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Springer
Constructive approximation 4 (1988), S. 321-339
ISSN: 1432-0940
Keywords: Primary 41A21 ; Primary 41A20 ; Secondary 41A25 ; Secondary 30E10 ; Padé approximants ; Smooth coefficients ; Uniform convergence
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract Letf(z):=Σ j=0 ∞ a j z j , where aj ≠ 0,j large enough, and for someq ε C such that ¦q¦〈I, $$q_j : = a_{j - 1} a_{j + 1} /a_j^2 \to q,j \to \infty .$$ Define for m,n = 0,1,2,..., the Toeplitz determinant $$D(m/n): = \det (a_{m - j + k} )_{j,k = 1}^n .$$ Given ɛ 〉 0, we show that form large enough, and for everyn = 1,2,3,..., $$(1 - \varepsilon )^n \leqslant \left| {{{D(m/n)} \mathord{\left/ {\vphantom {{D(m/n)} {\left\{ {a_m^n \mathop \Pi \limits_{j - 1}^{n - 1} (1 - q_m^j )^{n - j} } \right\}}}} \right. \kern-\nulldelimiterspace} {\left\{ {a_m^n \mathop \Pi \limits_{j - 1}^{n - 1} (1 - q_m^j )^{n - j} } \right\}}}} \right| \leqslant (1 + \varepsilon )^n .$$ We apply this to show that any sequence of Padé approximants {[m k /n k ]} 1 ∞ tof, withm k →∞ ask→ ∞, converges locally uniformly in C. In particular, the diagonal sequence {[n/n]} 1 ∞ converges throughout C. Further, under additional assumptions, we give sharper asymptotics forD(m/n).
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• 2
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Springer
Constructive approximation 7 (1991), S. 501-519
ISSN: 1432-0940
Keywords: Primary 41A20 ; 41A21 ; Secondary 30E10 ; Multipoint Padé approximation ; Rational approximation ; Near-best approximation ; Best uniform approximation ; Interpolation ; Walsh array ; Distribution of poles
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract We show that the most of the time, most poles of diagonal multipoint Padé or best rational approximants to functions admitting fast rational approximation, leave the region of meromorphy. Following is a typical result: Letf be single-valued and analytic in CS, where cap(S)=0. Let {n j } j=1 ∞ be an increasing sequence of positive integers withn j+1/n j →1 asj→∞. Then there exists an infinite sequenceL of positive integers such that asj→∞,j∈L the total multiplicity of poles of any sequence of type (n j ,n j ) multipoint Padé or best rational approximants tof, iso(n j ) in any compactK in whichf is meromorphic. The sequenceL is independent of the particular sequence of multipoint Padé or best approximants, and yields the same behavior for “near-best” approximants. If the errors of best approximation on some compact set satisfy a weak regularity condition, then we may takeL={1,2,3,⋯}.
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• 3
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Springer
Constructive approximation 8 (1992), S. 463-535
ISSN: 1432-0940
Keywords: Primary 41A17 ; 42C05 ; Secondary 41A10 ; Freud weights ; Exponential weights ; Orthonormal polynomials ; Christoffel functions ; Markov-Bernstein inequalities ; Potentials ; Discretization of potentials ; Nevai's conjecture
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract We obtain upper and lower bounds for Christoffel functions for Freud weights by relatively new methods, including a new way to estimate discretization of potentials. We then deduce bounds for orthogonal polynomials onℝ thereby largely resolving a 1976 conjecture of P. Nevai. For example, let W:=e −Q, whereQ:ℝ→ℝ is even and continuous inℝ, Q" is continuous in (0, ∞) andQ '〉0 in (0, ∞), while, for someA, B, $$1〈 A \leqslant \frac{{(d/dx)(xQ'(x))}}{{Q'(x)}} \leqslant B,x \in (0,\infty )$$ Leta n denote thenth Mhaskar-Rahmanov-Saff number forQ, andL〉0. Then, uniformly forn≥1 and |x|≤a n (1+Ln −2/3), $$\lambda _n (W^2 ,x) \sim \frac{{a_n }}{n}W^2 (x)\left( {\max \left\{ {n^{ - 2/3} ,1 - \frac{{|x|}}{{a_n }}} \right\}} \right)^{ - 1/2}$$ Moreover, for all x εℝ, we can replace ∼ by ≥. In particular, these results apply toW(x):=exp(-|x|α), α〉1. We also obtain lower bounds for allx εℝ, when onlyA〉0, but this necessarily requires a more complicated formulation. We deduce that thenth orthonormal plynomialp n (W 2, ·). forW 2 satisfies $$\mathop {\sup }\limits_{x \in \mathbb{R}} |p_n (W^2 ,x)|W(x)\left| {1 - \frac{{|x|}}{{a_n }}} \right|^{1/4} \sim a_n^{ - 1/2}$$ and $$\mathop {\sup }\limits_{x \in \mathbb{R}} |p_n (W^2 ,x)|W(x) \sim a_n^{ - 1/2} n^{1/6} .$$ In particular, this applies toW(x):=exp(-|x|α), α〉1.
