pinecone/core-2020-05-10-deduplication huggingface dataset:
Markov- and Bernstein-Type Inequalities for Müntz Polynomials and Exponential Sums in Lp

AbstractThe principal result of this paper is the following Markov-type inequality for Müntz polynomials. Theorem (Newman's Inequality in Lp[a, b] for [a, b]⊂(0, ∞)).LetΛ≔(λj)∞j=0be an increasing sequence of nonnegative real numbers. Supposeλ0=0 and there exists aδ>0 so thatλj⩾δjfor each j. Suppose 0<a<band 1⩽p⩽∞. Then there exists a constantc(a, b, δ) depending only ona, b, andδso that ‖P′‖Lp[a, b]⩽c(a, b, δ)(∑nj=0λj)‖P‖Lp[a, b]for everyP∈Mn(Λ), whereMn(Λ) denotes the linear span of {xλ0, xλ1, …, xλn} overR. When p=∞ this has been shown by P. B. Borwein and the author (1996, J. Approx. Theory85, 132–139). When [a, b]=[0, 1] and with ‖P′‖Lp[a, b] replaced with ‖xP′(x)‖Lp[a, b] this was proved by D. Newman (1976, J. Approx. Theory18, 360–362) for p=∞ and by P. Borwein and the author (1996, Proc. Amer. Math. Soc.124, 101–109) for 1⩽p⩽∞. Note that the interval [0, 1] plays a special role in the study of Müntz spaces Mn(Λ). A linear transformation y=αx+β does not preserve membership in Mn(Λ) in general (unless β=0). So the analogue of Newman's Inequality on [a, b] for a>0 does not seem to be obtainable in any straightforward fashion from the [0, b] case
A Better Approximation for Balls

AbstractUnexpectedly accurate and parsimonious approximations for balls in Rd and related functions are given using half-spaces. Instead of a polytope (an intersection of half-spaces) which would require exponentially many half-spaces (of order (1ε)d) to have a relative accuracy ε, we use T=c(d2/ε2) pairs of indicators of half-spaces and threshold a linear combination of them. In neural network terminology, we are using a single hidden layer perceptron approximation to the indicator of a ball. A special role in the analysis is played by probabilistic methods and approximation of Gaussian functions. The result is then applied to functions that have variation Vf with respect to a class of ellipsoids. Two hidden layer feedforward sigmoidal neural nets are used to approximate such functions. The approximation error is shown to be bounded by a constant times Vf/T1/21+Vfd/T1/42, where T1 is the number of nodes in the outer layer and T2 is the number of nodes in the inner layer of the approximation fT1, T2
Uniqueness of the Gaussian Quadrature for a Ball

AbstractWe construct a formula for numerical integration of functions over the unit ball in Rd that uses n Radon projections of these functions and is exact for all algebraic polynomials in Rd of degree 2n−1. This is the highest algebraic degree of precision that could be achieved by an n term integration rule of this kind. We prove the uniqueness of this quadrature. In particular, we present a quadrature formula for a disk that is based on line integrals over n chords and integrates exactly all bivariate polynomials of degree 2n−1
Surface Design using Locally Interpolating Subdivision Schemes

AbstractNon-uniform multivariate subdivision schemes are constructed, which generate limit functions interpolating some of the initial control points. Our schemes differ from the known interpolatory subdivision schemes, in that only some of the original control points are interpolated, and not the control points in every level. These new schemes are combinations of a non-interpolatory schemes with different local schemes near some of the original control points. They generate smooth surfaces interpolating given points, using stencils of small support. Next, it is shown how to modify known subdivision schemes so that the limit surfaces generated by them interpolate given normal vectors at given interpolation points
The N-Widths of Spaces of Holomorphic Functions on Bounded Symmetric Domains of Tube Type

