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1.
Let f: be a continuous, 2π-periodic function and for each n ε let tn(f; ·) denote the trigonometric polynomial of degree n interpolating f in the points 2kπ/(2n + 1) (k = 0, ±1, …, ±n). It was shown by J. Marcinkiewicz that limn → ∞0¦f(θ) − tn(f θ)¦p dθ = 0 for every p > 0. We consider Lagrange interpolation of non-periodic functions by entire functions of exponential type τ > 0 in the points kπ/τ (k = 0, ± 1, ± 2, …) and obtain a result analogous to that of Marcinkiewicz.  相似文献   

2.
In this paper, we determine the exact value of average n − K width n(Wrpq(R), Lq(R)) of Sobolev-Wiener class Wrpq(R) in the metric Lq(R) for 1 > qp > ∞ and get the value of n(Wrp(R), Lqp(R)) for the dual case. We also solve the optimal interpolation problems of Wrpq(R) in the metric Lq(R) and Wrp(R) in the metric Lqp(R) for 1 < qp < ∞.  相似文献   

3.
A remarkable theorem proved by Komlòs [4] states that if {fn} is a bounded sequence in L1(R), then there exists a subsequence {fnk} and f L1(R) such that fnk (as well as any further subsequence) converges Cesaro to f almost everywhere. A similar theorem due to Révész [6] states that if {fn} is a bounded sequence in L2(R), then there is a subsequence {fnk} and f L2(R) such that Σk=1 ak(fnkf) converges a.e. whenever Σk=1 | ak |2 < ∞. In this paper, we generalize these two theorems to functions with values in a Hilbert space (Theorems 3.1 and 3.3).  相似文献   

4.
We consider a strictly convex domain D n and m holomorphic functions, φ1,…, φm, in a domain . We set V = {z ε Ω: φ1(z) = ··· = φm(z) = 0}, M = VD and ∂M = V ∩ ∂D. Under the assumptions that the variety V has no singular point on ∂M and that V meets ∂D transversally we construct an explicit kernel K(ζ, z) defined for ζ ε ∂M and z ε D so that the integral operator Ef(z) = ∝ ζ ε ∂M f(ζ) K(ζ, z) (z ε D), defined for f ε H(M) (using the boundary values f(ζ) for a.e. ζ ε ∂M), is an extension operator, i.e., Ef(z) = f(z) for z ε M and furthermore E is a bounded operator from H to H(D).  相似文献   

5.
Oscillations of first-order neutral delay differential equations   总被引:1,自引:0,他引:1  
Consider the neutral delay differential equation (*) (d/dt)[y(t) + py(t − τ)] + qy(t − σ) = 0, t t0, where τ, q, and σ are positive constants, while p ε (−∞, −1) (0, + ∞). (For the case p ε [−1, 0] see Ladas and Sficas, Oscillations of neutral delay differential equations (to appear)). The following results are then proved. Theorem 1. Assume p < − 1. Then every nonoscillatory solution y(t) of Eq. (*) tends to ± ∞ as t → ∞. Theorem 2. Assume p < − 1, τ > σ, and q(σ − τ)/(1 + p) > (1/e). Then every solution of Eq. (*) oscillates. Theorems 3. Assume p > 0. Then every nonoscillatory solution y(t) of Eq. (*) tends to zero as t → ∞. Theorem 4. Assume p > 0. Then a necessary condition for all solutions of Eq. (*) to oscillate is that σ > τ. Theorem 5. Assume p > 0, σ > τ, andq(σ − τ)/(1 + p) > (1/e). Then every solution of Eq. (*) oscillates. Extensions of these results to equations with variable coefficients are also obtained.  相似文献   

6.
Let T be an ergodic automorphism of a probability space, f a bounded measurable function, . It is shown that the property that the probabilities μ(|Sn(f)|>n) are of order np roughly corresponds to the existence of an approximation in L of f by functions (coboundaries) ggT, gLp. Similarly, the probabilities μ(|Sn(f)|>n) are exponentially small iff f can be approximated by coboundaries ggT where g have finite exponential moments.

Résumé

Soit T un automorphisme ergodique d'un espace probabilisé, f une fonction bornée mesurable et . Une correspondance est établie entre l'existence de l'estimation des probabilités μ(|Sn(f)|>n) d'ordre np et l'existence de l'approximation dans L de la fonction f par des cobords ggTg est “presque” dans Lp. De manière similaire, les probabilités μ(|Sn(f)|>n) sont d'ordre ecn, pour un certain c>0, n=1,2… , si et seulement si f admet une approximation dans L par des cobords ggT avec g ayant des moments exponentiels.  相似文献   