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• 4
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Springer
Numerische Mathematik 54 (1988), S. 33-39
ISSN: 0945-3245
Keywords: AMS(MOS): 41A21, 30E10, 30B70 ; CR: G1.2
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Summary Letf(z) be a function analytic in a neighbourhood of zero. For each pair of non-negative integers (m, n), form then byn Toeplitz determinantD(m/n) whose entries are the Maclaurin series coefficients off, namely, $$D(m/n): = det[f^{(m + j - k)} (0)/(m + j - k)!]_{j,k = 1'}^n$$ where we definef (s) (0)/s!≔0, ifs〈0. A classical theorem of Kronecker asserts thatf(z) is a rational function if and only if there existm 0 andn 0 such thatD(m/n)=0 form≧m 0 andn≧n 0. In some important recent work, such as the solution of Meinardus's Conjecture, it has been found useful to form Padé approximants not at 0, but at different points near 0. In questions regarding normality of these Padé approximants with a shifting origin, one considers then byn determinantD(m/n; u) which is defined by (1), but with 0 replaced byu. In this spirit, we prove thatf(z) is a rational function if and only if there exists asingle pair of positive integers (m, n) such thatD(m/n; u) is identically zero foru in a neighbourhood of zero. Further, we deduce that except possibly for countably many values ofu, the Padé table of a non-rationalf(z) atz=u is normal, that isD(m/n; u)≠0, for allm, n=0, 1, 2,....
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• 5
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Springer
Constructive approximation 15 (1999), S. 577-610
ISSN: 1432-0940
Keywords: Key words. Lagrange interpolation, Marcinkiewicz—Zygmund inequalities. AMS Classification. Primary 41A05; Secondary 42A15.
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract. We obtain converse Marcinkiewicz—Zygmund inequalities such as $$\| P\nu \|_{L_{p}[-1,1]}\leq C\left( \sum_{j=1}^{n}\mu _{j}| P(t_{j})| ^{p}\right) ^{1/p}$$ for polynomials P of degree ≤ n-1 , under general conditions on the points {t j } n j=1 and on the function ν . The weights {μ j } n j=1 are appropriately chosen. We illustrate the results by applying them to extended Lagrange interpolation for exponential weights on [-1,1] .