AbstractLet D be a bounded symmetric domain of tube type and Σ be the Shilov boundary of D. Denote by H2(D) and A2(D) the Hardy and Bergman spaces, respectively, of holomorphic functions on D; and let B(H2(D)) and B(A2(D)) denote the closed unit balls in these spaces. For an integer l⩾0 we define the notion Rlf of the lth radial derivative of a holomorphic function f on D, and we prove the following results: Let 0<ρ<1. Denote by W the class of holomorphic functions f on D for which Rlf∈B(H2(D)) and set X=C(ρΣ). Then we show that the linear and Gelfand N-widths of W in X coincide, and we compute the exact value. We do the same for the case in which W is the class of holomorphic functions f for which Rlf∈B(A2(D)), and X=C(ρΣ). Next, let X=Lp(ρΣ) (respectively, Lp(ρD)) for 1⩽p⩽∞, and let W be a class of holomorphic functions f on D for which Rlf∈B(Hp(D)) (respectively, B(Ap(D))). We show that the Kolmogorov, linear, Gelfand, and Bernstein N-widths all coincide, we calculate the exact value, and we identify optimal subspaces or optimal linear operators. These results extend work of Yu. A. Farkov (1993, J. Approx. Theory75, 183–197) and K. Yu. Osipenko (1995, J. Approx. Theory82, 135–155), and initiate the study of N-widths of spaces of holomorphic functions on bounded symmetric domains
Dependence of α in Peak Norms and Best Peak Norms Approximation

AbstractLet C[0, 1] be the space of all continuous functions defined on [0, 1] and U be an n dimensional subspace of C[0, 1]. A peak norm, or α-norm for 0<α⩽1, α-norm is defined by ‖f‖α=1αsup{∫A|f|dμ∣μ(A)=α, A⊂[0, 1]}, where μ denotes the Lebesgue measure. We say p∈U is a best α-norm approximant to f from U if Dα(f)=‖f−p‖α=inf{‖f−u‖α∣u∈U}. In this paper we shall study ‖f‖α, Dα(f) and Pα(f)={p∈U∣‖f−p‖α=Dα(f)} as functions of α for fixed f. We shall show their continuous dependence on α and differentiability with respect to α
Holomorphic T-Monsters and Strongly Omnipresent Operators

AbstractAssume that G is a nonempty open subset of the complex plane and that T is an operator on the linear space of holomorphic functions in G, endowed with the compact-open topology. In this paper we introduce the notions of strongly omnipresent operator and of T-monster, which are related to the wild behaviour of certain holomorphic functions near the boundary of G. T-monsters extend a concept introduced by W. Luh and K.-G. Grosse-Erdmann. After showing that T is strongly omnipresent if and only if the set of T-monsters is residual, it is proved in this paper that certain kinds of infinite order differential and antidifferential operators are strongly omnipresent, which improves some earlier nice results due to the mentioned authors
On Mean Convergence of Lagrange Interpolation for General Arrays

AbstractFor n⩾1, let {xjn}nj=1 be n distinct points in a compact set K⊂R and let Ln[·] denote the corresponding Lagrange interpolation operator. Let v be a suitably restricted function on K. What conditions on the array {xjn}1⩽j⩽n, n⩾1 ensure the existence of p>0 such that limn→∞‖(f−Ln[f])v‖Lp(K)=0 for very continuous f:K→R? We show that it is necessary and sufficient that there exists r>0 with supn⩾1‖πnv‖Lr(K)∑nj=1(1/|π′n|(xjn))<∞. Here for n⩾1, πn is a polynomial of degree n having {xjn}nj=1 as zeros. The necessity of this condition is due to Ying Guang Shi
On the Rate of Convergence of Two Bernstein–Bézier Type Operators for Bounded Variation Functions, II

AbstractThe rates of convergence of two Bernstein–Bézier type operators B(α)n and L(α)n for functions of bounded variation have been studied for the case α⩾1 by the author and A. Piriou (1998, J. Approx. Theory95, 369–387). In this paper the other case 0<α<1 is treated and asymptotically optimal estimations of B(α)n and L(α)n for functions of bounded variation are obtained. Besides, some interesting behaviors of the operators B(α)n and L(α)n (α>0) for monotone functions and functions of bounded variation are also given
Embeddings in Spaces of Lipschitz Type, Entropy and Approximation Numbers, and Applications

AbstractWe consider the embeddings of certain Besov and Triebel–Lizorkin spaces in spaces of Lipschitz type. The prototype of such embeddings arises from the result of H. Brézis and S. Wainger (1980, Comm. Partial Differential Equations5, 773–789) about the “almost” Lipschitz continuity of elements of the Sobolev spaces H1+n/pp(Rn) when 1<p<∞. Two-sided estimates are obtained for the entropy and approximation numbers of a variety of related embeddings. The results are applied to give bounds for the eigenvalues of certain pseudo-differential operators and to provide information about the mapping properties of these operators
Characterizations of Multivariate Differences and Associated Exponential Splines

