Kergin Interpolants of Holomorphic Functions

Let D be a C-convex domain in C ⁿ . Let , and d = 0,1,2, ..., be an array of points in a compact set . Let f be holomorphic on and let K _d (f) denote the Kergin interpolating polynomial to f at A _d0 ,... , A _dd . We give conditions on the array and D such that . The conditions are, in an appropriate sense, optimal. This result generalizes classical one variable results on the convergence of Lagrange—Hermite interpolants of analytic functions. Date received: October 21, 1995. Date revised: May 1, 1996.     

Extremal Polynomials with Preassigned Zeros and Rational Approximants

Let p _n be the n th orthonormal polynomial with respect to a positive finite measure μ supported by Δ=[-1,1] . It is well known that, uniformly on compact subsets of C/Δ , and, for a large class of measures μ , where g _Ω (z) is Green's function of with pole at infinity. It is also well known that these limit relations give convergence of the diagonal Padé approximants of the Markov function to f on Ω with a certain geometric speed measured by g _Ω (z) . We prove corresponding results when we restrict the freedom of p _n by preassigning some of the zeros. This means that the Padé approximants are replaced by Padé-type approximants where some of the poles are preassigned. We also replace Δ by general compact subsets of C. July 12, 1995. Date revised: October 1, 1996.     

On the Convergence in Capacity of Rational Approximants

Let {r _n } be a sequence of rational functions deg( r _n ≤ n) that converge rapidly in measure to an analytic function f on an open set in C ^N . We show that {r _n } converges rapidly in capacity to f on its natural domain of definition W _f (which, by a result of Goncar, is an open subset of C ^N ). In particular, for f meromorphic on C ^N and analytic near zero the sequence of Padé approximants {π _n (z, f, λ)} (as defined by Goncar) converges rapidly in capacity to f on C ^N . January 14, 1999. Date revised: October 7, 1999. Date accepted: November 1, 1999.     

A generalization of bounded variation

Generalized Wiener classes are considered. For these classes the exact order of Fourier coefficients with respect to the trigonometric system is established and the estimation of ‖S _n(·, f)-f(·)‖_{C [0,2π]} where S _n(·, f) are the Fourier partial sums, is given. In particular, a uniform convergence criterion for the Fourier trigonometric series is obtained. This revised version was published online in August 2006 with corrections to the Cover Date.     

A survey of truncation error analysis for Padé and continued fraction approximants

To compute the value of a functionf(z) in the complex domain by means of a converging sequence of rational approximants {f _n(z)} of a continued fraction and/or Padé table, it is essential to have sharp estimates of the truncation error ¦f(z)–f _n(z)¦. This paper is an expository survey of constructive methods for obtaining such truncation error bounds. For most cases dealt with, {f _n(z)} is the sequence of approximants of a continued fractoin, and eachf _n(z) is a (1-point or 2-point) Padé approximant. To provide a common framework that applies to rational approximantf _n(z) that may or may not be successive approximants of a continued fraction, we introduce linear fractional approximant sequences (LFASs). Truncation error bounds are included for a large number of classes of LFASs, most of which contain representations of important functions and constants used in mathematics, statistics, engineering and the physical sciences. An extensive bibliography is given at the end of the paper.Research supported in part by the U.S. National Science Foundation under Grants INT-9113400 and DMS-9302584.     

Propertyf −1(S) =g −1(S) for entire and meromorphicP-adic functions

LetW be an algebraically closed filed of characteristic zero, letK be an algebraically closed field of characteristic zero, complete for an ultrametric absolute value, and letA(K) (resp. ℳ(K)) be the set of entire (resp. meromorphic) functions inK. For everyn≥7, we show that the setS _n(b) of zeros of the polynomialx ⁿ−b (b≠0) is such that, iff, g ∈W[x] or iff, g ∈A(K), satisfyf ⁻¹(S _n(b))=g ⁻¹(S _n(b)), thenf ⁿ=g ⁿ. For everyn≥14, we show thatS _n(b) is such that iff, g ∈W({tx}) or iff, g ∈ ℳ(K) satisfyf ⁻¹(S _n(b))=g ⁻¹(S _n(b)), then eitherf ⁿ=g ⁿ, orfg is a constant. Analogous properties are true for complex entire and meromorphic functions withn≥8 andn≥15, respectively. For everyn≥9, we show that the setY _n(c) of zeros of the polynomial , (withc≠0 and 1) is an ursim ofn points forW[x], and forA(K). For everyn≥16, we show thatY _n(c) is an ursim ofn points forW(x), and for ℳ(K). We follow a method based on thep-adic Nevanlinna Theory and use certain improvement of a lemma obtained by Frank and Reinders.     

