Rates of Best Uniform Rational Approximation of Analytic Functions by Ray Sequences of Rational Functions

In this paper, problems related to the approximation of a holomorphic function f on a compact subset E of the complex plane C by rational functions from the class of all rational functions of order (n,m) are considered. Let ρ _n,m = ρ _n,m (f;E) be the distance of f in the uniform metric on E from the class . We obtain results characterizing the rate of convergence to zero of the sequence of the best rational approximation { ρ _n,m(n) } _n=0 ^∞ , m(n)/n → θ ∈ (0,1] as n → ∞ . In particular, we give an upper estimate for the liminf _{n →∞} ρ _n,m(n) ^1/(n+m(n)) in terms of the solution to a certain minimum energy problem with respect to the logarithmic potential. The proofs of the results obtained are based on the methods of the theory of Hankel operators. June 16, 1997. Date revised: December 1, 1997. Date accepted: December 1, 1997. Communicated by Ronald A. DeVore.     

On Best Rational Approximations to Certain Functions Defined as Integrals

Rational approximations are constructed for certain real-valuedfunctions of t [–k, k] denned in terms of the analyticfunction where w(x) is a non-negative weight-function and (x) is a polynomialwith distinct zeros, all in (–1,1). The method is similarto that described in Barrett (1971), using Gaussian quadratureafter a suitable change of variable. Asymptotic upper and lowerbounds are found for the maximum error of a best Chebyshev rationalapproximation of degree n.     

Uniform Continuity of Generalized Rational Approximations

In this paper, it is shown that there are no uniformly continuous multiplicative $\varepsilon $ -selections from $C[0;1]$ on the set of generalized rational fractions of sufficiently general form for small $\varepsilon $ .     

On the Problem of Describing Sequences of Best Trigonometric Rational Approximations

For a strictly decreasing sequence a_{n n=0} of nonnegative real numbers converging to zero, we construct a continuous 2-periodic function f such that R^T _n(f) = a_n, n=0,1,2,..., where R^T _n(f) are best approximations of the function f in uniform norm by trigonometric rational functions of degree at most n.     

On the Best Approximation by Ridge Functions in the Uniform Norm

We consider the best approximation of some function classes by the manifold M _n consisting of sums of n arbitrary ridge functions. It is proved that the deviation of the Sobolev class W _p ^r,d from the manifold M _n in the space L _q for any 2≤ q≤ p≤∈fty behaves asymptotically as n ^-r/(d-1) . In particular, we obtain this asymptotic estimate for the uniform norm p=q=∈fty . January 10, 2000. Date revised: March 1, 2001. Date accepted: March 12, 2001.     

Rates of Best Rational Approximation of Analytic Functions

Let E be a compact set in the extended complex plane C and let f be holomorphic on E. Denote by ρ_n the distance from f to the class of all rational functions of order at most n, measured with respect to the uniform norm on E. We obtain results characterizing the relationship between estimates of lim inf_n→∞ ρ^1/n_n and lim sup_n→∞ ρ^1/n_n.     

Best Uniform Approximation of Complex-Valued Functions by Generalized Polynomials Having Restricted Ranges

We study the uniform best restricted ranges approximations of complex-valued functions by generalized polynomials. The theory, generalizing the real-valued case, embraces the theorems of existence, characterization, uniqueness, and strong uniqueness.     

Uniform Approximations of Bernoulli and Euler Polynomials in Terms of Hyperbolic Functions

Bernoulli and Euler polynomials are considered for large values of the order. Convergent expansions are obtained for B _n ( nz +1/2) and E _n ( nz +1/2) in powers of n ⁻¹, and coefficients are rational functions of z and hyperbolic functions of argument 1/(2 z ). These expansions are uniformly valid for | z ± i /2π|>1/2π and | z ± i /π|1/π, respectively. For a real argument, the accuracy of these approximations is restricted to the monotonic region. The range of validity of the uniformity parameter z is enlarged, respectively, to regions of the form | z ± i /2( m +1)π|>1/2( m +1)π and | z ± i /(2 m +1)π|>1/(2 m +1)π, m =1,2,3,…, by adding certain combinations of incomplete gamma functions to these uniform expansions. In addition, the convergence of these improved expansions is stronger, and for a real argument, the accuracy of these improved approximations is also better in the oscillatory region.     

