A survey of weighted polynomial approximation with

1999

2. The Weierstrass approximation theorem 3. Estimates for the Bernstein polynomials 4. Weierstrass' original proof 5. The Stone-Weierstrass approximation theorem 6. Chebyshev's theorems 7. Approximation by polynomials and trigonometric polynomials 8. The nonexistence of a continuous linear projection 9. Approximation of functions of higher regularity 10. Inverse theorems References Introductory remarks These notes comprise the main part of a course on approximation theory presented at Uppsala University in the Fall of 1998, viz. the part on polynomial approximation. The material is mainly classical. As sources I used Cheney [1966], Dzjadyk [1977], Korovkin [1959], and Lorentz [1953], as well as papers listed in the bibliography. The emphasis is on explaining the main ideas behind the most important techniques. The last part of the course was on rational approximation and is not included here. I followed mainly Cheney [1966, Chapter 5, pp. 150-167]. I also discussed Padé approximation briefly, following Cheney [1966, Chapter 5, pp. 173-177] and the introduction in Rudälv [1998]. I am grateful to Tsehaye Kahsu Araaya for remarks to the manuscript. 2 a k 2 − d(0, A) 2 , since 1 2 a j + 1 2 a k belongs to A in view of the convexity. We see that the right-hand side tends to zero as j, k → ∞. This implies that (a j) is a Cauchy sequence, and it must therefore have a limit in H. The limit cannot depend on the sequence, for if we take two sequences (a j) and (b j) and mix them, the new sequence (a 0 , b 0 , a 1 , b 1 , a 2 , ...) must converge by the same argument. We call the limit π(0); by translation we define π(x) ∈ A. Exercise 1.3. Prove that the mapping π: H → A is continuous; more precisely that π(x) − π(y) x − y , x, y ∈ H. Prove also that the set A is contained in a half-space as soon as x / ∈ A: in the real case every a ∈ A must satisfy (a − x|π(x) − x) π(x) − x 2. What about the complex case? Exercise 1.4. Prove that if A is a closed linear subspace, then x−π(x) is orthogonal to π(x). Prove that we get two idempotent mappings π and I − π, and determine all possible relations between the subspaces ker π, ker(I − π), im π, im(I − π). So Hilbert space is an easy case where the best approximant is unique. However, there are other interesting cases when we can prove uniqueness of the best approximant. Now we may perhaps dare to say that approximation theory is the study of approximating sequences, best approximants and their uniqueness or nonuniqueness in cases where X is a space of interesting functions, and A is some subspace of nice functions, like polynomials, trigonomentric polynomials,... 2. The Weierstrass approximation theorem The best starting point for these lectures is the classical Weierstrass 1 approximation theorem. It says that for any continuous real-valued function f on the interval [0, 1] and any integer k 1 there is a polynomial p k such that |f (x) − p k (x)| 1/k for all

Abstract and Applied Analysis, 1996

Abstract. Recently, Z. Ditzian gave an interesting direct estimate for Bernstein polynomials. In this paper we give direct and inverse results of this type for linear combinations of Bernstein polynomials. ... (1.2) Bn(f,x) − f(x) = O (( 1 √ n δn(x) )α) ⇐⇒ ω1(f,t) = O(tα),

Journal of Approximation Theory, 1989

Proceedings of the American Mathematical Society, 2017

In 1924 S. Bernstein [4] asked for conditions on a uniformly bounded on R Borel function (weight) w : R → [0, +∞) which imply the denseness of algebraic polynomials P in the seminormed space C 0 w defined as the linear set {f ∈ C(R) | w(x)f (x) → 0 as |x| → +∞} equipped with the seminorm f w := sup x∈R w(x)|f (x)|. In 1998 A. Borichev and M. Sodin [6] completely solved this problem for all those weights w for which P is dense in C 0 w but there exists a positive integer n = n(w) such that P is not dense in C 0 (1+x 2) n w. In the present paper we establish that if P is dense in C 0 (1+x 2) n w for all n ≥ 0 then for arbitrary ε > 0 there exists a weight Wε ∈ C ∞ (R) such that P is dense in C 0 (1+x 2) n Wε for every n ≥ 0 and Wε(x) ≥ w(x) + e −ε|x| for all x ∈ R.