Convergence of Rational Interpolants with Preassigned Poles

Let Ω be a domain in the extended complex plane such that ∞∈Ω . Further, let K= C / Ω and, for each n , let Q _n be a monic polynomial of degree n with all its zeros in K . This paper is concerned with whether (Q _n ) can be chosen so that, if f is any holomorphic function on Ω and P _n is the polynomial part of the Laurent expansion of Q _n f at ∞ , then (P _n /Q _n ) converges to f locally uniformly on Ω . It is shown that such a sequence (Q _n ) can be chosen if and only if either K has zero logarithmic capacity or Ω is regular. January 21, 1999. Date accepted: August 17, 1999.     

LIMIT OF ITERATES FOR BERNSTEIN POLYNOMIALS DEFINED ON A TRIANGLE

Let f∈C[0,1],and Bn(f,x) be the a-th Bernstein polynomial associated with function f.ln 1967,the limit of iterates for B.(f,x) was given by Kelisky and Rivlin.After this,Many mathematicians studied and generalized this result.But anyway,all these discussions are only for univariate case ,In this paper,the main contrlbution is that the limit of lterates for Bernstein polynomial defined on a triangle is given completely.     

On q-Parametric Szász-Mirakjan Operators

In the present paper, we introduce q-parametric Szász-Mirakjan operators. We study convergence properties of these operators S _n,q (f). We obtain inequalities for the weighted approximation error of q-Szász-Mirakjan operators. Such inequalities are valid for functions of polynomial growth and are expressed in terms of weighted moduli of continuity. We also discuss Voronovskaja-type formula for q-Szász-Mirakjan operators.     

Lagrange Interpolation in Weighted Besov Spaces

The behavior of the Lagrange polynomial L _m (w,f) , based on the zeros of the orthogonal polynomials, is studied in some weighted Besov spaces B ^p _r,q (u) . It is proved that L _m (w) is a uniformly bounded map under suitable conditions on the weight functions and the parameters p , r , and q . December 11, 1996. Date revised: October 29, 1997. Date accepted: June 15, 1998.     

New Findings on the Bank–Sauer Approach in Oscillation Theory

In 1988, S. Bank showed that if {z _n } is a sparse sequence in the complex plane, with convergence exponent zero, then there exists a transcendental entire A(z) of order zero such that f″+A(z)f=0 possesses a solution having {z _n } as its zeros. Further, Bank constructed an example of a zero sequence {z _n } violating the sparseness condition, in which case the corresponding coefficient A(z) is of infinite order. In 1997, A. Sauer introduced a condition for the density of the points in the zero sequence {z _n } of finite convergence exponent such that the corresponding coefficient A(z) is of finite order.     

Growth Behavior and Zero Distribution of Rational Approximants

We investigate the growth and the distribution of zeros of rational uniform approximations with numerator degree ≤n and denominator degree ≤m _n for meromorphic functions f on a compact set E of ℂ where m _n =o(n/log n) as n→∞. We obtain a Jentzsch–Szegő type result, i.e., the zero distribution converges weakly to the equilibrium distribution of the maximal Green domain E _ρ(f) of meromorphy of f if f has a singularity of multivalued character on the boundary of E _ρ(f). The paper extends results for polynomial approximation and rational approximation with fixed degree of the denominator. As applications, Padé approximation and real rational best approximants are considered.     

On certain positive linear operators in polynomial weight spaces

We consider certain modified Szász-Mirakyan operators A _n (f;r) in polynomial weight spaces of functions of one variable and we study approximation properties of these operators. This revised version was published online in June 2006 with corrections to the Cover Date.     

Laurent-Padé approximants to four kinds of Chebyshev polynomial expansions. Part I. Maehly type approximants

Laurent Padé-Chebyshev rational approximants,A _m (z,z ⁻¹)/B _n (z, z ⁻¹), whose Laurent series expansions match that of a given functionf(z,z ⁻¹) up to as high a degree inz, z ⁻¹ as possible, were introduced for first kind Chebyshev polynomials by Clenshaw and Lord [2] and, using Laurent series, by Gragg and Johnson [4]. Further real and complex extensions, based mainly on trigonometric expansions, were discussed by Chisholm and Common [1]. All of these methods require knowledge of Chebyshev coefficients off up to degreem+n. Earlier, Maehly [5] introduced Padé approximants of the same form, which matched expansions betweenf(z,z ⁻¹)B _n (z, z ⁻¹)). The derivation was relatively simple but required knowledge of Chebyshev coefficients off up to degreem+2n. In the present paper, Padé-Chebyshev approximants are developed not only to first, but also to second, third and fourth kind Chebyshev polynomial series, based throughout on Laurent series representations of the Maehly type. The procedures for developing the Padé-Chebyshev coefficients are similar to that for a traditional Padé approximant based on power series [8] but with essential modifications. By equating series coefficients and combining equations appropriately, a linear system of equations is successfully developed into two sub-systems, one for determining the denominator coefficients only and one for explicitly defining the numerator coefficients in terms of the denominator coefficients. In all cases, a type (m, n) Padé-Chebyshev approximant, of degreem in the numerator andn in the denominator, is matched to the Chebyshev series up to terms of degreem+n, based on knowledge of the Chebyshev coefficients up to degreem+2n. Numerical tests are carried out on all four Padé-Chebyshev approximants, and results are outstanding, with some formidable improvements being achieved over partial sums of Laurent-Chebyshev series on a variety of functions. In part II of this paper [7] Padé-Chebyshev approximants of Clenshaw-Lord type will be developed for the four kinds of Chebyshev series and compared with those of the Maehly type.     