AbstractThe subject of this investigation is the class of difference functionals—linear combinations of finitely many function and/or derivative evaluations—which annihilate the nullspace of a certain constant coefficient differential operator. Any such functional can be viewed as an integral-differential operator whose Peano kernel is a compactly supported exponential spline. Besides extending some earlier results (1996, T. Kunkle, J. Approx. Theory84, 290–314), we show that these functionals are the only ones whose convolutions with the associated exponential truncated powers have compact support. It is then proven that, in case the functional depends entirely on function values at rational points, it must be a linear combination of forward differences. These results have applications in the areas of (a) placing compactly supported exponential splines in the span of the box splines, and (b) interpolation by exponential polynomials to function values at the support points of the forward difference functional
On Rational Interpolation to Meromorphic Functions in Several Variables

AbstractIn this paper, a new approach to construct rational interpolants to functions of several variables is considered. These new families of interpolants, which in fact are particular cases of the so-called Padé-type approximants (that is, rational interpolants with prescribed denominators), extend the classical Padé approximants (for the univariate case) and provide rather general extensions of the well-known Montessus de Ballore theorem for several variables. The accuracy of these approximants and the sharpness of our convergence results are analyzed by means of several examples
The Eigenstructure of the Bernstein Operator

AbstractThe Bernstein operator Bn reproduces the linear polynomials, which are therefore eigenfunctions corresponding to the eigenvalue 1. We determine the rest of the eigenstructure of Bn. Its eigenvalues areλ(n)k≔n!(n−k)!1nk,k=0, 1, …, n,and the corresponding monic eigenfunctions p(n)k are polynomials of degree k, which have k simple zeros in [0, 1]. By using an explicit formula, it is shown that p(n)k converges as n→∞ to a polynomial related to a Jacobi polynomial. Similarly, the dual functionals to p(n)k converge as n→∞ to measures that we identity. This diagonal form of the Bernstein operator and its limit, the identity (Weierstrass density theorem), is applied to a number of questions. These include the convergence of iterates of the Bernstein operator and why Lagrange interpolation (at n+1 equally spaced points) fails to converge for all continuous functions whilst the Bernstein approximants do. We also give the eigenstructure of the Kantorovich operator. Previously, the only member of the Bernstein family for which the eigenfunctions were known explicitly was the Bernstein–Durrmeyer operator, which is self adjoint
On Generalized Hermite–Fejér Interpolation of Lagrange Type on the Chebyshev Nodes

AbstractFor f∈C[−1,1], let Hm,n(f,x) denote the (0, 1, …,anbsp;m) Hermite–Fejér (HF) interpolation polynomial of f based on the Chebyshev nodes. That is, Hm,n(f,x) is the polynomial of least degree which interpolates f(x) and has its first m derivatives vanish at each of the zeros of the nth Chebyshev polynomial of the first kind. In this paper a precise pointwise estimate for the approximation error |H2m, n(f, x)−f(x)| is developed, and an equiconvergence result for Lagrange and (0, 1, …, 2m) HF interpolation on the Chebyshev nodes is obtained. This equiconvergence result is then used to show that a rational interpolatory process, obtained by combining the divergent Lagrange and (0, 1, …, 2m) HF interpolation methods on the Chebyshev nodes, is convergent for all f∈C[−1, 1]
Polynomial Interpolation to Data on Flats in Rd

AbstractStimulated by recent work of Hakopian and Sahakian, polynomial interpolation to data at all the s-dimensional intersections of an arbitrary sequence of hyperplanes in Rd is considered, and reduced, by the adjunction of an additional s hyperplanes in general position with respect to the given sequence, to the case s=0 solved much earlier by two of the present authors. In particular, interpolation is from the very same polynomial spaces already used earlier. The difficult question of multiplicity and corresponding matching of derivative information is completely solved, with the number of independent derivative conditions at an intersection exactly equal to that intersection's multiplicity. Also, the consistency requirements placed on the data are minimal in the sense that they need to be checked only at the finitely many 0-dimensional intersections of the hyperplanes involved. The arguments used provide, incidentally, further insights into the two polynomial spaces, P(Ξ) and D(Ξ), of basic interest in box spline theory
Approximation in Lp(Rd) from Spaces Spanned by the Perturbed Integer Translates of a Radial Function

AbstractThe problem of approximating smooth Lp-functions from spaces spanned by the integer translates of a radially symmetric function φ is very well understood. In case the points of translation, Ξ, are scattered throughout Rd, the approximation problem is only well understood in the “stationary” setting. In this work, we provide lower bounds on the obtainable approximation orders in the “non-stationary” setting under the assumption that Ξ is a small perturbation of Zd. The functions which we can approximate belong to certain Besov spaces. Our results, which are similar in many respects to the known results for the case Ξ=Zd, apply specifically to the examples of the Gauss kernel and the generalized multiquadric
Estimates for the Discrepancy of a Signed Measure Using Its Energy Norm

