# MULTIVARIATE APPROXIMATION

@inproceedings{Boor1986MULTIVARIATEA, title={MULTIVARIATE APPROXIMATION}, author={Carl de Boor}, year={1986} }

The lecture addresses topics in multivariate approximation which have caught the author’s interest in the last ten years. These include: the approximation by functions with fewer variables, correct points for polynomial interpolation, the B(ernstein,-ézier, -arycentric)-form for polynomials and its use in understanding smooth piecewise polynomial (pp) functions, approximation order from spaces of pp functions, multivariate Bsplines, and surface generation by subdivision. AMS (MOS) Subject…

## 44 Citations

Approximation of functions with small mixed smoothness in the uniform norm

- Computer Science, MathematicsArXiv
- 2020

The focus will be on the behavior of the best m-term trigonometric approximation as well as the decay of Kolmogorov and entropy numbers in the uniform norm, which have direct implications for the problem of sampling recovery in L2.

On universal sampling representation

- Computer Science, MathematicsArXiv
- 2022

This work replaces the normalized Lebesgue measure by a discrete measure in such a way, which preserves the convolution properties and provides sampling discretization of integral norms, that in the two-variate case the Fibonacci point sets provide an ideal solution.

On exact estimates of the order of approximation of functions of several variables in the anisotropic Lorentz-Zygmund space

- Mathematics
- 2021

Abstract. In this paper we consider L p,α,τ (T ) anisotropic Lorentz-Zygmund space 2π of periodic functions of m variables and Nikol’skii–Besov’s class S r̄ p,α,τ,θ̄ B. In this paper, we establish…

High-dimensional nonlinear approximation by parametric manifolds in Hölder-Nikol'skii spaces of mixed smoothness

- Computer Science, MathematicsArXiv
- 2021

We study high-dimensional nonlinear approximation of functions in Hölder-Nikol’skii spaces H ∞ (I) on the unit cube I := [0, 1] having mixed smoothness, by parametric manifolds. The approximation…

Estimates of the order of approximation of functions of several variables in the generalized Lorentz space

- Mathematics
- 2021

Abstract. In this paper we consider X(φ̄) anisotropic symmetric space 2π of periodic functions of m variables, in particular, the generalized Lorentz space L ψ̄,τ̄ (T) and Nikol’skii–Besov’s class S…

On estimates of the order of approximation of functions of several variables in the anisotropic Lorentz-Karamata space

- Mathematics
- 2021

Abstract. In this paper we consider anisotropic Lorentz-Karamata space 2π of periodic functions of m variables and Nikol’skii–Besov’s class . In this paper, we establish order-sharp estimates of the…

Best $n$-term approximation of diagonal operators and application to function spaces with mixed smoothness

- Mathematics
- 2021

In this paper we give exact values of the best n-term approximation widths of diagonal operators between lp(N) and lq(N) with 0 < p, q ≤ ∞. The result will be applied to obtain the asymptotic…

Sampling discretization of integral norms

- Mathematics, Computer ScienceArXiv
- 2020

A conditional theorem for all integral norms of functions from a given finite dimensional subspace is obtained, which is an extension of known results for q=1 and a new Marcinkiewicz type discretization for the multivariate trigonometric polynomials with frequencies from the hyperbolic crosses is derived.

On optimal approximation in periodic Besov spaces

- MathematicsJournal of Mathematical Analysis and Applications
- 2019

Sampling discretization and related problems

- Mathematics
- 2021

This survey addresses sampling discretization and its connections with other areas of mathematics. We present here known results on sampling discretization of both integral norms and the uniform norm…

## References

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- MathematicsNature
- 1965

Theory of Approximation of Functions of a Real VariableBy A. F. Timan. Translated by J. Berry. English translation edited and editorial preface by J. Cossar. (International Series of Monographs on…

Approximation by functions of fewer variables

- Mathematics
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There are different possibilities to approximate a continuous function of n independent real variables by functions of fewer veriables and their combinations. Here we consider several special types…

Shortest Path Algorithms for the Approximation by Nomographic Functions

- Mathematics
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Continuous functions of two real variables are approximated on compact domains in the uniform norm by the classes NOM of nomographic functions which appear as approximation subspaces in bivariate…

Controlled approximation and a characterization of the local approximation order

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Abstract : Document characteristics the local approximation order from a scale (S sub h) of approximating functions on R to the m power is characterized in terms of the linear span (and its Fourier…

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- Mathematics
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Multivariate interpolation at arbitrary points made simple

- Mathematics
- 1979

The concrete method of ‘surface spline interpolation’ is closely connected with the classical problem of minimizing a Sobolev seminorm under interpolatory constraints; the intrinsic structure of…

The computational cost of simplex spline functions

- Mathematics
- 1987

This paper presents a negative result concerning the stable evaluation of simplex spline functions. It has been conjectured that a great deal of computational effort might be saved by implementing…

On Pólya frequency functions IV: The fundamental spline functions and their limits

- Mathematics
- 1966

The present paper was written in 1945 and completed by 1947 (see the abstract [3]) but for no good reason has so far not been published. It appears now in a somewhat revised and improved form.