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• 6
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Springer
Constructive approximation 3 (1987), S. 331-361
ISSN: 1432-0940
Keywords: Padé approximant ; Theta function ; Rogers-Szegö polynomial ; Convergence regions ; Distribution of zeros ; poles ; Primary ; 41A21 ; 33A65 ; Secondary ; 30E05 ; 30E10
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract We investigate the convergence of sequences of Padé approximants for the partial theta function $$h_q (z): = \sum\limits_{j = 0}^\infty { q^{j(j - 1)/2_{Z^j } } } , q = e^{i\theta } , \theta \in [0,2\pi ).$$ Whenθ/(2π) is irrational, this function has the unit circle as its natural boundary. We determine subrogions of ¦z¦ 〈 1 in which sequences of Padé approximants converge uniformly, and subrogions in which they converge in capacity, but not uniformly. In particular, we show that only a proper subsequence of the diagonal sequence {[n/n]} n=1 ∞ converges locally uniformly in all of ¦z¦〈 l; in contrast, no subsequence of any Padé row {[m/n]} m=1 ∞ (withn ≥ 2 fixed) can converge locally uniformly in all of ¦z¦ 〈 1. Further, we obtain the zero and pole distributions of sequences of Padé approximants by analyzing the zero distribution of the Rogers-Szegö polynomials $$G_n (z): = \sum\limits_{j = 0}^n {\left[ {\begin{array}{*{20}c} n \\ j \\ \end{array} } \right]} z^j , n = 0,1,2,....$$
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• 7
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Springer
Constructive approximation 4 (1988), S. 9-20
ISSN: 1432-0940
Keywords: 42C05 ; Orthogonal polynomials ; Leading coefficients ; Freud's conjecture ; Recurrence relation coefficients ; Weighted polynomial approximation
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract LetW(x) be a function that is nonnegative inR, positive on a set of positive measure, and such that all power moments ofW 2 (x) are finite. Let {p n (W 2;x)} 0 ∞ denote the sequence of orthonormal polynomials with respect to the weightW 2, and let {α n } 1 ∞ and {β n } 1 ∞ denote the coefficients in the recurrence relation $$xp_n (W^2 ,x) = \alpha _{n + 1} p_{n + 1} (W^2 ,x) + \beta _n p_n (W^2 ,x) + \alpha _n p_{n - 1} (W^2 ,x).$$ We obtain a sufficient condition, involving mean approximation ofW −1 by reciprocals of polynomials, for $$\mathop {\lim }\limits_{n \to \infty } {{\alpha _n } \mathord{\left/ {\vphantom {{\alpha _n } {c_n }}} \right. \kern-\nulldelimiterspace} {c_n }} = \tfrac{1}{2}and\mathop {\lim }\limits_{n \to \infty } {{\beta _n } \mathord{\left/ {\vphantom {{\beta _n } {c_{n + 1} }}} \right. \kern-\nulldelimiterspace} {c_{n + 1} }} = 0,$$ wherec n 1 ∞ is a certain increasing sequence of positive numbers. In particular, we obtain a sufficient condition for Freud's conjecture associated with weights onR.
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• 8
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Springer
Constructive approximation 4 (1988), S. 21-64
ISSN: 1432-0940
Keywords: Primary 41A25 ; Primary 42C05 ; Weighted polynomials ; Exponential weights ; L ∞ approximation ; L p approximation ; Freud's conjecture ; Christoffel functions ; Zeros of orthogonal polynomials
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract LetW(x):= exp(-{tiQ(x})), where, for example, Q(x) is even and convex onR, and Q(x)/logx → ∞ asx → ∞. A result of Mhaskar and Saff asserts that ifa n =a n (W) is the positive root of the equation $$n = ({2 \mathord{\left/ {\vphantom {2 \pi }} \right. \kern-\nulldelimiterspace} \pi })\int_0^1 {{{a_n xQ'(a_n x)} \mathord{\left/ {\vphantom {{a_n xQ'(a_n x)} {\sqrt {1 - x^2 } }}} \right. \kern-\nulldelimiterspace} {\sqrt {1 - x^2 } }}dx,}$$ then, given any polynomialP n(x) of degree at mostn, the sup norm ofP n(x)W(a n x) overR is attained on [-1, 1]. In addition, any sequence of weighted polynomials {p n (x)W(a n x)} 1 ∞ that is uniformly bounded onR will converge to 0, for ¦x¦〉1. In this paper we show that under certain conditions onW, a function g(x) continuous inR can be approximated in the uniform norm by such a sequence {p n (x)W(a n x)} 1 ∞ if and only if g(x)=0 for ¦x¦⩾ 1. We also prove anL p analogue for 0〈p〈∞. Our results confirm a conjecture of Saff forW(x)=exp(−|x| α ), when α 〉1. Further applications of our results are upper bounds for Christoffel functions, and asymptotic behavior of the largest zeros of orthogonal polynomials. A final application is an approximation theorem that will be used in a forthcoming proof of Freud's conjecture for |x| p exp(−|x| α ),α 〉 0,p 〉 −1.
