Semialgebraic complexity of functions

In this paper we study the rate of the best approximation of a given function by semialgebraic functions of a prescribed “combinatorial complexity”. We call this rate a “Semialgebraic Complexity” of the approximated function. By the classical Approximation Theory, the rate of a polynomial approximation is determined by the regularity of the approximated function (the number of its continuous derivatives, the domain of analyticity, etc.). In contrast, semialgebraic complexity (being always bounded from above in terms of regularity) may be small for functions not regular in the usual sense. We give various natural examples of functions of low semialgebraic complexity, including maxima of smooth families, compositions, series of a special form, etc. We show that certain important characteristics of the functions, in particular, the geometry of their critical values (Morse–Sard Theorem) are determined by their semialgebraic complexity, and not by their regularity.     

Rational functions of best L p- approximation and analytic continuability

Sufficient conditions for the analytic (resp. meromorphic) continuability of a function f ∈ L _p([?1, l]), p > 0, in terms of rational functions of best weighted L _p- approximation with an unbounded number of finite poles are established.     

An example of a complete orthonormal system in a Hilbert space of generalized functions

We construct a complete orthonormal system of generalized functions in a Hilbert space W ^?1. We obtain an estimate of the error of approximation in W ^?1, which is expressed in terms of the integral modulus of continuity of a function from L ₂.     

Weak and quasi-polynomial tractability of approximation of infinitely differentiable functions

We comment on recent results in the field of information based complexity, which state (in a number of different settings), that the approximation of infinitely differentiable functions is intractable and suffers from the curse of dimensionality. We show that renorming the space of infinitely differentiable functions in a suitable way allows weakly tractable uniform approximation by using only function values. Moreover, the approximating algorithm is based on a simple application of Taylor’s expansion about the center of the unit cube. We discuss also the approximation on the Euclidean ball and the approximation in the _L1$L_1$-norm.     

Asymptotics and numerics of polynomials used in Tricomi and Buchholz expansions of Kummer functions

Expansions in terms of Bessel functions are considered of the Kummer function ₁ F ₁(a; c, z) (or confluent hypergeometric function) as given by Tricomi and Buchholz. The coefficients of these expansions are polynomials in the parameters of the Kummer function and the asymptotic behavior of these polynomials for large degree is given. Tables are given to show the rate of approximation of the asymptotic estimates. The numerical performance of the expansions is discussed together with the numerical stability of recurrence relations to compute the polynomials. The asymptotic character of the expansions is explained for large values of the parameter a of the Kummer function.     

Metrics defined via discrepancy functions

We introduce the notion of a discrepancy function, as an extended real-valued function that assigns to a pair (A,U) of sets a nonnegative extended real number ω(A,U), satisfying specific properties. The pairs (A,U) are certain pairs of sets such that A⊆U, and for fixed A, the function ω takes on arbitrarily small nonnegative values as U varies. We present natural examples of discrepancy functions and show how they can be used to define traditional pseudo-metrics, quasimetrics and metrics on hyperspaces of topological spaces and measure spaces.     

The curse of dimensionality for the class of monotone functions and for the class of convex functions

We study the integration and approximation problems for monotone or convex bounded functions that depend on $d$ variables, where $d$ can be arbitrarily large. We consider the worst case error for algorithms that use finitely many function values. We prove that these problems suffer from the curse of dimensionality. That is, one needs exponentially many (in $d$) function values to achieve an error $\epsilon$.     

Optimally uniform approximation by meromorphic functions in the strip

The paper studies the uniform approximation problem of functions f, which are continuous in a closed strip S _h and holomorphic in its interior. Such functions are approximated on S _h by meromorphic functions g, the growth of which is estimated in the terms of the Nevanlinna characteristic T (r, g) and depends on the growth of f in the strip and the differential properties of f on the boundary of the strip. Also, the possible location of the poles of g in the complex plane is studied.     

On obtaining close estimates in the approximation of functions of many variables by sums of functions of a fewer number of variables

We present a simple method for finding the values of the best approximation of a function of n variables of a given class by means of sums of two functions of a fewer number of variables; we establish close upper and lower bounds for the value of the best approximation to the functionf(x₁, ..., x_n), having the mixed derivativef x₁ ... x_n, by means of sums of a function of n–1 variables.Translated from Matematicheskie Zametki, Vol. 12, No. 1, pp. 105–114, July, 1972.     

Approximation in the mean by convex functions

We give elementary proofs for the existence and uniqueness of the best L₁-approximation to a continuous function from the class of convex functions on a closed interval, and describe thebest approximation in terms of certain piecewise linear functions.     

Approximation of functions of finite variation by superpositions of a Sigmoidal function

The aim of this note is to generalize a result of Barron [1] concerning the approximation of functions, which can be expressed in terms of the Fourier transform, by superpositions of a fixed sigmoidal function. In particular, we consider functions of the type h(x) = ∫ℝ^d ƒ (〈t, x〉)dμ(t), where μ is a finite Radon measure on ℝ^d and ƒ : ℝ → ℂ is a continuous function with bounded variation in ℝ We show (Theorem 2.6) that these functions can be approximated in L₂-norm by elements of the set G_n = {Σ_i=0^staggeredn c_ig(〈a_i, x〉 + b_i) : a_iℝ^d, b_i, c_iℝ}, where g is a fixed sigmoidal function, with the error estimated by C/n^1/2, where C is a positive constant depending only on f. The same result holds true (Theorem 2.9) for f : ℝ → ℂ satisfying the Lipschitz condition under an additional assumption that ∫ℝ^d6t6e^d|u(t)| > ∞     

On some extremal problems of approximation theory of functions on the real axis II

The work is devoted to the solution of a number of extremal problems of approximation theory of functions on the real axis $ \mathbb{R} $ . In the space L ₂( $ \mathbb{R} $ ), the exact constants in Jackson-type inequalities are calculated. The exact values of average ν-widths are obtained for the classes of functions from L ₂( $ \mathbb{R} $ ) that are defined by averaged k-order moduli of continuity and for the classes of functions defined by K-functionals. In the chronological order, the sufficiently complete analysis of the final results related to the solution of extremal problems of approximation theory in the periodic case and on the whole real axis is carried out.     

