Reconstruction of density functions by sk-splines

Reconstruction of density functions and their characteristic functions by radial basis functions with scattered data points is a popular topic in the theory of pricing of basket options. Such functions are usually entire or admit an analytic extension into an appropriate tube and "bell-shaped" with rapidly decaying tails. Unfortunately, the domain of such functions is not compact which creates various technical difficulties. We solve interpolation problem on an infinite rectangular grid for a wide range of kernel functions and calculate explicitly their Fourier transform to obtain representations for the respective density functions

OPTIMAL SK-SPLINE APPROXIMATION OF SOBOLEV’S CLASSES ON THE 2-SPHERE

Abstract The space of-splines is the linear span of shifts of a single kernel. In this article we introduce-splines on. It is shown that, with suitably chosen kernel, the subspace of-splines realizes sharp orders of Kolmogorov’s-widths in different important situations. 1

Optimal cubature formulas on compact homogeneous manifolds

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)We find lower bounds for the rate of convergence of optimal cubature formulas on sets of differentiable functions on compact homogeneous manifolds of rank I or two-point homogeneous Spaces. It is Shown that these lower bounds are sharp in the power scale in the case of S(2), the unit sphere in R(3). (C) 2009 Elsevier Inc. All rights reserved.257516211629Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)FAPESP [2007/56162-8

Quasi--Interpolation on the 2--Sphere using Radial Polynomials

In this paper we consider a simple method of radial quasi-interpolation by polynomials on S 2 and present rates of convergence for this method on a wide range of smooth functions. 1 Harmonic analysis and sets of smooth functions on S 2 Let hx; yi = x 1 y 1 + x 2 y 2 + x 3 y 3 This research has been partially supported by EPSRC (UK) Grant GL41639 and CNPq (Brazil). be the usual scalar product in Euclidean 3--space IR 3 , and S 2 be the 2dimensional unit sphere in IR 3 , i.e. S 2 = fx j x 2 IR 3 ; hx; xi = 1g: Let d¯ be the normalized rotation invariant measure on the sphere, and k'k p = ( ( R S 2 j'(x)j p d¯(x)) 1=p ; 1 p ! 1; ess supfj'(x)j j x 2 S 2 g; p = 1: Let L p = f' j k'k p ! 1g, and U p = f' j k'k p 1g. The space L 2 has the orthogonal decomposition L 2 = 1 M k=0 H k where H k is the space of spherical harmonic polynomials of degree k. It is known that H k has dimension 2k+1. Let fY (k) \Gammak ; : : : ; Y (k) k g be an orthonormal b..

