An approximation property of exponential functions

We solve the inhomogeneous linear first order differential equations of the form y′(x) ? λy(x) = Σ _m=0 ^∞ a _m (x ? c) ^m , and prove an approximation property of exponential functions. More precisely, we prove the local Hyers-Ulam stability of linear first order differential equations of the form y′(x) = λy(x) in a special class of analytic functions.     

An improved estimate of PSWF approximation and approximation by Mathieu functions

In this paper, an error estimate of spectral approximations by prolate spheroidal wave functions (PSWFs) with explicit dependence on the bandwidth parameter and optimal order of convergence is derived, which improves the existing result in [Chen et al., Spectral methods based on prolate spheroidal wave functions for hyperbolic PDEs, SIAM J. Numer. Anal. 43 (5) (2005) 1912-1933]. The underlying argument is applied to analyze spectral approximations of periodic functions by Mathieu functions, which leads to new estimates featured with explicit dependence on the intrinsic parameter.     

An approximation of analytical functions by local splines

Local splines are presented for the approximation of functions of one and many variables, which are analytic in the domains , where U_i(z_i) is a unit disk in the complex plane C_i,i=1,2,…,l, l=1,2, …. Results are given for functions whose r-order derivatives belong to the Hardy's class H_p,1≤p≤∞. It is shown that the approximation converge to the function at the rate for functions of one variable and An^{−(r−1/p)/(l−1)} for functions of l variables, where n is the number of points of local splines and A and C are positive constants. This work was supported by Russian Foundation of Fundumental Inverstigations     

An approximation for the zeros of Bessel functions

Summary LetC _vk be thekth positive zero of the cylinder functionC _v(x)=cosJ _v(x)–sinY _v(x), whereJ _v(x),Y _v(x) are the Bessel functions of first kind and second kind, resp., andv>0, 0<. Definej _vk byj _vk=C _vk with . Using the notation 1/K=, we derive the first two terms of the asymptotic expansion ofj _vk in terms of the powers of at the expense of solving a transcendental equation. Numerical examples are given to show the accuracy of this approximation.Dedicated to the memory of Professor Lothar CollatzThis work has been supported by the Hungarian Scientific Grant No. 6032/6319     

An ultimate extremely accurate formula for approximation of the factorial function

We prove in this paper that for every x ≥ 0,

An alternative computational method for the approximation of random functions

Summary This paper shows that a computational procedure for approximation of random functions can be accomplished using purely linear programming techniques. This contrasts with previous results which use a twostage approach for the computation, one of which requires linear programming techniques. Computational results are given.     

An optimal adaptive algorithm for the approximation of concave functions

Motivated by the study of parametric convex programs, we consider approximation of concave functions by piecewise affine functions. Using dynamic programming, we derive a procedure for selecting the knots at which an oracle provides the function value and one supergradient. The procedure is adaptive in that the choice of a knot is dependent on the choice of the previous knots. It is also optimal in that the approximation error, in the integral sense, is minimized in the worst case. This work was partially supported by NSERC (Canada) and FCAR (Québec).     

One-sided approximation of functions

The best one-sided approximation of periodic functions by trigonometric polynomials of classW _p ⁰ (K) in the metric ofL is obtained. Translated fromMatematicheskie Zametki, Vol. 67, No. 1, pp. 136–140, January, 2000.     

Comonotone approximation of periodic functions

Suppose that a continuous 2π-periodic function f on the real axis ? changes its monotonicity at different ordered fixed points y _i ∈ [? π, π), i = 1, …, 2s, s ∈ ?. In other words, there is a set Y:= {y _i } _i∈? of points y _i = y _i+2s + 2π on ? such that, on [y _i , y _i?1], f is nondecreasing if i is odd and nonincreasing if i is even. For each n ≥ N(Y), we construct a trigonometric polynomial P _n of order ≤ n changing its monotonicity at the same points y _i ∈ Y as f and such that $$ \left\| {f - P_n } \right\| \leqslant c\left( s \right)\omega _2 \left( {f,\frac{\pi } {n}} \right), $$ where N(Y) is a constant depending only on Y, c(s) is a constant depending only on s, ω ₂(f, ·) is the modulus of continuity of second order of the function f, and ∥ · ∥ is the max-norm.     

