On the degree of approximation of continuous functions (original) (raw)

ON THE DEGREE OF APPROXIMATION IV

So far only finite summability operators have been used in determining the degree of approximation of functions. Huzoor H. Khan and Sarfaraz Umar [12] discussed the problem for the first time for infinite summability operators (Logarithmic operators) and determined the degree of approxima­ tion of functions belonging to the class Lip(ty(t),p) , obtaining (under certain conditions) the same order as for other finite operators. In this paper, however, we have succeeded in determining the degree of approxima­ tion of functions belonging to the class Ltp(i|/(fc) ,p) for Abel-type-operators (of infinite operators). Suppose /(x) is a 2n-periodic function integrable iP (p > 1) , and OO (0.1) f(x) ~ j a Q + £ (a cos nx + b sin nx) , n= l n 0 0 00 (0.2) /(x) ~ £ (a sin nx-b cos nx) = \ B (x) n=l n n n= 1 are respectively the Fourier series and conjugate Fourier series of /(x). Let n (0.3) 5 (x) = J a 0 + 1 (a cos vx + sin v x) » n v=\ V and ^ n (0.4) S (x) = I (a v sin vx-b cos vx) K v=i

ON THE DEGREE OF APPROXIMATION

2003

In the present paper we obtain the degree of approximation using the Euler's means, of functions belonging to Lip (ψ(t), p) class. It is also proved that the order of approximation arrived at is best possible and is free from the means generating sequences.

Some direct and inverse theorems in approximation of functions

Journal of the Australian Mathematical Society, 1983

The paper is concerned with the determination of the degree of convergence of a sequence of linear operators connected with the Fourier series of a function of class L p (p > 1) to that function and some inverse results in relating the convergence to the classes of functions. In certain cases one can obtain the saturation results too. In all cases L p norm is used. 1980 Mathematics subject classification (Amer. Math. Soc): 41 A 40. 00 00 2 B n (x) = 2 (b k coskx -a k sin kx). n=\ k=\

A survey on the Weierstrass approximation theorem

2008

The celebrated and famous Weierstrass approximation theorem characterizes the set of continuous functions on a compact interval via uniform approximation by algebraic polynomials. This theorem is the first significant result in Approximation Theory of one real variable and plays a key role in the development of General Approximation Theory. Our aim is to investigate some new results relative to such theorem, to present a history of the subject, and to introduce some open problems. Key words and phrases: Approximation Theory, Weierstrass' theorem, Sobolev spaces, weighted Sobolev spaces, G-valued polynomials, G-valued smooth functions.

Some Topics in Fourier Analysis and Approximation Theory

arXiv: Functional Analysis, 1996

This manuscript presents shortly the results obtained by participants of the scientific seminar which is held more than twenty years under leadership of the author at Donetsk University. In the list of references main publications are given. These results are published in serious scientific journals and reported at various conferences, including international ones at Moscow,ICM66; Kaluga,1975; Kiev,1983; Haifa,1994; Z\"urich,ICM94; Moscow,1995. The area of investigation is the Fourier analysis and the theory of approximation of functions. Used are methods of classical analysis including special functions, Banach spaces, etc., of harmonic analysis in finitedimensional Euclidean space, of Diophantine analysis, of random choice, etc. The results due to the author and active participants of the seminar, namely E. S. Belinskii, O. I. Kuznetsova, E. R. Liflyand, Yu. L. Nosenko, V. A. Glukhov, V. P. Zastavny, Val. V. Volchkov, V. O. Leontyev, and others, are given. Besides the partici...