On polyharmonic interpolation

Haussmann, Werner; Kounchev, Ognyan

doi:10.1016/j.jmaa.2006.09.038

Cited by 5 publications

(6 citation statements)

References 12 publications

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“…The main barrier here is non-denseness of Ψ G,p (R n ) in Besov and Lizorkin-Triebel type spaces, if p = ∞. ., m, is an important question in many applications, including polyharmonic interpolation[HK07], wavelet analysis[BRV05], etc. Nazarova[Naz97] studied multi-point boundary value problems generated by a singular Bessel type operators.…”

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confidence: 99%

Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols

Umarov

2015

Developments in Mathematics

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Fractional order differential equations are an efficient tool to model various processes arising in science and engineering. Fractional models adequately reflect subtle internal properties, such as memory or hereditary properties, of complex processes that the classical integer order models neglect. In this chapter we will discuss the theoretical background of fractional modeling, that is the fractional calculus, including recent developments -distributed and variable fractional order differential operators.Fractional order derivatives interpolate integer order derivatives to real (not necessarily fractional) or complex order derivatives. There are different types of fractional derivatives not always equivalent. The first attempt to develop the fractional calculus systematically was taken by Liouville (1832) and Riemann (1847) in the first half of the nineteenth century, even though discussions on non-integer order derivatives had been started long ago. 1 In the 1870s Letnikov and Grünwald independently used an approach for the definition of the fractional order derivative and integral different from that of Riemann and Liouville. The Cauchy problem for fractional order differential equations with the Riemann-Liouville derivative is not well posed (Section 3.3), that is the Cauchy problem in this case is unphysical. In the 1960s Caputo and Djrbashian introduced independently, so-called, a regularization of the Riemann-Liouville fractional derivative, which was later named a fractional derivative in the sense of Caputo-Djrbashian (Section 3.5). The usefulness of the Caputo-Djrbashian derivative is that the Cauchy problem for fractional order differential equations with the Caputo-Djrbashian derivative is well posed. In the operators language one can write the latter in the form DJ = I, where I is the identity operator, which means that the operator D is a left inverse to the operator J. One can easily check that D is not a right inverse to J, since, according to the same fundamental theorem of calculus, for any differentiable function f the equality JD f (t) = f (t) − f (0) holds. The similar relations are true by induction for operators D n and J n , where D n = d n dt n , "n-th derivative," and J n is the n-fold integration operator. Namely,andThus D n is the left inverse to J n , and is the right inverse to J n in the class of functions satisfying additional conditions: f (k) (0) = 0, k = 0,..., n − 1. These relations between "differentiation" and "integration" operators valid for n = 1, 2, . . . , form the basis for the definitions of fractional derivatives in the sense of RiemannLiouville and Caputo-Djrbashian, as soon as the fractional order integration operator is defined. Fractional order integration operator 123 Fractional order integration operatorIn this section we introduce the fractional order integration operator of order α > 0 (ℜ(α) > 0). One can verify (by changing order of integration) that the n-fold integration operatorTaking into account the relationship Γ (n) = (n − 1)!, whereis the Euler's ...

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confidence: 99%

Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols

Umarov

2015

Developments in Mathematics

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“…In [18] polyharmonic interpolation has been considered for functions defined in the ball in R d . In the same spirit, in [27] and [25] we have introduced a new multivariate cubature formulae C N (f ) in the ball depending on a parameter N ∈ N which approximates the integral…”

Section: Gauss-almansi Formulamentioning

confidence: 99%

“…Our main framework of Interpolation and Cubature was defined in [27], [18], [25]. It has further brought to life the multivariate complexification and the polyharmonic Hardy spaces.…”

Section: Error Estimate Of Polyharmonic Interpolation and Cubature Fo...mentioning

confidence: 99%

Polyharmonic Hardy Spaces on the Klein-Dirac Quadric with Application to Polyharmonic Interpolation and Cubature Formulas

Kounchev,

Render

2012

Preprint

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“…By Theorem 2.2, it is sufficient to show BD −1 C = 0. Under the condition (a), D = diag( f 1 (x 1 , y 1 ),... , f n (x n , y n )) is a nonsingular matrix and D −1 = diag( f 1 (x 1 , y 1 ) −1 , ··· , f n (x n , y n ) −1 ) [9] . Let E = BD −1 , hence Anal.…”

Section: Proof Supposementioning

confidence: 99%

“…In recent years, the bivariate and multivariate interpolations have been studied in the papers [3,8,9,12,13,16]. In [6,7] the inherited interpolation of matrices has been introduced by using the LU inherited factorization of a matrix.…”

Section: Introductionmentioning

confidence: 99%

Semi Inherited Bivariate Interpolation

Araghi

Fallahzadeh

2011

ATA

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The bivariate interpolation in two dimensional space R 2 is more complicated than that in one dimensional space R, because there is no Haar space of continuous functions in R 2 . Therefore, the bivariate interpolation has not a unique solution for a set of arbitrary distinct pairwise points. In this work, we suggest a type of basis which depends on the points such that the bivariate interpolation has the unique solution for any set of distinct pairwise points. In this case, the matrix of bivariate interpolation has the semi inherited factorization.

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On polyharmonic interpolation

Cited by 5 publications

References 12 publications

Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols

Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols

Polyharmonic Hardy Spaces on the Klein-Dirac Quadric with Application to Polyharmonic Interpolation and Cubature Formulas

Semi Inherited Bivariate Interpolation

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