On the size of lemniscates of polynomials in one and several variables (original) (raw)
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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) ; ...
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Notation. Let Rm be the Euclidean m-dimensional space with elements x = (x1, . . . , xm), y = (y1, . . . , ym), t = (t1, . . . , tm), s = (s1, . . . , sm), v = (v1, . . . , vm), the inner product (t, x) := ∑m j=1 tjxj , and the norm |x| := √ (x, x). Next, Cm := Rm + iRm is the m-dimensional complex space with elements z = (z1, . . . , zm) = x+ iy and the norm |z| := √ |x|2 + |y|2; Zm denotes the set of all integral lattice points in Rm; Z+ is a subset of Z m of all points with nonnegative coordinates; and N := {1, 2, . . .}. We also use multi-indices k = (k1, . . . , km) ∈ Z+ with |k| := ∑m j=1 kj and xk := x1 1 · · · xkm m . Given σ ∈ Rm \ {0}, let Πm(σ) := {t ∈ Rm : |tj | ≤ |σj | , 1 ≤ j ≤ m} be the l-dimensional parallelepiped in Rm, where l ≥ 1 is the number of nonzero coordinates of σ. Given M > 0, let Qm(M) := {t ∈ Rm : |tj| ≤ M, 1 ≤ j ≤ m}, Bm(M) := {t ∈ Rm : |t| ≤ M}, Om(M) := {t ∈ Rm : ∑mj=1 |tj | ≤ M}, and Sm−1 := {t ∈ Rm : |t| = 1} be the m-dimensional cube, ball, octa...
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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.
On the Lebesgue constant of Berrut’s rational interpolant at equidistant nodes
Journal of Computational and Applied Mathematics, 2011
It is well known that polynomial interpolation at equidistant nodes can give bad approximation results and that rational interpolation is a promising alternative in this setting. In this paper we confirm this observation by proving that the Lebesgue constant of Berrut's rational interpolant grows only logarithmically in the number of interpolation nodes. Moreover, the numerical results show that the Lebesgue constant behaves similarly for interpolation at Chebyshev as well as logarithmically distributed nodes.
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We define F -polynomials as linear combinations of dilations by some frequencies of an entire function F . In this paper we use Padé interpolation of holomorphic functions in the unit disk by F -polynomials to obtain explicitly approximating F -polynomials with sharp estimates on their coefficients. We show that when frequencies lie in a compact set K ⊂ C then optimal choices for the frequencies of interpolating polynomials are similar to Fekete points. Moreover, the minimal norms of the interpolating operators form a sequence whose rate of growth is determined by the transfinite diameter of K.
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Let θ be a real number satisfying 1 < θ < 2, and let A(θ) be the set of polynomials with coefficients in {0, 1}, evaluated at θ. Using a result of Bugeaud, we prove by elementary methods that θ is a Pisot number when the set (A(θ) − A(θ) − A(θ)) is discrete; the problem whether Pisot numbers are the only numbers θ such that 0 is not a limit point of (A(θ) − A(θ)) is still unsolved. We also determine the three greatest limit points of the quantities inf{c, c > 0, c ∈ C(θ)}, where C(θ) is the set of polynomials with coefficients in {−1, 1}, evaluated at θ , and we find in particular infinitely many Perron numbers θ such that the sets C(θ) are discrete.
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