Trigonometric Polynomials Of Order Research Articles

Modified Bernstein Function and a Uniform Approximation of Some Rational Fractions by Polynomials

P. L. Chebyshev posed and solved (1857, 1859) the problem of finding an improper rational fraction least deviating from zero in the uniform metric on a closed interval among rational fractions whose denominator is a fixed polynomial of a given degree m that is positive on the interval and whose numerator is a polynomial of a given degree n ≥ m with unit leading coefficient. In 1884, A.A.Markov solved a similar problem in the case when the denominator is the square root of a given positive polynomial. In the 20th century, this research direction was developed by S.N.Bernstein, N. I.Akhiezer, and other mathematicians. For example, in 1964, G. Szegő extended Chebyshev’s result to the case of trigonometric fractions using the methods of complex analysis. In this paper, using the methods of real analysis and developing Bernstein’s approach, we find the best uniform approximation on the period by trigonometric polynomials of a certain order for an infinite series of proper trigonometric fractions of a special form. It turned out that, in the periodic case, it is natural to formulate some results in terms of the generalized Poisson kernel Πρ,ξ(t) = (cosξ)Pρ(t) + (sinξ)Qρ(t), which is a linear combination of the Poisson kernel Pρ(t) = (1 − ρ2)/[2(1 + ρ2 − 2ρ cos t)] and the conjugate Poisson kernel Qρ(t) = (ρ sin t)/(1 + ρ2 − 2ρ cos t), where ρ ∈ (−1, 1) and ξ ∈ R. We find the best uniform approximation on the period by the subspace Tn of trigonometric polynomials of order at most n for the linear combination Πρ,ξ(t) + (−1)nΠρ,ξ(t + π) of the generalized Poisson kernel and its translation. For ξ = 0, this yields Bernstein’s known results on the best uniform approximation on [−1, 1] of the fractions 1/(x2 − a2) and x/(x2 − a2), |a| > 1, by algebraic polynomials. For ξ = π/2, we obtain the weight analogs (with the weight $$\sqrt {1 - {x^2}} $$ x2) of these results. In addition, we find the value of the best uniform approximation on the period by the subspace Tn of a special linear combination of the mentioned Poisson kernel Pρ and the Poisson kernel Kρ for the biharmonic equation in the unit disk.

Bernstein’s Inequality for the Weyl Derivatives of Trigonometric Polynomials in the Space L0

A logarithmic asymptotics for the behavior with respect to n of the exact constant in Bernstein’s inequality for the Weyl derivative of positive noninteger order of trigonometric polynomials of order n in the space L0 is obtained. It turns out that the order in n of the behavior of this constant for positive noninteger orders of the derivatives has exponential growth in contrast to the power growth in the well-studied case of classical derivatives of positive integer order.

Approximation of functions in variable-exponent Lebesgue and Sobolev spaces by de la Vallée-Poussin means

We consider the space formed by -periodic real measurable functions for which the integral exists and is finite, where , , is a -periodic measurable function (a variable exponent). If , then the space can be endowed with the structure of Banach space with the norm 0: \\int_{-\\pi}^{\\pi}\\biggl|\\frac{f(x)}{\\alpha}\\biggr|^{p(x)}\\,dx\\le1\\biggr\\}. \\end{equation*} ?> In the space we distinguish a subspace of Sobolev type. We investigate the approximation properties of the de la Vallée-Poussin means for trigonometric Fourier sums for functions in the space . In particular, we prove that if the variable exponent satisfies the Dini-Lipschitz condition and if , then the de la Vallée-Poussin means with satisfy where is a modulus of continuity of the function defined in terms of the Steklov functions. It is proved that if , , and the Dini-Lipschitz condition holds, then where stands for the best approximation to by trigonometric polynomials of order . Bibliography: 19 titles.

Approximating smooth functions using algebraic-trigonometric polynomials

The problem under consideration is that of approximating classes of smooth functions by algebraic-trigonometric polynomials of the form , where is an algebraic polynomial of degree and is a trigonometric polynomial of order . The precise order of approximation by such polynomials in the classes and an upper bound for similar approximations in the class with are found. The proof of these estimates uses mixed series in Legendre polynomials which the author has introduced and investigated previously.Bibliography: 13 titles.

A sharp constant in the Jackson-Stechkin inequality in the space L 2 on the period

In the space L2 of real-valued measurable 2π-periodic functions that are square summable on the period [0, 2π], the Jackson-Stechkin inequality $$ E_n (f) \leqslant \mathcal{K}_n (\delta ,\omega )\omega (\delta ,f), f \in L^2 $$ , is considered, where En(f) is the value of the best approximation of the function f by trigonometric polynomials of order at most n and ω(δ, f) is the modulus of continuity of the function f in L2 of order 1 or 2. The value $$ \mathcal{K}_n (\delta ,\omega ) = \sup \left\{ {\frac{{E_n (f)}} {{\omega (\delta ,f)}}:f \in L^2 } \right\} $$ is found at the points δ = 2π/m (where m ∈ ℕ) for m ≥ 3n2 + 2 and ω = ω1 as well as for m ≥ 11n4/3 − 1 and ω = ω2.