7.
Let T = {T(t)}t ≥ 0 be a C0-semigroup on a Banach space X. In this paper, we study the relations between the abscissa ωLp(T) of weak p-integrability of T (1 ≤ p < ∞), the abscissa ωpR(A) of p-boundedness of the resolvent of the generator A of T (1 ≤ p ≤ ∞), and the growth bounds ωβ(T), β ≥ 0, of T. Our main results are as follows.
1. (i) Let T be a C0-semigroup on a B-convex Banach space such that the resolvent of its generator is uniformly bounded in the right half plane. Then ω1 − ε(T) < 0 for some ε > 0.
2. (ii) Let T be a C0-semigroup on Lp such that the resolvent of the generator is uniformly bounded in the right half plane. Then ωβ(T) < 0 for all β>¦1/p − 1/p′¦, 1/p + 1/p′ = 1.
3. (iii) Let 1 ≤ p ≤ 2 and let T be a weakly Lp-stable C0-semigroup on a Banach space X. Then for all β>1/p we have ωβ(T) ≤ 0.
Further, we give sufficient conditions in terms of ωqR(A) for the existence of Lp-solutions and W1,p-solutions (1 ≤ p ≤ ∞) of the abstract Cauchy problem for a general class of operators A on X.  相似文献   

8.
Let f ε Cn+1[−1, 1] and let H[f](x) be the nth degree weighted least squares polynomial approximation to f with respect to the orthonormal polynomials qk associated with a distribution dα on [−1, 1]. It is shown that if qn+1/qn max(qn+1(1)/qn(1), −qn+1(−1)/qn(−1)), then fH[f] fn + 1 · qn+1/qn + 1(n + 1), where · denotes the supremum norm. Furthermore, it is shown that in the case of Jacobi polynomials with distribution (1 − t)α (1 + t)β dt, α, β > −1, the condition on qn+1/qn is satisfied when either max(α,β) −1/2 or −1 < α = β < −1/2.  相似文献   

9.
Let d−1{(x1,…,xd) d:x21+···+x2d=1} be the unit sphere of the d-dimensional Euclidean space d. For r>0, we denote by Brp (1p∞) the class of functions f on d−1 representable in the formwhere (y) denotes the usual Lebesgue measure on d−1, and Pλk(t) is the ultraspherical polynomial.For 1p,q∞, the Kolmogorov N-width of Brp in Lq( d−1) is given bythe left-most infimum being taken over all N-dimensional subspaces XN of Lq( d−1).The main result in this paper is that for r2(d−1)2,where ANBN means that there exists a positive constant C, independent of N, such that C−1ANBNCAN.This extends the well-known Kashin theorem on the asymptotic order of the Kolmogorov widths of the Sobolev class of the periodic functions.  相似文献   

10.
Es werden nichtlineare Differentialgleichungen der Form (∂u/∂t) + Au = f(u) in n × [0, ∞) betrachtet. Dabei ist A ein positiv definiter elliptischer Differentialoperator 2m-ter Ordnung und f eine nichtlineare glatte Funktion, die nicht schneller als ¦ u ¦q wächst, wobei q < 1 + [4m/(n − 2m)] für n > 2m und q < ∞ für n 2m, und deren Ableitungen polynomials Wachstum besitzen. Auβerdem besitze f eine nichtpositive Stammfunktion. Unter diesen Voraussetzungen wird durch Betrachtung der zugehörigen Integralgleichung in Lv n und unter Benutzung der Sobolevräume gebrochener Ordnung die globale klassische Lösbarkeit des Cauchyproblems für glatte Anfangswerte gezeigt.  相似文献   

11.
It is shown that for each convex bodyARnthere exists a naturally defined family AC(Sn−1) such that for everyg A, and every convex functionf: RRthe mappingySn−1 f(g(x)−yx) (x) has a minimizer which belongs toA. As an application, approximation of convex bodies by balls with respect toLpmetrics is discussed.  相似文献   

12.
For a functionfLp[−1, 1], 0<p<∞, with finitely many sign changes, we construct a sequence of polynomialsPnΠnwhich are copositive withfand such that fPnp(f, (n+1)−1)p, whereω(ft)pdenotes the Ditzian–Totik modulus of continuity inLpmetric. It was shown by S. P. Zhou that this estimate is exact in the sense that if f has at least one sign change, thenωcannot be replaced byω2if 1<p<∞. In fact, we show that even for positive approximation and all 0<p<∞ the same conclusion is true. Also, some results for (co)positive spline approximation, exact in the same sense, are obtained.  相似文献   

13.
For an integer k 1 and a geometric mesh (qi)−∞ with q ε (0, ∞), let Mi,k(x): = k[qi + k](· − x)+k − 1, Ni,k(x): = (qi + kqiMi,k(x)/k, and let Ak(q) be the Gram matrix (∝Mi,kNj,k)i,jεz. It is known that Ak(q)−1 is bounded independently of q. In this paper it is shown that Ak(q)−1 is strictly decreasing for q in [1, ∞). In particular, the sharp upper bound and lower bound for Ak (q)−1 are obtained: for all q ε (0, ∞).  相似文献   

