Near-best L1 approximations on circular and elliptical contours

Polynomial approximations are obtained to analytic functions on circular and elliptical contours by forming partial sums of order n of their expansions in Taylor series and Chebyshev series of the second kind, respectively. It is proved that the resulting approximations converge in the L₁ norm as n → ∞, and that they are near-best L₁ approximations within relative distances of the order of log n. Practical implications of the results are discussed, and they are shown to provide a theoretical basis for polynomial approximation methods for the evaluation of indefinite integrals on contours.     

Weighted simultaneous Chebyshev approximation

In this paper we discuss the problem of weighted simultaneous Chebyshev approximation to functions f₁,…f_m ε C(X) (1 m ∞), i.e., we wish to minimize the expression {∑^m_{j = 1} λ_j¦f_j − q¦^p}^1/p∞, where λ_j > 0, ∑^m_{j = 1} λ_j = 1, p 1. For this problem we establish the main theorems of the Chebyshev theory, which include the theorems of existence, alternation, de La Vallée Poussin, uniqueness, strong uniqueness, as well as that of continuity of the best approximation operator, etc.     

The Bernstein Constant and Polynomial Interpolation at the Chebyshev Nodes

In this paper, we establish new asymptotic relations for the errors of approximation in L_p[−1,1], 0<p∞, of x^λ, λ>0, by the Lagrange interpolation polynomials at the Chebyshev nodes of the first and second kind. As a corollary, we show that the Bernstein constant

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is finite for λ>0 and .     

L approximation with trigonometric n-nomials

We compare the degree of approximation to L²(−π, π) by nth degree trigonometric polynomials, with the degree of approximation by trigonometric n-nomials, which are linear combinations, with constant (complex) coefficients, of any 2n + 1 members of the sequence {exp (ikx)}, − ∞ < k < ∞.     

The Minimality Properties of Chebyshev Polynomials and Their Lacunary Series

By considering four kinds of Chebyshev polynomials, an extended set of (real) results are given for Chebyshev polynomial minimality in suitably weighted Hölder norms on [–1,1], as well as (L ) minimax properties, and best L ₁ sufficiency requirements based on Chebyshev interpolation. Finally we establish best L _p , L and L ₁ approximation by partial sums of lacunary Chebyshev series of the form _i=0 a ⁱ _b ⁱ(x) where _n (x) is a Chebyshev polynomial and b is an odd integer 3. A complete set of proofs is provided.     

Some new Chebyshev spaces

In this note we see another circumstance where Chebyshev polynomials play a significant role. In particular, we present some new extended Chebyshev spaces that arise in the asymptotic stability of the zero solution of first order linear delay differential equations with m commensurate delays where a_j,j=0,…,m, are constants and τ>0 is constant.     

On real and complex-valued bivariate Chebyshev polynomials

In the real uniform approximation of the function x^myⁿ by the space of bivariate polynomials of total degree m + n − 1 on the unit square, the product of monic univariate Chebyshev polynomials yields an optimal error. We exploit the fundamental Noether's theorem of algebraic curves theory to give necessary and sufficient conditions for unicity and to describe the set of optimal errors in case of nonuniqueness. Then, we extend these results to the complex approximation on biellipses. It turns out that the product of Chebyshev polynomials also provides an optimal error and that the same kind of uniqueness conditions prevail in the complex case. Yet, when nonuniqueness occurs, the characterization of the set of optimal errors presents peculiarities, compared to the real problem.     

Expansion of functions in a Fourier-Chebyshev series by shifted Chebyshev polynomials of the first kind

The paper considers an algorithm that develops functions in a Fourier—Chebyshev series by shifted Chebyshev polynomials of the first kind. The relationship of the algorithm with Fourier transform and discrete cosine transform is established.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 73, pp. 25–27, 1992.     

Decomposition and approximation of multivariate functions on the cube

In this paper, we present a decomposition method of multivariate functions. This method shows that any multivariate function f on [0, 1]d is a finite sum of the form ∑jφjψj , where each φj can be extended to a smooth periodic function, each ψj is an algebraic polynomial, and each φjψj is a product of separated variable type and its smoothness is same as f . Since any smooth periodic function can be approximated well by trigonometric polynomials, using our decomposition method, we find that any smooth multivariate function on [0, 1]d can be approximated well by a combination of algebraic polynomials and trigonometric polynomials. Meanwhile, we give a precise estimate of the approximation error.     

Constrained near-minimax approximation by weighted expansion and interpolation using Chebyshev polynomials of the second,third, and fourth kinds

Well known results on near-minimax approximation using Chebyshev polynomials of the first kind are here extended to Chebyshev polynomials of the second, third, and fourth kinds. Specifically, polynomial approximations of degreen weighted by (1–x ²)^1/2, (1+x)^1/2 or (1–x)^1/2 are obtained as partial sums of weighted expansions in Chebyshev polynomials of the second, third, or fourth kinds, respectively, to a functionf continuous on [–1, 1] and constrained to vanish, respectively, at ±1, –1 or +1. In each case a formula for the norm of the resulting projection is determined and shown to be asymptotic to 4^–2logn +A +o(1), and this provides in each case and explicit bound on the relative closeness of a minimax approximation. The constantA that occurs for the second kind polynomial is markedly smaller, by about 0.27, than that for the third and fourth kind, while the latterA is identical to that for the first kind, where the projection norm is the classical Lebesgue constant _n . The results on the third and fourth kind polynomials are shown to relate very closely to previous work of P.V. Galkin and of L. Brutman.Analogous approximations are also obtained by interpolation at zeros of second, third, or fourth kind polynomials of degreen+1, and the norms of resulting projections are obtained explicitly. These are all observed to be asymptotic to 2^–1logn +B +o(1), and so near-minimax approximations are again obtained. The norms for first, third, and fourth kind projections appear to be converging to each other. However, for the second kind projection, we prove that the constantB is smaller by a quantity asymptotic to 2^–1log2, based on a conjecture about the point of attainment of the supremum defining the projection norm, and we demonstrate that the projection itself is remarkably close to a minimal (weighted) interpolation projection.All four kinds of Chebyshev polynomials possess a weighted minimax property, and, in consequence, all the eight approximations discussed here coincide with minimax approximations when the functionf is a suitably weighted polynomial of degreen+1.     