A continuous function space with a Faber basis

Let be compact with #S=∞ and let C(S) be the set of all real continuous functions on S. We ask for an algebraic polynomial sequence (P_n)_n=0^∞ with deg P_n=n such that every fC(S) has a unique representation f=∑_i=0^∞ α_iP_i and call such a basis Faber basis. In the special case of , 0<q<1, we prove the existence of such a basis. A special orthonormal Faber basis is given by the so-called little q-Legendre polynomials. Moreover, these polynomials state an example with A(S_q)≠U(S_q)=C(S_q), where A(S_q) is the so-called Wiener algebra and U(S_q) is the set of all fC(S_q) which are uniquely represented by its Fourier series.     

On Measures Induced by Extremal Signatures in Best Real Polynomial Approximation

We consider the limit distribution of measures μ _n , that appear in extremal signatures in the best polynomial approximation of a real-valued function . Relations between structural properties of the function f and weak-star limit points of (μ _n ) _n are proved. April 4, 1996. Date revised: October 25, 1996.     

On the Uniform Approximation of a Class of Analytic Functions by Bruwier Series

For a class of analytic functions f(z) defined by Laplace–Stieltjes integrals the uniform convergence on compact subsets of the complex plane of the Bruwier series (B-series) ∑^∞_n=0 λ_n(f) , λ_n(f)=f⁽ⁿ⁾(nc)+cf⁽ⁿ⁺¹⁾(nc), generated by f(z) and the uniform approximation of the generating function f(z) by its B-series in cones |arg z|< is shown.     

Inversion and characterization of the hemispherical transform

Explicit inversion formulas are obtained for the hemispherical transform(FΜ)(x) = Μ{y ∃S ⁿ :x. y ≥ 0},x ∃S ⁿ, whereS ⁿ is thendimensional unit sphere in ℝⁿ⁺¹,n ≥ 2, and Μ is a finite Borel measure onS ⁿ. If Μ is absolutely continuous with respect to Lebesgue measuredy onS ⁿ, i.e.,dΜ(y) =f(y)dy, we write(F f)(x) = ∫ _{x.y> 0} f(y)dy and consider the following cases: (a)f ∃C ^∞(Sⁿ); (b)f ∃ L^p(S ⁿ), 1 ≤ p < ∞; and (c)f ∃C(Sⁿ). In the case (a), our inversion formulas involve a certain polynomial of the Laplace-Beltrami operator. In the remaining cases, the relevant wavelet transforms are employed. The range ofF is characterized and the action in the scale of Sobolev spacesL _p ^γ (Sⁿ) is studied. For zonalf ∃ L¹(S ²), the hemispherical transformF f was inverted explicitly by P. Funk (1916); we reproduce his argument in higher dimensions. Partially sponsored by the Edmund Landau Center for Research in Mathematical Analysis, supported by the Minerva Foundation (Germany).     

Polynomial interpolation and sums of powers of integers

In this note, we revisit the problem of polynomial interpolation and explicitly construct two polynomials in n of degree k + 1, P_k(n) and Q_k(n), such that P_k(n) = Q_k(n) = f_k(n) for n = 1, 2,…?, k, where f_k(1), f_k(2),…?, f_k(k) are k arbitrarily chosen (real or complex) values. Then, we focus on the case that f_k(n) is given by the sum of powers of the first n positive integers S_k(n) = 1^k + 2^k + ??? + n^k, and show that S_k(n) admits the polynomial representations S_k(n) = P_k(n) and S_k(n) = Q_k(n) for all n = 1, 2,…?, and k ≥ 1, where the first representation involves the Eulerian numbers, and the second one the Stirling numbers of the second kind. Finally, we consider yet another polynomial formula for S_k(n) alternative to the well-known formula of Bernoulli.     

Estimates of the Remainder in Taylor's Theorem Using the Henstock-Kurzweil Integral

When a real-valued function of one variable is approximated by its nth degree Taylor polynomial, the remainder is estimated using the Alexiewicz and Lebesgue p-norms in cases where f ⁽ⁿ⁾ or f ⁽ⁿ⁺¹⁾ are Henstock-Kurzweil integrable. When the only assumption is that f ⁽ⁿ⁾ is Henstock-Kurzweil integrable then a modified form of the nth degree Taylor polynomial is used. When the only assumption is that f ⁽ⁿ⁾ ∈ C ⁰ then the remainder is estimated by applying the Alexiewicz norm to Schwartz distributions of order 1. Research partially supported by the Natural Sciences and Engineering Research Council of Canada. An adjunct appointment in the Department of Mathematical and Statistical Sciences, University of Alberta, made valuable library and computer resources available.