Hyperbolic cross approximation is a special type of multivariate approximation. Recently, driven by applications in engineering, biology, medicine and other areas of science new challenging problems have appeared. The common feature of these problems is high dimensions. We present here a survey on classical methods developed in multivariate approximation theory, which are known to work very well for moderate dimensions and which have potential for applications in really high dimensions. The theory of hyperbolic cross approximation and related theory of functions with mixed smoothness are under detailed study for more than 50 years. It is now well understood that this theory is important both for theoretical study and for practical applications. It is also understood that both theoretical analysis and construction of practical algorithms are very difficult problems. This explains why many fundamental problems in this area are still unsolved. Only a few survey papers and monographs on the topic are published. This and recently discovered deep connections between the hyperbolic cross approximation (and related sparse grids) and other areas of mathematics such as probability, discrepancy, and numerical integration motivated us to write this survey. We try to put emphases on the development of ideas and methods rather than list all the known results in the area. We formulate many problems, which, to our knowledge, are open problems. We also include some very recent results on the topic, which sometimes highlight new interesting directions of research. We hope that this survey will stimulate further active research in this fascinating and challenging area of approximation theory and numerical analysis.

Hyperbolic Cross Approximation

No Child Left Behind Act (2001)

Some new conditions that arise naturally in the study of the Thresholding Greedy Algorithm are introduced for bases of Banach spaces. We relate these conditions to best n-term approximation and we study their duality theory. In particular, we obtain a complete duality theory for greedy bases.

The Thresholding Greedy Algorithm, Greedy Bases, and Duality

1. Introduction 2. Fundamentals of Fourier Analysis 3. Fourier Analysis in Systems Theory 4. Fourier Transforms in Probability, Random Variables and Stochastic Processes 5. The Sampling Theory 6. Generalizations of the Sampling Theorem 7. Noise and Error Effects 8. Multidimensional Signal Analysis 9. Time-Frequency Representations 10. Signal Recovery 11. Signal and Image Synthesis: Alternating Projections Onto Convex Sets 12. Mathematical Morphology and Fourier Analysis on Time Sales 13. Applications 14. Appendices 15. Reference

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Handbook of Fourier Analysis & Its Applications

In this survey, results on the representation of a function as an absolutely convergent Fourier integral are collected, classified and discussed. Certain applications are also given.

The Wiener algebra of absolutely convergent Fourier integrals: an overview

1. Representation Theorems.- 1.1 Theorems on representation at a point.- 1.2 Integral operators. Convergence in Lp-norm and almost everywhere.- 1.3 Multidimensional case.- 1.4 Further problems and theorems.- 1.5 Comments to Chapter 1.- 2. Fourier Series.- 2.1 Convergence and divergence.- 2.2 Two classical summability methods.- 2.3 Harmonic functions and functions analytic in the disk.- 2.4 Multidimensional case.- 2.5 Further problems and theorems.- 2.6 Comments to Chapter 2.- 3. Fourier Integral.- 3.1 L-Theory.- 3.2 L2-Theory.- 3.3 Multidimensional case.- 3.4 Entire functions of exponential type. The Paley-Wiener theorem.- 3.5 Further problems and theorems.- 3.6 Comments to Chapter 3.- 4. Discretization. Direct and Inverse Theorems.- 4.1 Summation formulas of Poisson and Euler-Maclaurin.- 4.2 Entire functions of exponential type and polynomials.- 4.3 Network norms. Inequalities of different metrics.- 4.4 Direct theorems of Approximation Theory.- 4.5 Inverse theorems. Constructive characteristics. Embedding theorems.- 4.6 Moduli of smoothness.- 4.7 Approximation on an interval.- 4.8 Further problems and theorems.- 4.9 Comments to Chapter 4.- 5. Extremal Problems of Approximation Theory.- 5.1 Best approximation.- 5.2 The space Lp. Best approximation.- 5.3 Space C. The Chebyshev alternation.- 5.4 Extremal properties for algebraic polynomials and splines.- 5.5 Best approximation of a set by another set.- 5.6 Further problems and theorems.- 5.7 Comments to Chapter 5.- 6. A Function as the Fourier Transform of A Measure.- 6.1 Algebras A and B. The Wiener Tauberian theorem.- 6.2 Positive definite and completely monotone functions.- 6.3 Positive definite functions depending only on a norm.- 6.4 Sufficient conditions for belonging to Ap and A*.- 6.5 Further problems and theorems.- 6.6 Comments to Chapter 6.- 7. Fourier Multipliers.- 7.1 General properties.- 7.2 Sufficient conditions.- 7.3 Multipliers of power series in the Hardy spaces.- 7.4 Multipliers and comparison of summability methods of orthogonal series.- 7.5 Further problems and theorems.- 7.6 Comments to Chapter 7.- 8. Summability Methods. Moduli of Smoothness.- 8.1 Regularity.- 8.2 Applications of comparison. Two-sided estimates.- 8.3 Moduli of smoothness and K-functionals.- 8.4 Moduli of smoothness and strong summability in Hp(D), 0erences.- Author Index.- Topic Index.

Fourier Analysis and Approximation of Functions

The main subject of this chapter is the study of best approximation either to separate functions or to classes of functions by polynomials of given degree as well as by approximants from other subspaces. In Section 5.1, we not only outline, in subsections A, B, and C, the precise setting of the investigated problems but also discuss questions of existence and uniqueness of best approximation and give a criterion of best approximation. In Section 5.2 we introduce a specific definition for the space L p (Ω, μ) , while for C on the compact the same procedure is done in Section 5.3. In Section 5.5 we discuss best approximation to classes of functions by polynomials and by entire functions of exponential type. In Section 5.4 extremal properties of splines (5.4.9 – 5.4.12) are given. We are going to use them further on, in Chapter 10. We also study the properties of polynomials (5.4.1 – 5.4.8 and 5.4.13 – 5.4.14) concerning best approximation to a constant by algebraic polynomials with integral coefficients (5.4.15 – 5.4.16).

Extremal Problems of Approximation Theory

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Мультипликаторы в пространствах Харди $H_p(D^m)$ при $p\in (0,1]$ и аппроксимативные свойства методов суммирования степенных рядов@@@Multipliers in the Hardy spaces $H_p(D^m)$ with $p\in (0,1]$ and approximation properties of summability methods for power series

We study the approximation of functions by linear polynomial means of their Fourier series with a function-multiplier φ that is equal to 1 not only at zero, in contrast with classical methods of summability. The exact order of convergence to zero of the sequence

$$ \mathop{\max}\limits_{{x\in \left[ {-\pi, \pi } \right]}}\left| {f(x)-\sum\limits_{{\left| k \right|\leq n}} {\varphi \left( {\frac{{k\pi }}{n}} \right){{\hat{f}}_k}{e^{ikx }}} } \right| $$

($ {{\hat{f}}_k} $ Fourier coefficients) as n→∞ is obtained. The answer is given in terms of the values of difference operators of a continuous function f and a special K-functional (step of $ \frac{\pi }{n} $). In addition, we obtain not only the sufficient conditions for φ but the necessary ones as well.

On the nonclassical approximation method for periodic functions by trigonometric polynomials

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Roald Mikhailovich Trigub

Papers

Fourier Analysis and Approximation of Functions

Extremal Problems of Approximation Theory

On the nonclassical approximation method for periodic functions by trigonometric polynomials

Положительно определенные сплайны специального вида@@@Positive-definite splines of special form