Numerical constructions involving Chebyshev polynomials

We propose a new algorithm for the character expansion of tensor products of finite-dimensional irreducible representations of simple Lie algebras. The algorithm produces valid results for the algebras B ₃, C ₃, and D ₃. We use the direct correspondence between Weyl anti-invariant functions and multivariate second-kind Chebyshev polynomials. We construct the triangular trigonometric polynomials for the algebra D ₃.     

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 function f(z,z ^–1) up to as high a degree in z,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 of f up to degree m+n. Earlier, Maehly [5] introduced Padé approximants of the same form, which matched expansions between f(z,z ^–1)B _n (z,z ^–1) and A _m (z,z ^–1). The derivation was relatively simple but required knowledge of Chebyshev coefficients of f up to degree m+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 degree m in the numerator and n in the denominator, is matched to the Chebyshev series up to terms of degree m+n, based on knowledge of the Chebyshev coefficients up to degree m+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.     

Approximation of classes of periodic functions with small smoothness

We prove that the approximations of classes of periodic functions with small smoothness in the metrics of the spaces C and L by different linear summation methods for Fourier series are asymptotically equal to the least upper bounds of the best approximations of these classes by trigonometric polynomials of degree not higher than (n - 1). We establish that the Fejér method is asymptotically the best among all positive linear approximation methods for these classes.     

Best <Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript> approximation of Heaviside-type functions from Chebyshev and weak-Chebyshev spaces

In this article, we study the problem of best L ₁ approximation of Heaviside-type functions from Chebyshev and weak-Chebyshev spaces. We extend the Hobby-Rice theorem (Proc. Am. Math. Soc., 16, 665–670, 1965) into an appropriate framework and prove the unicity of best L ₁ approximation of Heaviside-type functions from an even-dimensional Chebyshev space under some assumptions on the dimension of the subspaces composed of the odd and even functions. We also apply the results to compute best L ₁ approximations of Heaviside-type functions by polynomials and Hermite polynomial splines with fixed knots.     

The minimality properties of chebyshev polynomials and their lacunary series

By considering four kinds of Chebyshev polynomials, an extended set of (real) results are given for Chebyshev polynomial minimality in suitably weighted Hölder norms on [?1, 1], as well as (L _∞ minimax properties, and bestL ₁ sufficiency requirements based on Chebyshev interpolation. Finally we establish bestL _p,L _∞ andL ₁ approximation by partial sums of lacunary Chebyshev series of the form ∑ ₁₌₀ ^∞ a ⁱ ? _b ⁱ (x) where? _x (x) is a Chebyshev polynomial andb is an odd integer ≥3. A complete set of proofs is provided.     

Near-circularity of the error curve in complex Chebyshev approximation

Let f(z) be analytic on the unit disk, and let p^*(z) be the best (Chebyshev) polynomial approximation to f(z) on the disk of degree at most n. It is observed that in typical problems the “error curve,” the image of the unit circle under (f − p^*)(z), often approximates to a startling degree a perfect circle with winding number n + 1. This phenomenon is approached by consideration of related problems whose error curves are exactly circular, making use of a classical theorem of Carathéodory and Fejér. This leads to a technique for calculating approximations in one step that are roughly as close to best as the best approximation error curve is close to circular, and hence to strong theorems on near-circularity as the radius of the domain shrinks to 0 or as n increases to ∞. As a computational example, very tight bounds are given for approximation of e^z on the unit disk. The generality of the near-circularity phenomenon (more general domains, rational approximation) is discussed.     

Estimate for the Best Approximation of Summable Functions of Several Variables with a Certain Symmetry of Fourier Coefficients

An upper bound for the best approximation of summable functions of several variables by trigonometric polynomials in the metric of L is determined in terms of Fourier coefficients. We consider functions representable by trigonometric series with certain symmetry of coefficients satisfying a multiple analog of the Sidon–Telyakovskii conditions.     

The convergence of the best discrete linear Lp approximation as p → 1

It is well known that the best discrete linear L_p approximation converges to a special best Chebyshev approximation as p → ∞. In this paper it is shown that the corresponding result for the case p → 1 is also true. Furthermore, the special best L₁ approximation obtained as the limit is characterized as the unique solution of a nonlinear programming problem on the set of all L₁ solutions.     

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.     

Approximation by Nörlund means of Walsh-Fourier series

We study the rate of approximation by Nörlund means for Walsh-Fourier series of a function in L^p and, in particular, in Lip(α, p) over the unit interval [0, 1), where α > 0 and 1 p ∞. In case p = ∞, by L^p we mean C_W, the collection of the uniformly W-continuous functions over [0, 1). As special cases, we obtain the earlier results by Yano, Jastrebova, and Skvorcov on the rate of approximation by Cesàro means. Our basic observation is that the Nörlund kernel is quasi-positive, under fairly general assumptions. This is a consequence of a Sidon type inequality. At the end, we raise two problems.