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• 9
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Springer
Constructive approximation 4 (1988), S. 65-83
ISSN: 1432-0940
Keywords: Primary 41A25 ; Primary 42C05 ; Exponential weights ; Freud's conjecture ; Orthogonal polynomials ; Recurrence relation coefficients
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract LetW (x) be a function nonnegative inR, positive on a set of positive measure, and such that all power moments ofW 2(x) are finite. Let {p n (W 2;x)} 0 ∞ denote the sequence of orthonormal polynomials with respect to the weightW 2(x), and let {A n } 1 ∞ and {B n } 1 ∞ denote the coefficients in the recurrence relation $$xp_n (W^2 ,x) = A_{n + 1} p_{n + 1} (W^2 ,x) + B_n p_n (W^2 ,x) + A_n p_{n - 1} (W^2 ,x).$$ . WhenW(x) =w(x) exp(-Q(x)), xε(-∞,∞), wherew(x) is a “generalized Jacobi factor,” andQ(x) satisfies various restrictions, we show that $$\mathop {\lim }\limits_{n \to \infty } {{A_n } \mathord{\left/ {\vphantom {{A_n } {a_n }}} \right. \kern-\nulldelimiterspace} {a_n }} = \tfrac{1}{2}and\mathop {\lim }\limits_{n \to \infty } {{B_n } \mathord{\left/ {\vphantom {{B_n } {a_n }}} \right. \kern-\nulldelimiterspace} {a_n }} = 0,$$ where, forn large enough,a n is the positive root of the equation $$n = ({2 \mathord{\left/ {\vphantom {2 \pi }} \right. \kern-\nulldelimiterspace} \pi })\int_0^1 {a_n xQ'(a_n x)(1 - x^2 )^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} dx.}$$ In the special case, Q(x) = ¦x¦α, a 〉 0, this proves a conjecture due to G. Freud. We also consider various noneven weights, and establish certain infinite-finite range inequalities for weighted polynomials inL p(R).
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• 10
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Springer
Constructive approximation 6 (1990), S. 257-286
ISSN: 1432-0940
Keywords: Walsh array ; Best rational approximants ; Entire functions ; Smooth coefficients ; Asymptotics ; Padé approximants
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract Let $$f(z): = \sum\nolimits_{j = 0}^\infty {a_j z^J }$$ be entire, witha j≠0,j large enough, $$\lim _{J \to \infty } a_{j + 1} /a_J = 0$$ , and, for someq∈C, $$q_j : = a_{j - 1} a_{j + 1} /a_j^2 \to q$$ asj→∞. LetE mn(f; r) denote the error in best rational approximation off in the uniform norm on |z‖≤r, by rational functions of type (m, n). We study the behavior ofE mn(f; r) asm and/orn→∞. For example, whenq above is not a root of unity, or whenq is a root of unity, butq m has a certain asymptotic expansion asm→∞, then we show that, for each fixed positive integern, ,m→∞. In particular, this applies to the Mittag-Leffler functions $$f(z): = \sum\nolimits_{j = 0}^\infty {z^j /\Gamma (1 + j/\lambda )}$$ and to $$f(z): = \sum\nolimits_{j = 0}^\infty {z^j /(j!)^{I/\lambda } }$$ , λ〉0. When |q‖〈1, we also handle the diagonal case, showing, for example, that ,n→∞. Under mild additional conditions, we show that we can replace 1+0(1) n by 1+0(1). In all cases we show that the poles of the best approximants approach ∞ asm→∞.
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