Interpolatory properties of best L2-approximants (original) (raw)

Polynomial interpolation, an L-function, and pointwise approximation of continuous functions

Journal of Approximation Theory, 2008

We show that if {s k } ∞ k=1 is the sequence of all zeros of the L-function L(s,) := ∞ k=0 (−1) k (2k + 1) −s satisfying Re s k ∈ (0, 1), k = 1, 2,. .. , then any function from span {|x| s k } ∞ k=1 satisfies the pointwise rapid convergence property, i.e. there exists a sequence of polynomials Q n (f, x) of degree at most n such that f − Q n C[−1,1] C(f)E n (f), n=1, 2,. .. , and for every x ∈ [−1, 1], lim n→∞ (|f (x)−Q n (f, x)|)/E n (f)= 0, where E n (f) is the error of best polynomial approximation of f in C[−1, 1]. The proof is based on Lagrange polynomial interpolation to |x| s , Re s > 0, at the Chebyshev nodes. We also establish a new representation for |L(x,

Interpolatory Pointwise Estimates for Polynomial Approximation X1. Introduction and Main Results

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) ; ...

Approximation by polynomials

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

Interpolatory Pointwise Estimates for Polynomial Approximation

Constructive Approximation, 2000

N ov 2 01 7 Interpolatory pointwise estimates for monotone polynomial approximation

2018

Interpolatory pointwise estimates for monotone polynomial approximation

Journal of Mathematical Analysis and Applications, 2018

Given a nondecreasing function f on [−1, 1], we investigate how well it can be approximated by nondecreasing algebraic polynomials that interpolate it at ±1. We establish pointwise estimates of the approximation error by such polynomials that yield interpolation at the endpoints (i.e., the estimates become zero at ±1). We call such estimates "interpolatory estimates". In 1985, DeVore and Yu were the first to obtain this kind of results for monotone polynomial approximation. Their estimates involved the second modulus of smoothness ω2(f, •) of f evaluated at √ 1 − x 2 /n and were valid for all n ≥ 1. The current paper is devoted to proving that if f ∈ C r [−1, 1], r ≥ 1, then the interpolatory estimates are valid for the second modulus of smoothness of f (r) , however, only for n ≥ N with N = N(f, r), since it is known that such estimates are in general invalid with N independent of f. Given a number α > 0, we write α = r + β where r is a nonnegative integer and 0 < β ≤ 1, and denote by Lip * α the class of all functions f on [−1, 1] such that ω2(f (r) , t) = O(t β). Then, one important corollary of the main theorem in this paper is the following result that has been an open problem for α ≥ 2 since 1985: If α > 0, then a function f is nondecreasing and in Lip * α, if and only if, there exists a constant C such that, for all sufficiently large n, there are nondecreasing polynomials Pn, of degree n, such that |f (x) − Pn(x)| ≤ C √ 1 − x 2 n α , x ∈ [−1, 1].

On the Zeros of Polynomials of Best Approximation

Journal of Approximation Theory, 1999

Given a function f, uniform limit of analytic polynomials on a compact, regular set E/C N , we relate analytic extension properties of f to the location of the zeros of the best polynomial approximants to f in either the uniform norm on E or in appropriate L q norms. These results give multivariable versions of one-variable results due to Blatt Saff, Ples niak and Wo jcik.

Interpolatory properties of best L2-approximants (original) (raw)

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