In this note, the problem of degree of uniform approximation by polynomials on disjoint intervals is considered. It is interesting to note that the error estimates cannot be obtained by extending the given functions to functions on a single interval and applying the one-interval estimates.

Let / be a continuous real valued function defined on [-1, 1] and let £"(/) denote the degree of best uniform approximation to / by algebraic polynomial of degree at most n. The supremum norm on [a, b] is denoted by IMIio.ii and the polynomial of degree n of best uniform approximation is denoted by />". We find a class of functions / such that there exists a fixed a e ( -1, 1) with the following property 11/ -PJ[a-x. + ic\ > CE r (f), n = 1, 2,3,... for some positive constants C and N independent of n. Moreover the sequence (7) is optimal in the sense that if £ is replaced by b n = o(;) then the above inequality need not hold no matter how small C > 0 is chosen.We also find another, more general class a functions / for which 11/ -PJ[..e..+x\ > CE n (f) infinitely often.1991 Mathematics subject classification: 41al7, 41a25, 41a50.

Let W & R n be a smooth domain and let u P C 0 WX A classical result of potential theory states thatSr " x À uxdsx u" x for every "x P W and r b 0 if and only ifDu 0 in WX HereSr " x À uxdsx denotes the average of u on the sphere S r " x of center "x and radius rX Our main result, which is a ªlocalizedº version of the above result, states:Theorem. Let u P W 2Y1 W and let x P W be a Lebesgue point of Du such thatSr " x À u ds À a or 2 for some a P R and all sufficiently small r b 0X Then Dux 0X

Abstract. Let f (x) be a smooth function whose derivative of a given order must be computed. The signal f (x) is affected by two kinds of perturbation. The perturbation caused by the presence of the machine epsilon M of the computer may be considered to be an extremely high-frequency noise of very small amplitude. The way to minimize its effect consists of choosing an appropriate value for the step size of the difference quotient.The second perturbation, caused by the presence of noise, requires first the signal to be treated in some way. It is the purpose of this work to construct a wavelet-based band-pass filter that deals with the two cited perturbations simultaneously. In effect our wavelet acts like a "smoothed difference quotient" whose stepsize is of the same order as that of the usual difference quotient. Moreover the wavelet effectively removes the noise and computes the derivative with an accuracy equal to the one obtained by the corresponding difference quotient in the absence of noise.