Optimal Approximation On Sd

The asymptotic behavior of the n-widths of a wide range of sets of smooth functions on a d-dimensional sphere in Lq(Sd) is studied. Upper and lower bounds for the n-widths are established. Moreover, it is shown that these upper and lower bounds coincide for some important concrete examples. © 2000 Academic Press.162424458Askey, R., Wainger, S., On the behavior of special classes of ultraspherical expansions I, II (1965) J. Analyse Math., 15, pp. 193-244Bonami, A., Clerk, J.-L., Sommes de Cesàro et multiplicateurs des développements en harmoniques sphériques (1973) Trans. Amer. Math. Soc., 183, pp. 223-263Dunford, N., Schwartz, J., (1957) Linear Operators, , New York: InterscienceForst, W., Über die Breite von Klassen holomorpher periodischer Funktionen (1977) J. Approx. Theory, 19, pp. 325-331Gluskin, E.D., On some problem about n-widths (1974) Doklady Acad. Nauk. USSR, 219, pp. 527-530Gluskin, E.D., Norms of random matrices and n-widths of finite dimensional sets (1983) Mat. Sbornik, 120, pp. 180-189Gluskin, E.D., Extremal properties of orthogonal parallelepipeds and their application to the geometry of Banach spaces (1988) Mat. Sbornik, 136, pp. 85-96Ismagilov, R.S., On n-dimensional widths of compacts in Hilbert Space (1968) Funk. Anal. I Ego Prilog., 2, pp. 32-39Ismagilov, R.S., N-widths of sets in linear normed spaces and approximation of functions by trigonometric polynomials (1977) Uspekhi Mat. Nauk., 29, pp. 161-178Kamzolov, A.I., On the best approximation of sets of function Wα p(Sd) by polynomials (1982) Mat. Zametki, 32, pp. 285-293Kashin, B.S., N-widths of some finite dimensional sets and classes of smooth functions (1977) Izvestiya Acad. Nauk. USSR, 41, pp. 334-351Kashin, B.S., Some properties of the space of trigonometric polynomials with uniform norm (1980) Trudy. Mat. Inst. Steklov, 145, pp. 111-116Kashin, B.S., On n-widths of Sobolev's classes of small smoothness (1981) Vestnik MGU, 5, pp. 50-54Kogbetliantz, E., Recherches sur la sommabilité des séries ultrasphériques par la méthode des moyennes arithmétiques (1924) J. Math., 3, pp. 107-187Kolmogorov, A.N., Über die beste Annäherung von Funktionen einer gegebenen Funktionenklasse (1936) Ann. of Math., 37, pp. 107-110Kulanin, E.D., Estimates of n-widths of Sobolev's classes with small smoothness (1983) Vestnik MGU, 2, pp. 24-30Kushpel, A.K., (1987) N-widths of Sets of Smooth Functions in Lq, , Preprint 87.44, Kiev, Inst. of Math, [In Russian]Kushpel, A.K., N-widths of sets of analytic functions (1989) Ukrain. Math. Zh., 41, pp. 576-579Kushpel, A.K., (1989) Approximation of Sets of Smooth Functions and N-widths, , Preprint 89.77, Kiev, Inst. of Math, [In Russian]Kushpel, A.K., Bernstein's n-widths of sets of functions with super small smoothness (1993) Theory of Functions and Their Applications, , Voronezh University, 14Kushpel, A.K., Inequalities for discrete norms with scattered centers for polynomials on Sd (1998) 47° Seminário Brasileiro de Análise São José Dos Campos, , p. 283-292Kushpel, A.K., Levesley, J., Wilderotter, K., On the asymptotically optimal rate of approximation of multiplier operators from Lp into Lq (1998) Constructive Approx., 14, pp. 169-186Makovoz, J.I., On a method for estimation from below of n-widths of sets in Banach spaces (1972) Mat. Sbornik, 86, pp. 136-142Maiorov, V.E., Discretization of problem on n-widths (1975) Uspekhi Mat. Nauk, 30, pp. 179-190Marcinkiewicz, J., Sur l'interpolation (1936) Studia Math., 6, pp. 1-17Marcinkiewicz, J., Quelques remarques sur l'interpolation (1937) Acta Litt. Acad. Sci. Szeged, 8, pp. 127-130Müller, C., (1966) Spherical Harmonics, , Berlin: Springer-VerlagMurnaghan, F.D., (1962) The Unitary and Rotation Groups, , Washington: Spartan BooksPietsch, A., (1979) Operator Ideals, , Berlin: Deutscher Verlag der WissenschaftenPinkus, A., (1985) N-Widths in Approximation Theory, , Berlin: Springer-VerlagRudin, W., L2-approximation by partial sums of orthogonal developments (1952) Duke Math. J., 19, pp. 1-4Sobolev, S.L., (1974) Introduction into the Theory of Cubature Formulas, , Moscow: NaukaStechkin, S.B., On the best approximation by an arbitrary polynomials (1960) Uspekhi Mat. Nauk, 9, pp. 133-134Stein, E.M., Weiss, G., (1971) Introduction to Fourier Analysis on Euclidean Spaces, , Princeton: Princeton Univ. PressStepanets, A.I., (1987) Classification and Approximation of Periodic Functions, , Kiev: Naukova DumkaSzegö, G., (1939) Orthogonal Polynomials, , New York: AMSTemlyakov, V.N., On estimates of widths of sets of infinitely differentiable functions (1990) Mat. Zametki, 47, pp. 155-157Tikhomirov, V.M., N-widths in some functional spaces and approximation theory (1960) Uspekhi Mat. Nauk., 15, pp. 81-120Tikhomirov, V.M., (1976) Some Problems in Approximation Theory, , Moscow: NaukaTikhomirov, V.M., II. Approximation theory (1990) Encyclopaedia of Mathematical Sciences, , New York/London: Springer-Verlag. p. 93-255Timan, A.F., (1963) Theory of Approximation of Functions of Real Variable, , New York: Pergamon PressVilenkin, N.J., (1965) Special Functions and Theory of Representation of Groups, , Moscow: NaukaZygmund, A., (1959) Trigonometric Series, , Cambridge: Cambridge Univ. Pres

On the problem of optimal reconstruction

We find lower bounds for linear and Alexandrov's cowidths of Sobolev's classes on Compact Riemannian homogeneous manifolds M-d. Using these results we give an explicit solution of the problem of optimal reconstruction of functions from Sobolev's classes W-p(y) (M-d) in L-q (M-d), 1 <= q <= p <= infinity.134SI45947