in $ L_{2\pi }^{p\left( \cdot \right)} $ . We consider the moduli of smoothness of fractional order and obtain direct and inverse approximation theorems together with a constructive characterization of a Lipschitz-type class.