Two-sided polynomial approximation of functions

Conditions are established when the collocation polynomials P_m(x) and P_M(x), m M, constructed respectively using the system of nodes x_j of multiplicities a_j 1, j = O,, n, and the system of nodes x_-r,,x_o,,x_n,,x_n+r1, r O, r₁ O, of multiplicities a_-r,,(a_o + y_o),,(a_n + y_n),,a_n+r1, a_j + y_j 1, are two sided-approximations of the function f on the intervals , x_j[, j = O,...,n + 1, and on unions of any number of these intervals. In this case, the polynomials P_m (x), P_M ^(l) (x) with l a_j are two-sided approximations of the function f⁽¹⁾ in the neighborhood of the node x_j and the integrals of the polynomials P_m(x), P_M(x) over D_j are two-sided approximations of the integral of the function f (over D_j). If the multiplicities a_j a_j + y_j of the nodes x_j are even, then this is also true for integrals over the set _j= _µ ^k D_j µ 1, k n. It is shown that noncollocation polynomials (Fourier polynomials, etc.) do not have these properties.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 67, pp. 31–37, 1989.     

Comonotone approximation of periodic functions

Suppose that a continuous 2π-periodic function f on the real axis ? changes its monotonicity at different ordered fixed points y _i ∈ [?π,π), i = 1, …, 2s, s ∈ ?. In other words, there is a set Y: = {y _i } _i∈? of points y _i = y _i+2s + 2π on ? such that f is nondecreasing on [y _i ,y _i?1] if i is odd and not increasing if i is even. For each n ≥ N(Y), we construct a trigonometric polynomial P _n of order ≤ n changing its monotonicity at the same points y _i ∈ Y as f and such that $$ \parallel f - P_n \parallel \leqslant c(s) \omega _2 \left( {f,\frac{\pi } {n}} \right), $$ where N(Y) is a constant depending only on Y, c(s) is a constant depending only on s, ω₂(f,·) is the modulus of continuity of second order of the function f, and ∥ · ∥ is the max-norm.     

Coconvex approximation of continuous functions

Suppose thatf(x)C[–1, 1],f(x) is convex in [0,1] and concave in [–1,0] with the new concept of regular convexity-turning points,we obtain the following estimates of Jackson type for coconvex approximation off(x) by algebraic polynomials of degree n:

Rational approximation of decay-type functions

Numerically given functions with exponential asymptotic behavior are approximated by rational expressions. The numerators are obtained from the nodes of the functions, and the denominators are powers of polynomials, whose coefficients are obtained by expansion in orthogonal polynomials. Different weights are discussed for uniform absolute or relative errors.An iterative procedure is further described to improve the least-squares approximation to one of Chebyshev-type best fit.The research reported in this paper was sponsored in part by the King Gustaf VI Adolf's 70-Years Fund for Swedish Culture, Knut and Alice Wallenberg's Foundation, The Swedish Natural Science Research Council, The Swedish Technical Research Council, and in part by the Aeronautical Research Laboratory, OAR, through the European Office, Aerospace Research, United States Air Force.Part of the work on this paper was carried out while the author was at the Physics Department, University of Florida, Gainesville, Florida, U.S.A.     

Holomorphic approximation of ultradifferentiable functions

Mathematische Annalen -     

Fractal approximation of vector functions

We present a new approach to the approximation of continuous vector-valued functions by fractal interpolative vector-valued ones and find optimal values of their parameters. We illustrate the obtained results with examples.     

Uniform approximation of continuous functions by rational functions

Summary Viene data la condizione necessaria e sufficiente perchè le funzioni razionali di una variabile, aventi poli di ordine prefissato in assegnati punti del piano complesso, costituiscano un sistema completo iu C^o (0, 1). A Bruno Finzi nel suo70 ^mo compleanno. This research has been sponsored in part by the Aerospace Research Laboratoires through the European Office of Aerospace Research, OAR, United States Air Force, under Grant EOOAR-69-0066. Entrata in Redazione il 22 aprile 1970.