Approximating periodic functions by interpolation sums of Jackson type

Let

Approximation of periodic functions by trigonometric polynomials in the mean

For arbitrary summation methods we obtain inequalities between upper bounds of deviations in the L metric and corresponding upper bounds in the C metric with respect to a certain class of functions. These inequalities constitute a generalization of known relationships due to S. M. Nikol'skii. We consider the cases wherein these inequalities become exact or asymptotic equalities.Translated from Matematicheskie Zametki, Vol. 16, No. 1, pp. 15–26, July, 1974     

Best trigonometric approximations for some classes of periodic functions of several variables in the uniform metric

We study best M-term trigonometric approximations and best orthogonal trigonometric approximations for the classes B _pθ ^r and W _pα ^r of periodic functions of several variables in the uniform metric.     

Sharp approximations to the Bernoulli periodic functions by trigonometric polynomials

We obtain optimal trigonometric polynomials of a given degree N that majorize, minorize and approximate in the Bernoulli periodic functions. These are the periodic analogues of two works of Littmann [F. Littmann, Entire majorants via Euler–Maclaurin summation, Trans. Amer. Math. Soc. 358 (7) (2006) 2821–2836; F. Littmann, Entire approximations to the truncated powers, Constr. Approx. 22 (2) (2005) 273–295] that generalize a paper of Vaaler [J.D. Vaaler, Some extremal functions in Fourier analysis, Bull. Amer. Math. Soc. 12 (1985) 183–215]. As applications we provide the corresponding Erdös–Turán-type inequalities, approximations to other periodic functions and bounds for certain Hermitian forms.     

Approximation of periodic functions by interpolation polynomials in L1

Asymptotically precise estimates are obtained for the deviation, in the L₁-norm, of interpolation polynomials with equally-spaced nodes from certain classes of functions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 6, pp. 781–786, June, 1990.     

On the nonclassical approximation method for periodic functions by trigonometric polynomials

We study the approximation of functions by linear polynomial means of their Fourier series with a function-multiplier φ that is equal to 1 not only at zero, in contrast with classical methods of summability. The exact order of convergence to zero of the sequence $$ \mathop{\max}\limits_{{x\in \left[ {-\pi, \pi } \right]}}\left| {f(x)-\sum\limits_{{\left| k \right|\leq n}} {\varphi \left( {\frac{{k\pi }}{n}} \right){{\hat{f}}_k}{e^{ikx }}} } \right| $$ ( $ {{\hat{f}}_k} $ Fourier coefficients) as n→∞ is obtained. The answer is given in terms of the values of difference operators of a continuous function f and a special K-functional (step of $ \frac{\pi }{n} $ ). In addition, we obtain not only the sufficient conditions for φ but the necessary ones as well.     

Approximating probability density functions and their convolutions using orthogonal polynomials

This paper describes and tests methods for piecewise polynomial approximation of probability density functions using orthogonal polynomials. Empirical tests indicate that the procedure described in this paper can provide very accurate estimates of probabilities and means when the probability density function cannot be integrated in closed form. Furthermore, the procedure lends itself to approximating convolutions of probability densities. Such approximations are useful in project management, inventory modeling, and reliability calculations, to name a few applications. In these applications, decision makers desire an approximation method that is robust rather than customized. Also, for these applications the most appropriate criterion for accuracy is the average percent error over the support of the density function as opposed to the conventional average absolute error or average squared error. In this paper, we develop methods for using five well-known orthogonal polynomials for approximating density functions and recommend one of them as giving the best performance overall.     

Best approximation of periodic functions by trigonometric polynomials in L2

Estimates are gotten for the best approximations in L₂ (0, 2) of a periodic function by trigonometric polynomials in terms of its m-th continuity modulus or in terms of the continuity modulus of its r-th derivative. The inequality E_n–1(f)L₂<(C _2m ^m _m(2/n;f)L₂(1 const) is proved, where the constant (C _2m ^m )^–1/2 is unimprovable for the whole space L₂ (0, 2). Two titles are cited in the bibliography.Translated from Matematicheskie Zametki, Vol. 2, No. 5, pp. 513–522, November, 1967.     

Best approximations of the classes B p,θ r of periodic functions of many variables in uniform metric

We obtain estimates exact in order for the best approximations of the classes B _∞,θ ^r of periodic functions of two variables in the metric of L _∞ by trigonometric polynomials whose spectrum belongs to a hyperbolic cross. We also investigate the best approximations of the classes B _p,θ ^r , 1 ≤ p < ∞, of periodic functions of many variables in the metric of L _∞ by trigonometric polynomials whose spectrum belongs to a graded hyperbolic cross. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 10, pp. 1395–1406, October, 2006.     

Approximation of infinitely differentiable periodic functions by interpolation trigonometric polynomials

We establish asymptotically unimprovable interpolation analogs of Lebesgue-type inequalities on the classes of periodic infinitely differentiable functions C C whose elements can be represented in the form of convolutions with fixed generating kernels. We obtain asymptotic equalities for upper bounds of approximations by interpolation trigonometric polynomials on the classes C _, and C H .Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 4, pp. 495–505, April, 2004.     

On uniform approximations of almost periodic functions by entire functions of finite degree

We give a new proof of the well-known Bernshtein statement that, among entire functions of degree which realize the best uniform approximation (of degree ) of a periodic function on (–,), there is a trigonometric polynomial of degree . We prove an analog of the mentioned Bernshtein statement and the Jackson theorem for uniform almost periodic functions with arbitrary spectrum.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 9, pp. 1274–1279, September, 1995.