On the approximation of square-integrable functions by exponential series

The expansion of a real square-integrable function in a Legendre series is considered. Existence of best approximations from different sets of exponential functions and their mean convergence to the function in question are proved. As an extension of this result existence and mean convergence of some non-linear best approximations that have been developed by Longman are also demonstrated.     

A property of a system of functions close to exponential functions

We consider the system {f _n=xⁿ[l+_n]} in the interval [a,b] (0 a n > 0 and _n(x) such as the condition, we obtain a bound for the coefficients of the polynomial P(x)=#x2211;c_n f _n(x) in terms of P(x)L_p[a,b]. It is found that this bound is not valid without this condition (assuming the other conditions to remain the same).Translated from Matematicheskie Zametki, Vol. 12, No. 1, pp. 29–36, July, 1972.     

An approximation property of lacunary sequences

Let f ₁ and f ₂ be two positive numbers of the field , and let f _n+2 = f _n+1 + f _n for each n ≥ 1. Let us denote by {x} the fractional part of a real number x. We prove that, for each ξ ∉ K, the inequality {ξf _n } > 2/3 holds for infinitely many positive integers n. On the other hand, we prove a result which implies that there is a transcendental number ξ such that {ξf _n } < 39/40 for each n ≥ 1. Moreover, it is shown that, for every a > 1, there is an interval of positive numbers that contains uncountably many numbers ξ such that {a ⁿ } 6 min 2/(a − 1), (34a ² − 32a + 7)/(9(2a − 1)²) for each n > 1. Here, the minimum is strictly smaller than 1 for each a > 1. In contrast, by an old result of Weyl, for any a > 1, the sequence {ξa ⁿ }, n = 1, 2, ..., is uniformly distributed in [0, 1] (and so everywhere dense in [0, 1]) for almost all real numbers ξ.     

An approximation property of harmonic functions in Lipschitz domains and some of its consequences

An extension of an inequality of J. B. Garnett (1979), with improvements by B. E. J. Dahlberg (1980), on an approximation property of harmonic functions is proved. The weighted inequality proved here was suggested by the work of J. Pipher (1993) and it implies an extension of a result of S. Y. A. Chang, J. Wilson and T. Wolff (1985) and C. Sweezy (1991) on exponential square integrability of the boundary values of solutions to second-order linear differential equations in divergence form. This implies a solution of a problem left open by R. Bañuelos and C. N. Moore (1989) on sharp estimates for the area integral of harmonic functions in Lipschitz domains.

     

Form-preserving exponential approximation

In this paper we continue our research in constructing linear methods for approximating functions by parabolic splines with equidistant knots, inheriting such properties of approximated functions as monotony and convexity. The obtained results are generalized to the case of nonequidistant knots and to some kinds of exponential splines of the third order.     

An accurate approximation of zeta-generalized-Euler-constant functions

Zeta-generalized-Euler-constant functions,

Function approximation via particular input space partition and region-based exponential membership functions

This paper presents an algorithm to build a compact fuzzy model for approximating unknown nonlinear functions. According to the coordinates of input–output pairs in the Cartesian product-space, the proposed algorithm partitions the input space into several characteristic regions, whose boundaries are determined by the local-minimum, local-maximum, and turning points of the training data. The region-based exponential functions are chosen as membership functions of antecedents, with consequents of rules as singletons, to construct the fuzzy model. Finally, five examples demonstrate that the proposed fuzzy model indeed improves performance in the approximating of nonlinear functions.     

An exponential approximation for solutions of generalized pantograph-delay differential equations

In this paper, a new matrix method based on exponential polynomials and collocation points is proposed for solutions of pantograph equations with linear functional arguments under the mixed conditions. Also, an error analysis technique based on residual function is developed for the suggested method. Some examples are given to demonstrate the validity and applicability of the method and the comparisons are made with existing results.     

A note on uniform exponential convergence of sample average approximation of random functions

Shapiro and Xu (2008) [17] investigated uniform large deviation of a class of Hölder continuous random functions. It is shown under some standard moment conditions that with probability approaching one at exponential rate with the increase of sample size, the sample average approximation of the random function converges to its expected value uniformly over a compact set. This note extends the result to a class of discontinuous functions whose expected values are continuous and the Hölder continuity may be violated for some negligible random realizations. The extension entails the application of the exponential convergence result to a substantially larger class of practically interesting functions in stochastic optimization.     

The property (DN) and the exponential representation of holomorphic functions

It is shown that a nuclear Fréchet spaceE has the property (DN) if and only if every holomorphic function onE ^*, the strongly dual space ofE, with values in the strongly dual space of a Fréchet spaceF having the property ( ) can be represented in the exponential form. Moreover, it is shown that the space of holomorphic functions onC , the space of all complex number sequences, has a linearly absolutely exponential representation system. But the space of holomorphic functions onE ^* does not have such a system whenE is a nuclear Fréchet space that does not have the property (DN).Supported by the State Program for Fundamental Researches in Natural Sciences     

Weighted exponential polynomial approximation

A necessary and sufficient condition for completeness of systems of exponentials with a weight in Lp is established and a quantitative relation between the weight and the system of exponential in Lp is obtained by using a generalization of Malliavin's uniqueness theorem about Watson's problem.     

An extremal property of outer functions

Our main result is the following: iff (z) is in the space H², and F(z) is its outer part, then F⁽ⁿ⁾_H2F⁽ⁿ⁾_H2(n=1,2,...), the left side being finite if the right side is finite. Under certain essential restrictions, this. inequality was proved by B. I. Korenblyum and V. S. Korolevich [1].Translated from Matematicheskie Zametki, Vol. 10, No. 1, pp. 53–56, July, 1971.     

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.     

Weighted approximation of entire functions of exponential type of several variables by trigonometric polynomials

Letf be an entire function (in Cⁿ) of exponential type for whichf(x)=0(?(x)) on the real subspace \(\mathbb{R}^w (\phi \geqslant 1,{\mathbf{ }}\mathop {\lim }\limits_{\left| x \right| \to \infty } \phi (x) = \infty )\) and ?δ>0?C_δ>0 $$\left| {f(z)} \right| \leqslant C_\delta \exp \left\{ {h_s (y) + S\left| z \right|} \right\},z = x + iy$$ where h, (x)=sup〈3, x〉, S being a convex set in ?ⁿ. Then for any ?, ?>0, the functionf can be approximated with any degree of accuracy in the form p→ \(\mathop {\sup }\limits_{x \in \mathbb{R}^w } \frac{{\left| {P(x)} \right|}}{{\varphi (x)}}\) by linear combinations of functions x→expi〈λx〉 with frequenciesX belonging to an ?-neighborhood of the set S.     

Chebyshev approximation of the null function by an affine combination of complex exponential functions

We describe the theoretical solution of an approximation problem that uses a finite weighted sum of complex exponential functions. The problem arises in an optimization model for the design of a telescope array occurring within optical interferometry for direct imaging in astronomy. The problem is to find the optimal weights and the optimal positions of a regularly spaced array of aligned telescopes, so that the resulting interference function approximates the zero function on a given interval. The solution is given by means of Chebyshev polynomials.