Maps Between Uniform Algebras Preserving Norms of Rational Functions

Let A, B be uniform algebras. Suppose that A ₀, B ₀ are subgroups of A ⁻¹, B ⁻¹ that contain exp A, exp B respectively. Let α be a non-zero complex number. Suppose that m, n are non-zero integers and d is the greatest common divisor of m and n. If T : A ₀ → B ₀ is a surjection with ||T(f)^mT(g)ⁿ - a||_￥ = ||f^mgⁿ - a||_￥{\|T(f)^{m}T(g)^{n} - \alpha\|_{\infty} = \|f^{m}g^{n} - \alpha\|_{\infty}} for all f,g ? A₀{f,g \in A_0}, then there exists a real-algebra isomorphism [(T)\tilde] : A ? B{\tilde{T} : A \rightarrow B} such that [(T)\tilde](f)^d = (T(f)/T(1))^d{\tilde{T}(f)^d = (T(f)/T(1))^d} for every f ? A₀{f \in A_0}. This result leads to the following assertion: Suppose that S _A , S _B are subsets of A, B that contain A ⁻¹, B ⁻¹ respectively. If m, n > 0 and a surjection T : S _A → S _B satisfies ||T(f)^mT(g)ⁿ - a||_￥ = ||f^mgⁿ - a||_￥{\|T(f)^{m}T(g)^{n} - \alpha\|_{\infty} = \|f^{m}g^{n} - \alpha\|_{\infty}} for all f, g ? S_A{f, g \in S_A}, then there exists a real-algebra isomorphism [(T)\tilde] : A ? B{\tilde{T} : A \rightarrow B} such that [(T)\tilde](f)^d = (T(f)/T(1))^d{\tilde{T}(f)^d = (T(f)/T(1))^d} for every f ? S_A{f \in S_A}. Note that in these results and elsewhere in this paper we do not assume that T(exp A) = exp B.     

Norms of Multipliers and Best Approximations of Holomorphic Functions of Many Variables

We show that the Lebesgue–Landau constants of linear methods for summation of Taylor series of functions holomorphic in a polydisk and in the unit ball from over triangular domains do not depend on the number m. On the basis of this fact, we find a relation between the complete and partial best approximations of holomorphic functions in a polydisk and in the unit ball from .     

On the Convergence of Rational Functions of Best Lp-Approximation

In the present paper, for sequences of rational functions, analytic in some domain, a theorem of Montel’s type is proved. As an application, sequences of rational functions of best Lp — approximation of a function ?(z) ∈ L_p(Γ), with Γ being a closed rectifiable analytic curve, are considered. The case ?(z) ∈ Hp is discussed, too.     

On the Existence of Functions with Prescribed Best $L_1$ Approximations

This paper gives a partial answer to a problem of Rivlin in $L_1$ approximation.     

Asymptotic Two-Point Approximations by Rational Function

Continuous functions in 0 ≤ s ≤ ∞ with asymptotic power expansions both for t → +0 and t → +∞ are approximated by rational functions, being generalized Padé approximants both for s → +0 and s → +∞. The existence and the uniform convergence of such approximants are proved. As examples, piecewise rational functions, square roots of rational functions, and a modified error function are considered.     

On the Exact Asymptotics of the Best Relative Approximations of Classes of Periodic Functions by Splines

We obtain the exact asymptotics (as n ) of the best L ₁-approximations of classes of periodic functions by splines s S _{2n, r – 1} and s S _{2n, r + k – 1} (S _{2n, r} is the set of 2-periodic polynomial splines of order r and defect 1 with nodes at the points k/n, k Z) under certain restrictions on their derivatives.     

多元函数的最佳逼近递减的逼近阶

关于找一个必要与充分条件使得ωｉ１…ｋｉｍ（ｆ,１σｉ１,…,１σｉｍ）～Ｙσｉ１…σｉｍ（ｆ）,（σｉｊ→∞）成立的Ｔｉｍａｎ问题被解决。这条件是ωｋ１…ｋｉｍ（ｆ,δｉ１…δｉｍ）～ωｋｉ１＋１,…,ｋｉｍ＋１（ｆ,δｉ１…δｉｍ）,（δｉｊ→０）。