A nonparametric adaptation theory is developed for the construction of confidence intervals for linear functionals. A between class modulus of continuity captures the expected length of adaptive confidence intervals. Sharp lower bounds are given for the expected length and an ordered modulus of continuity is used to construct adaptive confidence procedures which are within a constant factor of ...

*Correspondence: [email protected] 2Department of Mathematics, Gauhati University, Guwahati, India Full list of author information is available at the end of the article Abstract The notion of statistical weighted B-summability was introduced very recently (Kadak et al. in Appl. Math. Comput. 302:80–96, 2017). In the paper, we study the concept of statistical deferred weighted B-summ...

Abstract. We introduce certain linear positive operators and study some approximation properties of these operators in the space of functions, continuous on a compact set, of two variables. We also find the order of this approximation by using modulus of continuity. Moreover we define an rth order generalization of these operators and observe its approximation properties. Furthermore, we study ...

Following D. B. H. Cline [Cl], we study the connections between the behavior of the spherical mean of an L modulus of continuity corresponding to the finite difference of an arbitrary order m > 0 of a function f ∈ L(R), 1 ≤ p ≤ ∞, d ∈ N, and the average decay of the Fourier transform f̂ . We obtain an estimate which refines one result from [Cl] for 1 ≤ p ≤ 2. Our main interest is in two-sided es...

We consider a compact set K in the form of a sequence of segments. By means of nearly Chebyshev polynomials for K, the modulus of continuity of the Green functions gC\K is estimated. Markov’s constants of the corresponding set are evaluated.

A detailed analysis of minimax estimates of arbitrary linear functionals based on infinite dimensional Gaussian models has been provided by Donoho and Liu. In particular it has been shown that if the parameter space is convex then linear estimates can always be found which have maximum mean squared error within a small constant multiple of the minimax value. These linear estimates do however ha...

We consider uniqueness theorems in classical analysis having the form (+) ∀u ∈ U, v1, v2 ∈ Vu ( G(u, v1) = 0 = G(u, v2) → v1 = v2 ) , where U, V are complete separable metric spaces, Vu is compact in V and G : U × V → IR is a constructive function. If (+) is proved by arithmetical means from analytical assumptions (++) ∀x ∈ X∃y ∈ Yx∀z ∈ Z ( F (x, y, z) = 0 ) only (where X, Y, Z are complete sep...

In this article we study the exponential behavior of the continuous stochastic Anderson model, i.e. the solution of the stochastic partial differential equation u(t, x) = 1+ ∫ t 0 κ∆xu (s, x) ds+ ∫ t 0 W (ds, x)u (s, x), when the spatial parameter x is continuous, specifically x ∈ R, and W is a Gaussian field on R+ × R that is Brownian in time, but whose spatial distribution is widely unrestric...

The applications of q calculus in operator theory is an active area of research in the last two decades. Several new q -operators were introduced and their approximation behavior were discussed. This paper is extension of the paper [South. Asian Bul. Math. (36) (2012), 343–352]. We propose here the Stancu variant of q -Beta-Szász type operators. We estimate the moments and also establish direct...