14.
On Hilbert''s Integral Inequality   总被引:5,自引:0,他引:5  
In this paper, we generalize Hilbert's integral inequality and its equivalent form by introducing three parameterst,a, andb.Iff, g L2[0, ∞), then[formula]where π is the best value. The inequality (1) is well known as Hilbert's integral inequality, and its equivalent form is[formula]where π2is also the best value (cf. [[1], Chap. 9]). Recently, Hu Ke made the following improvement of (1) by introducing a real functionc(x),[formula]wherek(x) = 2/π∫0(c(t2x)/(1 + t2)) dtc(x), 1 − c(x) + c(y) ≥ 0, andf, g ≥ 0 (cf. [[2]]). In this paper, some generalizations of (1) and (2) are given in the following theorems, which are other than those in [ [2]].  相似文献   

15.
Let Bn( f,q;x), n=1,2,… be q-Bernstein polynomials of a function f : [0,1]→C. The polynomials Bn( f,1;x) are classical Bernstein polynomials. For q≠1 the properties of q-Bernstein polynomials differ essentially from those in the classical case. This paper deals with approximating properties of q-Bernstein polynomials in the case q>1 with respect to both n and q. Some estimates on the rate of convergence are given. In particular, it is proved that for a function f analytic in {z: |z|<q+} the rate of convergence of {Bn( f,q;x)} to f(x) in the norm of C[0,1] has the order qn (versus 1/n for the classical Bernstein polynomials). Also iterates of q-Bernstein polynomials {Bnjn( f,q;x)}, where both n→∞ and jn→∞, are studied. It is shown that for q(0,1) the asymptotic behavior of such iterates is quite different from the classical case. In particular, the limit does not depend on the rate of jn→∞.  相似文献   

16.
Let (X, Y) be a random vector such that X is d-dimensional, Y is real valued, and θ(X) is the conditional αth quantile of Y given X, where α is a fixed number such that 0 < α < 1. Assume that θ is a smooth function with order of smoothness p > 0, and set r = (pm)/(2p + d), where m is a nonnegative integer smaller than p. Let T(θ) denote a derivative of θ of order m. It is proved that there exists estimate of T(θ), based on a set of i.i.d. observations (X1, Y1), …, (Xn, Yn), that achieves the optimal nonparametric rate of convergence nr in Lq-norms (1 ≤ q < ∞) restricted to compacts under appropriate regularity conditions. Further, it has been shown that there exists estimate of T(θ) that achieves the optimal rate (n/log n)r in L-norm restricted to compacts.  相似文献   

17.
Let be compact with #S=∞ and let C(S) be the set of all real continuous functions on S. We ask for an algebraic polynomial sequence (Pn)n=0 with deg Pn=n such that every fC(S) has a unique representation f=∑i=0 αiPi and call such a basis Faber basis. In the special case of , 0<q<1, we prove the existence of such a basis. A special orthonormal Faber basis is given by the so-called little q-Legendre polynomials. Moreover, these polynomials state an example with A(Sq)≠U(Sq)=C(Sq), where A(Sq) is the so-called Wiener algebra and U(Sq) is the set of all fC(Sq) which are uniquely represented by its Fourier series.  相似文献   

18.
In the paper sufficient conditions are given under which the differential equation y(n)=f(t,y,…,y(n−2))g(y(n−1)) has a singular solution y :[T,τ)→R, τ<∞ fulfilling
  相似文献   

19.
On a simplex SRd, the best polynomial approximation is En()Lp(S)=Inf{PnLp(S): Pn of total degree n}. The Durrmeyer modification, Mn, of the Bernstein operator is a bounded operator on Lp(S) and has many “nice” properties, most notably commutativity and self-adjointness. In this paper, relations between Mn−z.dfnc;Lp(S) and E[√n]()Lp(S) will be given by weak inequalities will imply, for 0<α<1 and 1≤p≤∞, En()Lp(S)=O(n-2α)Mn−z.dfnc;Lp(S)=O(n). We also see how the fact that P(DLp(S) for the appropriate P(D) affects directional smoothness.  相似文献   

20.
Discriminant analysis for locally stationary processes   总被引:1,自引:0,他引:1  
In this paper, we discuss discriminant analysis for locally stationary processes, which constitute a class of non-stationary processes. Consider the case where a locally stationary process {Xt,T} belongs to one of two categories described by two hypotheses π1 and π2. Here T is the length of the observed stretch. These hypotheses specify that {Xt,T} has time-varying spectral densities f(u,λ) and g(u,λ) under π1 and π2, respectively. Although Gaussianity of {Xt,T} is not assumed, we use a classification criterion D( f:g), which is an approximation of the Gaussian likelihood ratio for {Xt,T} between π1 and π2. Then it is shown that D( f:g) is consistent, i.e., the misclassification probabilities based on D( f:g) converge to zero as T→∞. Next, in the case when g(u,λ) is contiguous to f(u,λ), we evaluate the misclassification probabilities, and discuss non-Gaussian robustness of D( f:g). Because the spectra depend on time, the features of non-Gaussian robustness are different from those for stationary processes. It is also interesting to investigate the behavior of D( f:g) with respect to infinitesimal perturbations of the spectra. Introducing an influence function of D( f:g), we illuminate its infinitesimal behavior. Some numerical studies are given.  相似文献   

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