Integral approximation of the characteristic function of an interval and the Jackson inequality in C( % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb % L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe % pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam % aaeaqbaaGcbaWefv3ySLgznfgDOjdarCqr1ngBPrginfgDObcv39ga % iyaacqWFtcpvaaa!46CC! $$ \mathbb{T} $$ )

We apply the results on the integral approximation of the characteristic function of an interval by the subspace \( \mathcal{T}_{n - 1} \) of trigonometric polynomials of order at most n − 1, which were obtained by the authors earlier, to investigate the Jackson inequality between the best uniform approximation of a continuous periodic function by the subspace \( \mathcal{T}_{n - 1} \) and its modulus of continuity of the second order. The corresponding method of the uniform approximation of continuous periodic functions by trigonometric polynomials is constructed.

On the trigonometric interpolation and the entire interpolation

In this paper, we study a kind of interpolation problems on a given nodal set by trigonometric polynomials of order n and entire functions of exponential type according as the nodal set is $$\left\{ {\frac{{2k\pi }}{n}} \right\}_{k = 0}^{n - 1} or \left\{ {\frac{{2k\pi }}{\sigma }} \right\}_{k = - \infty }^{ + \infty } $$ respectively. We established some equivalent conditions and found the explicit forms of some interpolation functions on the interpolation problems. As a special case, the explicit forms of fundamential functions of (0,m)-interpolat on by trigonometric case or entire functions case (in B2 σ) respectively, if they exist, may follow from our results. Besides, we also considered the convergence of the interpolation functions at above stated.

ON THE DEPENDENCE OF THE PROPERTIES OF THE SET OF POINTS OF DISCONTINUITY OF A FUNCTION ON THE DEGREE OF ITS POLYNOMIAL HAUSDORFF APPROXIMATIONS

Let , where is the smallest deviation of a -periodic function from trigonometric polynomials of order in the Hausdorff -metric. It is shown that for arbitrary there exists a function such that and the set of points of discontinuity of has Hausdorff dimension 1. The concept of the -equiporosity coefficient of a set is introduced, and a best possible lower estimate is obtained for the -equiporosity coefficient of the set of points of discontinuity of a function in terms of the quantity , : Dolzhenko, Sevast'yanov, Petrushev, and Tashev proved earlier that the condition implies that is continuous almost everywhere, and implies continuity at all points. Petrushev and Tashev constructed an example of a discontinuous function for which , but, in contrast to the example mentioned above, had only one point of discontinuity on a period.Bibliography: 11 titles.

ON THE DEPENDENCE OF PROPERTIES OF FUNCTIONS ON THEIR DEGREE OF APPROXIMATION BY POLYNOMIALS

Let be a bounded -periodic function with modulus of continuity ; let and be the minimum deviations of from the trigonometric polynomials of order , in the uniform metric and the Hausdorff metric of order , respectively; letThen If as , then if the function is continuous almost everywhere; if it is continuous everywhere, and if we have , 0$ SRC=http://ej.iop.org/images/0025-5726/12/2/A05/tex_im_1853_img18.gif/>.Approximation by algebraic polynomials is also considered, and some corollaries are given.Bibliography: 13 titles.

Localizing the Lozinski-Harshiladze theorem on projections into the space of trigonometric polynomials

w 1. We think of periodic continuous functions as functions of the complex variable z = e ~x on the unit circle with Fourier series f(z)~ ~ akz k k ~ ~ and norm ]]f]l = Max ]f(z)l. By a projection T. we mean a linear operator that makes correspond to every such function a trigonometric polynomial of order at most n (i.e. an f with vanishing Fourier coefficients for [k[>n) but leaves each trigonometric polynomial of order =<n invariant, f-~ ~ a k z k is such an operator and the theorem referred to in k= --n

Best-possible one-sided approximations in the classes 688-01688-01688-01by trigonometric polynomials in the metric of L1

The quantities Open image in new window are calculated, where Open image in new window is the best approximation from above of the function f by trigonometric polynomials of order ⩽n—1 in the metric of L1.

The correctness problem for best approximations by trigonometric polynomials in the class W o r H[?

Suppose thatk, rez+, W o r H[ω]C= {f∶f is a 2π-periodic function,fe Cr [−π, π], ω (f(r), δ) ⩽ω (δ)}, Tk is the space of trigonometric polynomials of order k, pk(f)eTk is the polynomial of best uniform approximation to f, and Ek(f) is the error of the best approximation. It is shown that for an arbitrary e > 0 we have , where for 0 0.R (e) is the root of the equation $$R = (\varepsilon ^l R)^{r^l (2k)} \omega ((\varepsilon \cdot R)^{1(2k)} )$$ , and for k = 0 or e > ω(1) we have R(e)=e.

