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

In this paper we study the model of heat transfer in a porous medium with a critical diffusion. We obtain global existence and uniqueness of solutions to the equations of heat transfer of incompressible fluid in Besov spaces Ḃ 3/p p,1 (R ) with 1 ≤ p ≤ ∞ by the method of modulus of continuity and Fourier localization technique. AMS Subject Classification 2000: 76S05, 76D03

We obtain the best possible constants in preservation inequalities concerning the usual first modulus of continuity for the classical Szász–Mirakyan operator. The probabilistic representation of this operator in terms of the standard Poisson process is used. © 2007 Elsevier Inc. All rights reserved.

n` k ́ 1 k ̇ xk p1` xqn`k , bn,kptq “ 1 Bpn` 1, kq tk ́1 p1` tqn`k`1 . Gupta et al. [12] studied pointwise convergence, an asymptotic formula and error estimation for the operators Bnpf, ̈q. Recently, Govil and Gupta [7] studied some approximation properties for the operators Bnpf, ̈q and estimated local results in terms of modulus of continuity. Generalization of the operators (1.1) with parameter ...

In this paper, we define and study a new class of random fields called harmonizable multi-operator scaling stable random fields. These fields satisfy a local asymptotic operator scaling property which generalizes both the local asymptotic self-similarity property and the operator scaling property. Actually, they locally look like operator scaling random fields whose order is allowed to vary alo...

In this paper we give a detailed description of the random wavelet series representation of real-valued linear fractional stable sheet introduced in [2]. By using this representation, in the case where the sample paths are continuous, an anisotropic uniform and quasi-optimal modulus of continuity of these paths is obtained as well as an upper bound for their behavior at infinity and around the ...

In 1934, Whitney posed the problem of how to recognize whether a function f defined on a closed subset X of R is the restriction of a function of class C. Whitney himself solved the one-dimensional case (i.e., for n = 1) in terms of finite differences [W1, W2, W3], giving the classical Whitney’s extension theorem. A geometrical solution for the case C(R) was given by G. Glaeser [G], who introdu...

Let C be a subset of R (not necessarily convex), f : C → R be a function, and G : C → R be a uniformly continuous function, with modulus of continuity ω. We provide a necessary and sufficient condition on f , G for the existence of a convex function F ∈ C(R) such that F = f on C and ∇F = G on C, with a good control of the modulus of continuity of ∇F in terms of that of G. On the other hand, ass...

A Korovkin type theorem is established in the space C̃b[0,∞) of all uniformly continuous and bounded functions on [0,∞) for a sequence of positive linear operators, the approximation error being estimated with the aid of the usual modulus of continuity. As applications we obtain quantitative results for q-Baskakov operators. Mathematics Subject Classification (2010): 41A36, 41A25.

zi being independent and identically distributed standard normal random variables and M a finite or countably infinite index set. In particular, for these models minimax theory for mean squared error, confidence intervals and probabilistic error can all be precisely characterized by a modulus of continuity introduced by Donoho and Liu (1991). More specifically, for any linear functional T and c...