Majorization of polynomials on the plane

We prove that for every χ[−1, 1] and every real algebraic polynomial f of degree n such that |f(t): 1 on [−1, 1], the following inequality takes place on the complex plane |f(x+iy)||T_n(1+iy)|,−∞y∞ where T_n is the Tchebycheff polynomial. This implies easily Vladimir Markov inequality.     

On the almost automorphy of bounded solutions of differential equations with piecewise constant argument

In this paper we give sufficient spectral conditions for the almost automorphy of bounded solutions to differential equations with piecewise constant argument of the form x^′(t)=Ax([t])+f(t), t∈R, where A is a bounded linear operator in X and f is an X-valued almost automorphic function.     

Rate of pointwise approximation for locally bounded functions by Szász operators

In this paper the asymptotic behavior of Szász operators for locally bounded functions f is studied at points x where f(x+) and f(x−) exist. An asymptotic estimate of this type approximation is obtained by using some techniques and results of probability theory. The estimate essentially is the best possible for continuous points of f.     

On the convergence of averaging Hermite interpolators

We investigate two sequences of polynomial operators, H_{2n − 2}(A₁,f; x) and H_{2n − 3}(A₂,f; x), of degrees 2n − 2 and 2n − 3, respectively, defined by interpolatory conditions similar to those of the classical Hermite-Féjer interpolators H_{2n − 1}(f, x). If H_{2n − 2}(A₁,f; x) and H_{2n − 3}(A₂,f; x) are based on the zeros of the jacobi polynomials P_n^(α,β)(x), their convergence behaviour is similar to that of H_{2n − 1}(f;, x). If they are based on the zeros of (1 − x²)T_{n − 2}(x), their convergence behaviour is better, in some sense, than that of H_{2n − 1}(f, x).     

Reverse Martingales and Approximation Operators

Let {ξ_n, _n, n ≥ m ≥ 1} be a reverse martingale such that the distribution of ξ_n depends on x I R =(− ∞, ∞)x. for each n ≥ m, and ξ_n[formula] For a continuous bounded function f on R let L_n(f, x) = Ef(ξ_n) be the associated positive linear operator. The properties of ξ_n are used to obtain the convergence properties of L_n(f, x), and some more details are given when ξ_n is a reverse martingale sequence of -statistics. Lipschitz properties for a subclass of these operators resulting from an exponential Family of distributions are also given. It is further shown that this class of operators of convex functions preserves convexity also. An example of a reverse supermartingale related to the Bleimann-Butzer-Hahn operator is also discussed.     

q-Bernstein polynomials and their iterates

Let B_n( f,q;x), n=1,2,… be q-Bernstein polynomials of a function f : [0,1]→C. The polynomials B_n( f,1;x) are classical Bernstein polynomials. For q≠1 the properties of q-Bernstein polynomials differ essentially from those in the classical case. This paper deals with approximating properties of q-Bernstein polynomials in the case q>1 with respect to both n and q. Some estimates on the rate of convergence are given. In particular, it is proved that for a function f analytic in {z: |z|<q+} the rate of convergence of {B_n( f,q;x)} to f(x) in the norm of C[0,1] has the order q⁻ⁿ (versus 1/n for the classical Bernstein polynomials). Also iterates of q-Bernstein polynomials {B_n^j_n( f,q;x)}, where both n→∞ and j_n→∞, are studied. It is shown that for q(0,1) the asymptotic behavior of such iterates is quite different from the classical case. In particular, the limit does not depend on the rate of j_n→∞.     

Minimum asymptotic error of algorithms for solving ODE

We deal with algorithms for solving systems z′(x) = f(x, z(x)), x ε [0, c], z(0) = η where f has r continuous bounded derivatives in [0, c) × ^s. We consider algorithms whose sole dependence on f is through the values of n linear continuous functionals at f. We show that if these functionals are defined by partial derivatives off then, roughly speaking, the error of an algorithm (for a fixed f) cannot converge to zero faster than n^−r as n → +∞. This minimal error is achieved by the Taylor algorithm. If arbitrary linear continuous functionals are allowed, then the error cannot converge to zero faster than n^−(r+1) as n → +∞. This minimal error is achieved by the Taylor-integral algorithm which uses integrals of f.     