AbstractW. Kleiner established (1964, Ann. Polon. Math.14, 117–130) for smooth curves and arcs an estimate for the discrepancy of a signed measure by using its energy norm. We extend this result to quasiconformal curves and arcs. The proof uses the theory of quasiconformal mappings and condenser theory. In a first step, the discrepancy of a signed measure can be estimated from above in terms of its energy norm and the capacity of a special condenser. This result is valid on every Jordan curve. Examples show the sharpness of the results from various points of view
The Sobolev Regularity of Refinable Functions

AbstractRefinable functions underlie the theory and constructions of wavelet systems on the one hand and the theory and convergence analysis of uniform subdivision algorithms on the other. The regularity of such functions dictates, in the context of wavelets, the smoothness of the derived wavelet system and, in the subdivision context, the smoothness of the limiting surface of the iterative process. Since the refinable function is, in many circumstances, not known analytically, the analysis of its regularity must be based on the explicitly known mask. We establish in this paper a formula that computes, for isotropic dilation and in any number of variables, the sharp L2-regularity of the refinable function φ in terms of the spectral radius of the restriction of the associated transfer operator to a specific invariant subspace. For a compactly supported refinable function φ, the relevant invariant space is proved to be finite dimensional and is completely characterized in terms of the dependence relations among the shifts of φ together with the polynomials that these shifts reproduce. The previously known formula for this compact support case requires the further assumptions that the mask is finitely supported and that the shifts of φ are stable. Adopting a stability assumption (but without assuming the finiteness of the mask), we derive that known formula from our general one. Moreover, we show that in the absence of stability, the lower bound provided by that previously known formula may be abysmal. Our characterization is further extended to the FSI (i.e., vector) case, to the unisotropic dilation matrix case, and to even snore general setups. We also establish corresponding results for refinable distributions
On the Domain of Divergence of Hermite–Fejér Interpolating Polynomials

AbstractRecently, the study of the behavior of the Hermite–Fejér interpolants in the complex plane was initiated by L. Brutman and I. Gopengauz (1999, Constr. Approx.15, 611–617). It was shown that, for a broad class of interpolatory matrices on [−1, 1], the sequence of polynomials induced by Hermite–Fejér interpolation to f(z)≡z diverges everywhere in the complex plane outside the interval of interpolation [−1, 1]. In this note we amplify this result and prove that the divergence phenomenon takes place without any restriction on the interpolatory matrices
Approximating Solutions of Maximal Monotone Operators in Hilbert Spaces

AbstractLet H be a real Hilbert space and let T:H→2H be a maximal monotone operator. In this paper, we first introduce two algorithms of approximating solutions of maximal monotone operators. One of them is to generate a strongly convergent sequence with limit v∈T−10. The other is to discuss the weak convergence of the proximal point algorithm. Next, using these results, we consider the problem of finding a minimizer of a convex function. Our methods are motivated by Halpern's iteration and Mann's iteration
Best Approximation by Normal and Conormal Sets

AbstractThe aim of the present paper is to develop a theory of best approximation by elements of so-called normal sets and their complements—conormal sets—in the non-negative orthant RI+ of a finite-dimensional coordinate space RI endowed with the max-norm. A normal (respectively, conormal) set arises as the set of all solutions of a system of inequalities fα(x)⩽0 (α∈A), x∈RI+ (respectively, fα(x)⩾0 (α∈A), x∈RI+), where fα is an increasing function and A is an arbitrary set of indices. We consider these sets as analogues (in a certain sense) of convex sets, and we use the so-called min-type functions as analogues of linear functions. We show that many results on best approximation by convex and reverse convex sets and corresponding separation theory (but not all of them) have analogues in the case under consideration. At the same time there are no convex analogues for many results related to best approximation by normal sets
On Exact Values of n-Widths in a Hilbert Space