Refined theorems of approximation theory in the space of p-absolutely continuous functions

In this paper, we prove direct and inverse theorems of approximation theory in the space of p-absolutely continuous functions which generalize Terekhin’s results in the same way as Timan’s results in L _p generalize the classical theorems of approximation theory. The main theorems are refined for functions with quasimonotone Fourier coefficients and, in a number of cases, the resulats are shown to be sharp.     

On isocline lines for functions and convex stratifications of two variables

Let the isoclines of a function u be the level lines of the function θ = arg(Du). Formulas for the curvature and the length of isocline lines in terms of the curvatures k, j of the level curves and of the steepest descent lines of u are given. The special case when all isoclines are straight lines is studied: in this case the steepest descent lines bend proportionally to the level lines; the support function of the level lines is linear function on the isoclines parameterized by the level values, possibly changing them. This characterization gives a new proof of a property of the developable surfaces found in [A. Fialkow, Geometric characterization of invariant partial differential equations, Amer. J. Math. 59(4) (1937), pp. 833–844]. When u is in the class of quasi convex functions, the L _p norm of the length function I _θ of the isoclines has minimizers with isoclines straight lines; the same occurs for other functionals on u depending on k, j. For a strictly regular quasi convex function isoclines may have arbitrarily large length and arbitrarily large L ₁ norm of I _θ.     

Optimal approximation of convex curves by functions which are piecewise linear

In this paper an efficient method is presented for solving the problem of approximation of convex curves by functions that are piecewise linear, in such a manner that the maximum absolute value of the approximation error is minimized. The method requires the curves to be convex on the approximation interval only. The boundary values of the approximation function can be either free or specified. The method is based on the property of the optimal solution to be such that each linear segment approximates the curve on its interval optimally while the optimal error is uniformly distributed among the linear segments of the approximation function. Using this method the optimal solution can be determined analytically to the full extent in certain cases, as it was done for functions x² and $\text{x}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. In general, the optimal solution has to be computed numerically following the procedure suggested in the paper. Using this procedure, optimal solutions were computed for functions sin x, tg x, and arc tg x. Optimal solutions to these functions were used in practical applications.     

On some extremal problems of approximation theory of functions on the real axis I

A number of extremal problems of approximation theory of functions have been solved on the real line $ \mathbb{R} $ . Exact constants in the Jackson-type inequalities in the space L ₂( $ \mathbb{R} $ ) are calculated. The exact values of average ν-widths are obtained for the classes of functions from L ₂( $ \mathbb{R} $ ) defined by averaged moduli of continuity of the k-th order, as well as for the classes of functions defined by K-functionals. The quite complete analysis of the final results related to the solution of extremal problems of approximation theory in the periodic case and for the whole real axis is carried out in the chronological order.     

Regions of meromorphy and value distribution of geometrically converging rational functions

Let D be a region, {rn}_n∈N a sequence of rational functions of degree at most n and let each rn have at most m poles in D, for m∈N fixed. We prove that if {rn}_n∈N converges geometrically to a function f on some continuum S⊂D and if the number of zeros of rn in any compact subset of D is of growth o(n) as n→∞, then the sequence {rn}_n∈N converges m₁-almost uniformly to a meromorphic function in D. This result about meromorphic continuation is used to obtain Picard-type theorems for the value distribution of m₁-maximally convergent rational functions, especially in Padé approximation and Chebyshev rational approximation.     

Polynomial operators for spectral approximation of piecewise analytic functions

We propose the construction of a mixing filter for the detection of analytic singularities and an auto-adaptive spectral approximation of piecewise analytic functions, given either spectral or pseudo-spectral data, without knowing the location of the singularities beforehand. We define a polynomial frame with the following properties. At each point on the interval, the behavior of the coefficients in our frame expansion reflects the regularity of the function at that point. The corresponding approximation operators yield an exponentially decreasing rate of approximation in the vicinity of points of analyticity and a near best approximation on the whole interval. Unlike previously known results on the construction of localized polynomial kernels, we suggest a very simple idea to obtain exponentially localized kernels based on a general system of orthogonal polynomials, for which the Cesàro means of some order are uniformly bounded. The boundedness of these means is known in a number of cases, where no special function properties are known.     

On rational approximation of algebraic functions

We construct a new scheme of approximation of any multivalued algebraic function f(z) by a sequence {r_n(z)}_n∈N of rational functions. The latter sequence is generated by a recurrence relation which is completely determined by the algebraic equation satisfied by f(z). Compared to the usual Padé approximation our scheme has a number of advantages, such as simple computational procedures that allow us to prove natural analogs of the Padé Conjecture and Nuttall's Conjecture for the sequence {r_n(z)}_n∈N in the complement CP¹?D_f, where D_f is the union of a finite number of segments of real algebraic curves and finitely many isolated points. In particular, our construction makes it possible to control the behavior of spurious poles and to describe the asymptotic ratio distribution of the family {r_n(z)}_n∈N. As an application we settle the so-called 3-conjecture of Egecioglu et al. dealing with a 4-term recursion related to a polynomial Riemann Hypothesis.