Corrigendum to “Discrepancy principle for the dynamical systems method” [Communications in Nonlinear Science and Numerical Simulation 10 (2005) 95–101]

Consider an operator equation B(u) − f = 0 in a real Hilbert space. Let us call this equation ill-posed if the operator B′(u) is not boundedly invertible, and well-posed otherwise. The dynamical systems method (DSM) for solving this equation consists of a construction of a Cauchy problem, which has the following properties: (1) it has a global solution for an arbitrary initial data, (2) this solution tends to a limit as time tends to infinity, (3) the limit is the minimal-norm solution to the equation B(u) = f. A global convergence theorem is proved for DSM for equation B(u) − f = 0 with monotone operators B.     

An estimate on the convergence of Baskakov-Bézier operators

In the present paper we estimate the rate of convergence on functions of bounded variation for the Bézier variant of the Baskakov operators Bn,α(f,x). Here we have studied the rate of convergence of Bn,α(f,x) for the case 0<α<1.     

Global Convergence of a Modified Gradient Projection Method for Convex Constrained Problems

In this paper, the continuously differentiable optimization problem min{f（x） ： x∈Ω}, where Ω ∈ R^n is a nonempty closed convex set, the gradient projection method by Calamai and More （Math. Programming, Vol.39. P.93-116, 1987） is modified by memory gradient to improve the convergence rate of the gradient projection method is considered. The convergence of the new method is analyzed without assuming that the iteration sequence {x^k} of bounded. Moreover, it is shown that, when f（x） is pseudo-convex （quasiconvex） function, this new method has strong convergence results. The numerical results show that the method in this paper is more effective than the gradient projection method.     

Positive Bernstein-Sheffer Operators

Let h(t) = Σ_{n ≥ 1}h_nt_n, h₁ > 0, and exp(xh(t)) = Σ_{n ≥ 0}P_n(x) tⁿ/n!. For f C[0,1], the associated Bernstein-Sheffer operator of degree n is defined by B^h_nf(x) = P_n^{− 1} Σⁿ_{k = 0}f(k/n)(ⁿ_k) P_k(x) P_{n − k}(1 − x) where p_n = p_n(1). We characterize functions h for which B^h_n is a positive operator for all n ≥ 0. Then we give a necessary and sufficient condition insuring the uniform convergence of B^h_nf to f. When h is a polynomial, we give an upper bound for the error f − B^h_nf _∞. We also discuss the behavior of B^h_nf when h is a series with a finite or infinite radius of convergence.     

Ground state solutions for some indefinite variational problems

We consider the nonlinear stationary Schrödinger equation −Δu+V(x)u=f(x,u) in . Here f is a superlinear, subcritical nonlinearity, and we mainly study the case where both V and f are periodic in x and 0 belongs to a spectral gap of −Δ+V. Inspired by previous work of Li et al. (2006) [11] and Pankov (2005) [13], we develop an approach to find ground state solutions, i.e., nontrivial solutions with least possible energy. The approach is based on a direct and simple reduction of the indefinite variational problem to a definite one and gives rise to a new minimax characterization of the corresponding critical value. Our method works for merely continuous nonlinearities f which are allowed to have weaker asymptotic growth than usually assumed. For odd f, we obtain infinitely many geometrically distinct solutions. The approach also yields new existence and multiplicity results for the Dirichlet problem for the same type of equations in a bounded domain.     

Characterizations of certain classes of Hankel operators on the Bergman spaces of the unit disk

Let f be an integrable function on the unit disk. The Hankel operator H_f is densely defined on the Bergman space A^p by H_fg = fg − P(fg), where g is a bounded analytic function and P is the Bergman projection (orthogonal projection from L² to A²) extended to L¹ via its integral formula. In this paper, the functions f for which H_f extends to a bounded operator from A^p to L^p are characterized, 1 < p < ∞. Also characterized are the functions f for which H_f extends to a compact or Schatten class operator on A². The proofs can be extended to handle any smoothly bounded domain in C in place of the unit disk.     

