A Sharp Jackson Inequality for Best Trigonometric Approximation
نویسنده
چکیده
The paper presents a sharp Jackson inequality and a corresponding inverse one for best trigonometric approximation in terms of moduli of smoothness that are equivalent to zero on the trigonometric polynomials up to a certain degree. Sharp relations between such moduli of different order are also considered.
منابع مشابه
Sharp Jackson-type Inequalities for Approximations of Classes of Convolutions by Entire Functions of Exponential Type
In this paper, a new method is introduced for the proof of sharp Jacksontype inequalities for approximation of convolution classes of functions defined on the real line. These classes are approximated by linear operators with values in sets of entire functions of exponential type. In particular, a sharp Jackson-type inequality for the even-order derivatives of the conjugate function is proved. ...
متن کاملSharp Jackson inequalities
For trigonometric polynomials on [− , ] ≡ T , the classical Jackson inequalityEn(f )p C r (f, 1/n)p was sharpened by M. Timan for 1<p<∞ to yield n−r { n ∑ k=1 ksr−1Ek(f )p }1/s C r (f, n−1)p where s =max(p, 2). In this paper a general result on the relations between systems or sequences of best approximation and appropriate measures of smoothness is given. Approximation by algebraic polynomials...
متن کاملOn the exact constant in Jackson-Stechkin inequality for the uniform metric
The classical Jackson-Stechkin inequality estimates the value of the best uniform approximation of a periodic function f by trigonometric polynomials of degree ≤ n− 1 in terms of its r-th modulus of smoothness ωr(f, δ). It reads En−1(f) ≤ cr ωr ( f, 2π n ) , where cr is some constant that depends only on r. It was known that cr admits the estimate cr < r ar and, basically, nothing else could be...
متن کامل2 00 6 On the exact constant in Jackson - Stechkin inequality for the uniform metric
The classical Jackson-Stechkin inequality estimates the value of the best uniform approximation of a periodic function f by trigonometric polynomials of degree ≤ n− 1 in terms of its r-th modulus of smoothness ωr(f, δ). It reads En−1(f) ≤ cr ωr ( f, 2π n ) , where cr is some constant that depends only on r. It was known that cr admits the estimate cr < r ar and, basically, nothing else could be...
متن کاملJackson on Approximation 501 Dunham Jackson on Approximation
The Theory of Approximation. By Dunham Jackson. New York (American Mathematical Society Colloquium Publications, Volume 11), published by this Society, 1930. v+178 pp. In 1885 Weierstrass announced his now famous theorem: Any continuous function j•(#)> defined over a finite interval (a, b), can be approximated in (a, b) uniformly and indefinitely by a sequence of polynomials Pn(x), whose degree...
متن کامل