The inclusion of certain classes of functions

For any sequence {Nk} with {Nk} ↓ O we find sharp theorems on the inclusion of the classes {f: f ∈ l (0, 2π),e k (1) (f) = O(Nk)¦, whereE k (1) (f) is the best approximation (in L) of f by trigonometric polynomials of order no greater than k, in the classL ϕ with slowly growing ϕ and in the class Lv, 1 <v < ∞.

A note on a theorem of Alexits and Králik

where S , ( f ) is the n-th partial sum of (1). It is known (cf. ZYGMtYNO [4] Vol. I. p. 115) that [V.(f)-f l<=4e.(f) where E, ( f ) represents the best approximation of f(x) by trigonometric polynomials of order n. We denote by Z the class of all continuous 2n periodic functions satisfying the inequality sup [f(x +t ) -2 f (x )+f (xt ) t<= <-Kit[ for all x and t where K is an absolute positive constant. Earlier, G. AL~XITS and D. KgALIK [2] gave the necessary and sufficient condition for r-th derivative f(r)(x) of a function f(x) to belong to the class Z in terms of the de la ValiSe Poussin sum in strong sense: 1 2n--1 ~ I S k ( f ) f [ = O ( n , 1 ) . n k=n

Local properties of functions and approximation by trigonometric polynomials

Suppose Φp, E (p>0 an integer, E ⊂[0, 2π]) is a family of positive nondecreasing functionsϕx(t) (t>0, x∈ E) such thatϕx(nt)≤nPϕx(t) (n=0,1,...), tn is a trigonometric polynomial of order at most n, and Δhl(f, x) (l>0 an integer) is the finite difference of orderl with step h of the functionf.THEOREM. Supposef (x) is a function which is measurable, finite almost everywhere on [0, 2π], and integrable in some neighborhood of each point xe E,ϕXeΦp,E and $$\overline {\mathop {\lim }\limits_{\delta \to \infty } } |(2\delta )^{ - 1} \smallint _{ - \delta }^\delta \Delta _u^l (f,x)du|\varphi _x^{ - 1} (\delta ) \leqslant C(x)< \infty (x \in E).$$ . Then there exists a sequence {tn}n=1∞ which converges tof (x) almost everywhere, such that for x e E $$\overline {\mathop {\lim }\limits_{n \to \infty } } |f(x) - l_n (x)|\varphi _x^{ - 1} (l/n) \leqslant AC(x),$$ where A depends on p andl.

Lebesgue's inequality in a uniform metric and on a set of full measure

Letf be a continuous periodic function with Fourier sums Sn(f), and let En(f)=En be the best approximation tof by trigonometric polynomials of order n. The following estimate is proved: $$||f - S_n (f)|| \leqslant c\sum\nolimits_{v = n}^{2u} {\frac{{E_v }}{{v - n + 1}}} .$$ (Here c is an absolute constant.) This estimate sharpens Lebesgue's classical inequality for “fast” decreasing E v . The sharpness of this estimate is proved for an arbitrary class of functions having a given majorant of best approximations. Also investigated is the sharpness of the corresponding estimate for the rate of convergence of a Fourier series almost everywhere.

The best one-sided approximation of one class of functions by another

We concern ourselves with problems of the best one-sided approximation of classes of continuous functions. We obtain estimates of the best one-sided approximation of one class of functions by another, and we find exact values of the upper bounds of the best one-sided approximations on the classes H Ω of 2π-periodic functions [given by an arbitrary convex modulus of continuityΩ(t)] by trigonometric polynomials of order not higher thann−1 in the L2π metric.

Exact constants of approximation for differentiable periodic functions

For all odd r we construct a linear operator Br,r(f) which maps the set of 2π-periodic functionsf(t)e X(r) (X(r)=C(r) or L1(r)) into a set of trigonometric polynomials of order not higher than n-1 such that where X is the C or L1 metric, En(f)X and ω(f, δ)X are the best approximation by means of trigonometric polynomials of order not higher than n-1 and the modulus of continuity of the functionf in the X metric, respectively; Kr are the known Favard constants.

A note on interpolation

where Dn(u)= 2-1's eiYu is the Dirichlet kernel and w02n+1(t) is a step function constant in the interior of the intervals [xo+kh, xo+(k+l)h] and having a jump h at the points xo+kh.' Thus W2n+1(t) is determined up to an arbitrary additive constant. It is well known that In(x; f) need not tend tof(x) even if f is everywhere continuous. S. Bernstein2 pointed out that the situation is considerably improved if we consider the trigonometric polynomial of order n assuming at the points Xk the values