Adaptive confidence intervals for regression functions are constructed under shape constraints of monotonicity and convexity. A natural benchmark is established for the minimum expected length of confidence intervals at a given function in terms of an analytic quantity, the local modulus of continuity. This bound depends not only on the function but also the assumed function class. These benchm...

Two extensions of classical Shepard operators to the fuzzy case are presented. We study Shepard-type interpolation/approximation operators for functions with domain and range in the fuzzy number’s space and max-product approximation operators. Error estimates are obtained in terms of the modulus of continuity.

1 Introduction Contour and solid modulus of continuity. The one-dimensional case There has been an extensive study of the relation between the " contour " and the " solid " moduli of continuity for analytic functions in planar domains, starting with the following beautiful result of Hardy and Littlewood. For a domain G in C n (n ≥ 1) we denote as usual by A(G) the algebra of analytic functions ...

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 ...

In this paper, we prove some approximation results in statistical sense and establish some direct theorems for the positive linear operators constructed by the means Lagrange type polynomials. We compute error estimation by using modulus of continuity with the help of Matlab and give its algorithm. Furthermore, we show graphically the convergence of our operators to various functions. AMS Subje...

This is a slightly expanded version of my lectures at the 2010 Clay Mathematics Institute summer (winter) school in Buzios, Brazil. The theme is the fine properties of Schramm-Loewner evolution (SLE) curves with an emphasis on recent work I have done with a number of co-authors on fractal and multifractal properties. I assume basic knowledge of SLE at the level of foundational course presented ...

which is the precision to which F approximates K. An instance of classical interest is that in which L is C([0, 1]), K is A"—i.e. those functions bounded by one, and whose modulus of continuity is dominated by the modulus of continuity function o>—and F is the family Pn of polynomials of degree n 1. For this case it is known that there are positive constants A and B, independent of to and n, su...

In this work, we construct Stancu type modification of the generalization of Meyer-König and Zeller operators (MKZ) defined in [12]. We show that the Lipschitz constant of a Lipschitz continuous function and the properties of the function of modulus of continuity can be retained by these operators.

We find a bound for the modulus of continuity of the blow-up time for the semilinear parabolic problem ut = ∆u + |u|u, with respect to the initial data.

We make use of the method of modulus of continuity [7] and Fourier localization technique [1] to prove the global well-posedness of the critical Burgers equation ∂tu + u∂xu + Λu = 0 in critical Besov spaces Ḃ 1 p p,1(R) with p ∈ [1,∞), where Λ = √ −△. 2000 Mathematics Subject Classification: 35K55, 35Q53