Best trigonometric and bilinear approximations of functions of several variables from the classes Bp,θr· I

Approximations are studied of the classes B_p,^r of periodic functions of several variables with the help of trigonometric polynomials with a given number of harmonics. The results obtained are used to establish order estimates of the approximation of functions of the form f(x–y), f(x) B_p,^r, by combinations of products of functions of a fewer number of variables.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 11, pp. 1535–1547, November, 1992.     

Linear widths of Hölder-Nikol'skii classes of periodic functions of several variables

We consider the linear widths _N (W _p ^r (Tⁿ), L_q) and _N (H _p ^r (Tⁿ), L_q) of the classesW _p ^r (Tⁿ) andH _p ^r (Tⁿ) of periodic functions of one or several variables in the spaceL _q. For the Sobolev classesW _p ^r (Tⁿ) of functions of one or several variables, we state some well-known results without proof; for the Hölder-Nikol'skii classesH _p ^r (Tⁿ), we state some well-known results, prove some new results, and present some previously unpublished proofs.Translated fromMatematicheskie Zametki, Vol. 59, No. 2, pp. 189–199, February, 1996.This research was partially supported by the Russian Foundation for Basic Research under grant No. 93-01-00237 and by the International Science Foundation under grant No. MP1000.     

Estimates of the approximation characteristics of the classes B p,θ Ω of periodic functions of several variables with given majorant of mixed moduli of continuity

We obtain order-sharp estimates of the orthogonal projection widths of the classes B _p,θ ^Ω of periodic functions of several variables whose majorant of the mixed moduli of continuity contains both exponential and logarithmic multipliers.     

Best constants for approximations of periodic functions by Fejér operators

We obtain the best estimate for approximations of continuous 2-periodic functions by Fejér summation operators. The estimate is expressed in terms of the moduli of continuity of functions to be approximated. In the class of functions satisfying Lipschitz condition, we obtain the best constant for the approximation by Fejer integral and summation operators.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 1, pp. 75–82, 1990.     

The best trigonometric and bilinear approximations for functions of many variables from the classesB p, θ r . II

Order estimates are obtained for the best trigonometric and bilinear approximations of the classesB _p, ^r of functions of many variables in the metricL _q, wherep andq connected by certain relations.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 10, pp. 1411–1423, October, 1993.     

Conjugate functions of several variables in the class Lip α

It is known that if a function f of a single variable belongs to the class Lip (, C(T)) (0<<1), then its conjugate function also belongs to the same class; in other words, the class Lip (, C(T)) (0<<1) is invariant with respect to the operator of conjugation acting in it. In the two-dimensional case the class Lip (, C(T²)) (0<<1) is no longer invariant with respect to conjugate functions of two variables. Here a final result elucidating the full character of violation of invariance of the class Lip (, C(T^N)) (0<<1) with respect to the multidimensional conjugation operator acting in it is established.Translated from Matematicheskie Zametki, Vol. 23, No. 3, pp. 361–372, March, 1978.     

Best polynomial approximations in L 2 of classes of 2π-periodic functions and exact values of their widths

We find sharp inequalities between the best approximations of periodic differentiable functions by trigonometric polynomials and moduli of continuity ofmth order in the space L ₂ as well as present their applications. For some classes of functions defined by these moduli of continuity, we calculate the exact values of n-widths in L ₂.     

On Kolmogorov widths of classesB p,θ r of periodic functions of many variables with low smoothness in the spacel q

We study the Kolmogorov widths of Besov classesB _p, ^r of periodic functions of many variables with low smoothness in the spaceLq, 1相似文献   

On Taylor’s formula for functions of several variables

Elementary courses in mathematical analysis often mention some trick that is used to construct the remainder of Taylor’s formula in integral form. The trick is based on the fact that, differentiating the difference $f(x) - f(t) - f'(t)\frac{{(x - t)}} {{1!}} - \cdots - f^{(r - 1)} (t)\frac{{(x - t)^{r - 1} }} {{(r - 1)!}} $ between the function and its degree r ? 1 Taylor polynomial at t with respect to t, we obtain $ - f^{(r)} (t)\frac{{(x - t)^{r - 1} }} {{(r - 1)!}} $ , so that all derivatives of orders below r disappear. The author observed previously a similar effect for functions of several variables. Differentiating the difference between the function and its degree r ? 1 Taylor polynomial at t with respect to its components, we are left with terms involving only order r derivatives. We apply this fact here to estimate the remainder of Taylor’s formula for functions of several variables along a rectifiable curve.     

Bestm-term trigonometric approximations of classes of (Ψ, β)-differentiable functions of one variable

We obtain an estimate exact in order for the best trigonometric approximation of classes of functions of one variable in the spaceL _q in the case where 1<p≤2≤q<∞. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 52, No. 6, pp. 850–855, June, 2000.     

Widths of classes of periodic differentiable functions in the space L 2[0, 2π]

We obtain exact values of different n-widths for classes of differentiable periodic functions in the space L ₂[0, 2π] satisfying the constraint $$ \left( {\int_0^h {\omega _m^p \left( {f^{\left( r \right)} ;t} \right)dt} } \right)^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} \leqslant \Phi \left( h \right) $$ , where 0 < h < ∞, 1/r < p ≤ 2, r ∈ ?, and ω _m (f ^(r); t) is the modulus of continuity of mth order of the derivative f ^(r)(x) ∈ L ₂[0, 2π].     

Approximation by Riesz sums of periodic functions of Hölder classes

We have found asymptotic equalities for the least upper bounds of the deviations of Riesz sums on the Hölder classes W^rH, r is a nonnegative integer, (t) is an arbitrary convex modulus of continuity.Translated from Matematicheskie Zametki, Vol. 21, No. 3, pp. 341–354, March, 1977.     

Generalizations of Young’s Theorem to real functions of several variables

In 1907 W. H. Young classified the real-valued Baire one functions on the line which have the Darboux (intermediate-value) property as those which are bilaterally approachable. Here we investigate generalizations of this theorem to the setting of real-valued Baire one functions of several variables which possess various “Darboux-like” properties.      

Sharp estimates of the norms of fractional derivatives of functions of several variables satisfying Hölder conditions

We prove a new sharp Kolmogorov-type inequality that estimates the uniform norm of a mixed derivative of fractional order (in the sense of Marchaud) of a function of several variables via the uniform norm of the function and its norm on Hölder spaces.     

On the bestm-term trigonometric and orthogonal trigonometric approximations of functions from the classesL Ψ β,ρ

We obtain estimates exact in order for the best trigonometric and orthogonal trigonometric approximations of the classesL _Ψ^β,ρ of functions of one variable in the spaceL _q in the case 2<p <q < ∞.