Stability of the geometric Ekeland variational principle: Convex case

In geometric terms, the Ekeland variational principle says that a lower-bounded proper lower-semicontinuous functionf defined on a Banach spaceX has a point (x ₀,f(x ₀)) in its graph that is maximal in the epigraph off with respect to the cone order determined by the convex coneK _λ = {(x, α) ∈X × ?:λ ∥x∥ ≤ ? α}, where λ is a fixed positive scalar. In this case, we write (x ₀,f(x ₀))∈λ-extf. Here, we investigate the following question: if (x ₀,f(x ₀))∈λ-extf, wheref is a convex function, and if 〈f _n 〉 is a sequence of convex functions convergent tof in some sense, can (x ₀,f(x ₀)) be recovered as a limit of a sequence of points taken from λ-extf _n ? The convergence notions that we consider are the bounded Hausdorff convergence, Mosco convergence, and slice convergence, a new convergence notion that agrees with the Mosco convergence in the reflexive setting, but which, unlike the Mosco convergence, behaves well without reflexivity.     

Some Convergence Properties of Descent Methods

In this paper, we discuss the convergence properties of a class of descent algorithms for minimizing a continuously differentiable function f on R ⁿ without assuming that the sequence { x _k } of iterates is bounded. Under mild conditions, we prove that the limit infimum of is zero and that false convergence does not occur when f is convex. Furthermore, we discuss the convergence rate of { } and { f(x _k )} when { x _k } is unbounded and { f(x _k )} is bounded.     

Mappings of finite distortion: the degree of regularity

This paper investigates the self-improving integrability properties of the so-called mappings of finite distortion. Let K(x)1 be a measurable function defined on a domain ΩRⁿ, n2, and such that exp(βK(x))L_loc¹(Ω), β>0. We show that there exist two universal constants c₁(n),c₂(n) with the following property: Let f be a mapping in W_loc^1,1(Ω,Rⁿ) with |Df(x)|ⁿK(x)J(x,f) for a.e. xΩ and such that the Jacobian determinant J(x,f) is locally in L¹ log^−c₁(n)βL. Then automatically J(x,f) is locally in L¹ log^c₂(n)βL(Ω). This result constitutes the appropriate analog for the self-improving regularity of quasiregular mappings and clarifies many other interesting properties of mappings of finite distortion. Namely, we obtain novel results on the size of removable singularities for bounded mappings of finite distortion, and on the area distortion under this class of mappings.     

Approximation by Durrmeyer–Bezier operators

In the present paper, we consider the Bezier variant M_n,α(f,x) of the generalized Durrmeyer type operators, and obtain an estimate on the rate of convergence of M_n,α(f,x) for the decomposition technique of functions of bounded variation. In the end we propose an open problem for the readers and give an asymptotic formula for these generalized Durrmeyer type operators.     

Comonotone Jackson's Inequality

Let 2s points y_i=−πy_2s<…<y₁<π be given. Using these points, we define the points y_i for all integer indices i by the equality y_i=y_i+2s+2π. We shall write fΔ⁽¹⁾(Y) if f is a 2π-periodic continuous function and f does not decrease on [y_i, y_i−1], if i is odd; and f does not increase on [y_i, y_i−1], if i is even. In this article the following Theorem 1—the comonotone analogue of Jackson's inequality—is proved. 1. If fΔ⁽¹⁾(Y), then for each nonnegative integer n there is a trigonometric polynomial τ_n(x) of order n such that τ_nΔ⁽¹⁾(Y), and |f(x)−π_n(x)|c(s) ω(f; 1/(n+1)), x , where ω(f; t) is the modulus of continuity of f, c(s)=const. Depending only on s.     

A note on a Bernstein-type operator of Bleimann, Butzer, and Hahn

Let L_n(f, x)(f ε C)[0, ∞)) be a Bernstein-type approximation operator as defined and studied by Bleimann, Butzer, and Hahn. Probabilistic arguments are used to simplify and sharpen some of their results. The rates of convergence are given in terms of the first and second moduli of continuity. Moreover, an appropriate limit of L_n is shown to be the well-known Szasz operator.     

Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces

In this paper, we achieve the general solution and the generalized Hyers–Ulam–Rassias stability of the following functional equation

f(x+ky)+f(x−ky)=k²f(x+y)+k²f(x−y)+2(1−k²)f(x)