ETNA - Electronic Transactions on Numerical Analysis, 2018

This paper summarizes recent results on weighted polynomial approximations for functions defined on the real semiaxis. The function may grow exponentially both at 0 and at +∞. We discuss orthogonal polynomials, polynomial inequalities, function spaces with new moduli of smoothness, estimates for the best approximation, Gaussian rules, and Lagrange interpolation with respect to the weight w(x) = x γ e −x −α −x β (α > 0, β > 1, γ ≥ 0).

The present paper is a study of L p − approximation by a new type of Bernstein-Durrmeyer type operators. It turns out the order of approximation by these operators is at best O(n −1 ), however smooth the function may be. In order to speed up the rate of convergence by the operators P n , we apply the technique of iterative combinations given by Micchelli [9] and prove that the order of approximation by these operators is O(n −k ) for sufficiently smooth functions.

2007

We discuss whether or not it is possible to have interpolatory pointwise estimates in the approximation of a function f 2 C 0; 1], by polynomials. For the sake of completeness as well as in order to strengthen some existing results, we discuss brieey the situation in unconstrained approximation. Then we deal with positive and monotone constraints where we show exactly when such interpolatory estimates are achievable by proving aarmative results and by providing the necessary counterexamples in all other cases. The eeect of the endpoints of the nite interval on the quality of approximation of continuous functions by algebraic polynomials, was rst observed by Nikolski Nik46]. Later pointwise estimates of this phenomenon were given by Timan Tim51] (k = 1), Dzjadyk Dzj58, Dzj77] (k = 2), Freud Fre59] (k = 2), and Brudny Bru63] (k 2), who proved that if f 2 C r 0; 1], then for each n N = r + k ? 1, a polynomial p n 2 n exists, such that (1.1) jf(x) ? p n (x)j c(r; k) r n (x)! k (f (r) ; ...

Pacific Journal of Mathematics, 1968

Let A be a discrete subring of C of rank 2. Let X be a compact subset of C with transfinite diameter not less than unity or with transfinite diameter less than unity, void interior, and connected complement. In an earlier paper we characterized the complex valued functions on X which can be uniformly approximated by elements from the ring of polynomials A[z], In this paper the same problem is studied where X is a compact subset of C with transfinite diameter d(X) less than unity and with nonvoid interior. It is also studied for certain compact subsets of C n where n is any positive integer. These subsets will have the property that every continuous function holomorphic on the interior is uniformly approximable by complex polynomials. A large class of sets of this type is shown to exist.

Monatshefte für Mathematik, 1995

Let w be a suitable weight function, B.,p denote the polynomial of best approximation to a function fin L~ [-1, 13, v. be the measure that associates a mass of 1/(n + 1) with each of the n+ 1 zeros of B.+~,p-B.,p and # be the arcsine measure defined by d/l:= (Tr~l -x2)l dx. We estimate the rate at which the sequence v. converges to t z in the weak-* topology. In particular, our theorem applies to the zeros of monic polynomials of minimal L~ norm.

Japan Journal of Industrial and Applied Mathematics, 1984

This work is motivated by the analysis of stability and convergence of spectral methods using Chebyshev and Legendre polynomials. We investigate the properties of the polynomial approximation to a functionu in the norms of the weightedL wp (− 1, 1) spaces. p is any real number between 1 and∞, andw (x) is either the Chebyshev or the Legendre weight. The estimates are given in terms of the degreeN of the polynomials and of the smoothness ofu. They include and generalize some theorems of Jackson. Some Bernstein ...