AbstractThe exact values of Kolmogorov n-widths have been calculated for two basic classes of functions. They are, on the one hand, classes of real functions defined by variation diminishing kernels and similar classes of analytic functions, and, on the other hand, classes of functions in a Hilbert space which are elliptical cylinders or generalized octahedra. This second case is surveyed and new results are presented. For n-widths of ellipsoids, elliptic cylinders, and generalized octahedra, upper bounds for the n-widths are based on the Fourier method. The lower bounds are based on the method of “embedded balls” for ellipsoids and the method of averaging for generalized octahedra. General theorems concerning elliptical cylinders and generalized octahedra are proved, various corollaries from these general theorems are considered, and some additional problems (average n-widths, extremal spaces for an ellipsoids and octahedra, etc.) are discussed
Fourier Approximation and Hausdorff Convergence

AbstractFor a function f on Rd we consider its Fourier transform Ff and the (integrable) Cesáro averages FMf of suitable truncations of Ff, described by the formula (FMf)(a)=M−d(M−|a1|)+…(M−|ad|)+(Ff)(a). We study the speed of the convergence F−1FMf→f, (M→∞), under a metric that is somewhere between the L1- and the L∞-metrics. In this metric (which is appropriate to problems of pattern recognition), the distance between two functions is, more or less, the Hausdorff distance between their graphs. We describe a class of functions f for which the distance between F−1FMf and f is O(M−1/2), the fastest rate of converges one can have for discontinuous f
Schur's Algorithm, Orthogonal Polynomials, and Convergence of Wall's Continued Fractions in L2(T)

AbstractA function f in the unit ball B of the Hardy algebra H∞ on the unit disc D={z∈C:|z|<1} is a non-exposed point of B (|f|<1 a.e. on T={ζ∈C:|ζ|=1}) iff limn∫T|fn|2dm=0, where m is the Lebesgue measure on T and (fn)n⩾0 are the Schur functions of f. This result easily implies Rakhmanov's well-known theorem which states that limnan=0 if σ′>0 a.e. on T, (an)n⩾0 being the parameters of the orthogonal polynomials (ϕn)n⩾0 in L2(dσ). We prove that fnbn is the Schur function of the probability measure |ϕn|2dσ, which leads to an important formula relating |ϕn|2σ′ to fn and bn=ϕn/ϕ*n. A probability measure σ is called a Rakhmanov measure if (*)−limn|ϕn|2dσ=dm. We show that a probability measure σ with parameters (an)n⩾0 is a Rakhmanov measure iff the an's satisfy the Máté–Nevai condition limnanan+κ=0 for every κ=1, 2, …. Next, we prove that even approximants An/Bn of the Wall continued fraction for f converge in L2(T) iff either f is an inner function or limnan=0. This implies that measures satisfying limnanan+κ=0, κ=1, 2, …, and limn|an|>0 are all singular
On Well Posed Generalized Best Approximation Problems

AbstractLet C be a closed bounded convex subset of X with 0 being an interior point of C and pC be the Minkowski functional with respect to C. Let G be a nonempty closed, boundedly relatively weakly compact subset of a Banach space X. For a point x∈X, we say the minimization problem minC(x, G) is well posed if there exists a unique point z such that pC(z−x)=λC(x, G) and every sequence {zn}⊂G satisfying limn→∞pC(zn−x)=λC(x, G) converges strongly to the point z, where λC(x, G)=infz∈GpC(z−x). Under the assumption that C is both strictly convex and Kadec, we prove that the set Xo(G) of all x∈X such that the problem minC(x, G) is well posed is a residual subset of X extending the results in the case that the modulus of convexity of C is strictly positive due to Blasi and Myjak. In addition, we also prove these conditions are necessary
θ-Summation and Hardy Spaces

AbstractA general summability method of Fourier series and Fourier transforms is given with the help of an integrable function θ having integrable Fourier transform. Under some weak conditions on θ we show that the maximal operator of the θ-means of a distribution is bounded from Hp(T) to Lp(T) (p0<p<∞) and is of weak type (1,1), where Hp(T) is the classical Hardy space and p0<1 is depending only on θ. As a consequence we obtain that the θ-means of a function f∈L1(T) converge a.e. to f. For the endpoint p0 we get that the maximal operator is of weak type (Hp0(T), Lp0(T)). Moreover, we prove that the θ-means are uniformly bounded on the spaces Hp(T) whenever p0<p<∞ and are uniformly of weak type (Hp0(T), Hp0(T)). Thus, in the case f∈Hp(T), the θ-means converge to f in Hp(T) norm (p0<p<∞). The same results are proved for the conjugate θ-means and for Fourier transforms, too. Some special cases of the θ-summation are considered, such as the Weierstrass, Picar, Bessel, Fejér, Riemann, de La Vallée-Poussin, Rogosinski and Riesz summations