Hyperbolic Cross Approximation

1 Introduction

• Hyperbolic cross polynomials play the multivariate role of univariate trigonometric polynomials and are closely connected to mixed derivatives and mixed differences.This connection motivates the study of function classes with dominating mixed smoothness and their approximation by hyperbolic crosses.Source: §1 · pp. 4–5
• Korobov’s cubature formulas achieve accuracy of order m^-r(log m)^rd for functions with bounded mixed derivatives, nearly matching the Sobolev-class rate despite the broader class.Fibonacci cubature formulas are exact on associated hyperbolic cross polynomials and are optimal in order for bounded mixed derivatives and mixed differences.Source: §1 · p. 5
• Hyperbolic cross problems remain substantially harder than univariate analogues, especially in L1 and L∞, because key kernel bounds and discretization tools fail or grow with dimension.These difficulties leave open problems for Bernstein inequalities, widths, sampling recovery, entropy numbers, and numerical integration.Source: §1 · pp. 5, 7, 10
• The survey covers widths, entropy numbers, nonlinear approximation, numerical integration, and sparse grids, with mixed-derivative classes serving as models for scientific-computing problems.Applications include elliptic variational problems, mathematical-physics equations, stochastic PDEs, data mining, and finance.Source: §1 · pp. 6–8
• For 2 ≤ p < q < ∞, best m-term approximation is substantially better than hyperbolic cross approximation, while wavelet-type bases are often order-optimal among orthogonal bases.Greedy algorithms are described as effective constructive methods for many dictionaries.Source: §1 · pp. 8–9
• The cited discussion gives a typical error bound m^-1(log m)^(d−1), compared with regular-grid error of order m^-1/d, and relates these results to numerical integration.Source: §1 · p. 9
• The survey generalizes classical univariate approximation to multivariate hyperbolic cross approximation and distinguishes W-type derivative classes from H- and B-type mixed-difference classes.H-type classes are presented as a special case of B-type classes.Source: §1 · p. 11
• For d ≥2, asymptotic characteristics measured in Lq can differ across mixed-smoothness classes, with H-type classes substantially larger than their W-type counterparts.The discussion motivates studying B-type classes and the influence of their secondary parameter θ.Source: §1 · p. 11
• The survey develops anisotropic and quasi-Banach mixed-smoothness theory, direct and inverse theorems, widths, nonlinear approximation, and links to contemporary numerical analysis including Monte Carlo methods.It also covers other widths such as Weyl and Bernstein numbers and continuous algorithms in m-term approximation.Source: §1 · p. 11
• High-dimensional analysis examines dimension-dependent constants, convergence rates, preasymptotic Hilbert-space estimates, and ε-dimensions for mixed-smoothness classes and hyperbolic cross approximations, including parametric and stochastic elliptic PDEs.The section notes applications in areas such as quantum chemistry and meteorology, with d potentially very large or infinite.Source: §1 · pp. 11–12

Outstanding open problems

• The outstanding problems ask for sharp orders of Bernstein inequalities, the Small Ball Inequality, Kolmogorov widths, sampling recovery, entropy numbers, best m-term approximation, and numerical integration errors.The listed questions include dimensions d ≥3, several ranges of p and q, and small smoothness regimes.Source: Outstanding open problems · p. 12

2.1 Univariate polynomials

• Trigonometric polynomials of order n are defined using complex coefficients c_k, a_k, and b_k.The displayed form includes cosine and sine terms, as shown in (2.1).Source: §2.1 · p. 14
• The notation T(n) denotes all such trigonometric polynomials, while RT(n) denotes the subset of real polynomials.Source: §2.1 · p. 14
• The section proceeds by considering concrete polynomials that are important in approximation theory, including the Dirichlet kernel of order n.Source: §2.1 · p. 14, §1 · p. 14

• The Dirichlet kernel is an even trigonometric polynomial satisfying ∥D_n∥_1 ≤ C ln n for n = 2, 3, … .Source: §1 · p. 14
• The Fourier partial-sum operator is represented by convolution with the Dirichlet kernel: S_n(f) = f ∗ D_n.Convolution is denoted by f ∗ g.Source: §1 · pp. 14–15
• Theorem 2.1 states that S_n preserves polynomials from T(n) and provides endpoint results for p = 1 or ∞.The theorem is derived from the Dirichlet-kernel bound and the Marcinkiewicz multiplier theorem.Source: §1 · p. 15

• The Fejér kernel is an even trigonometric polynomial in T (n −1).Source: §2 · p. 15
• It is nonnegative and comes with a majorant.Source: §2 · p. 15
• The section also invokes obvious relations for the kernel.Source: §2 · p. 15

• The kernels are represented in terms of the Fejér kernels, yielding an associated estimate.Source: §3 · p. 16
• The de la Vallée Poussin kernels are even trigonometric polynomials of order n −1 with a stated majorant.Source: §3 · p. 16
• The section specializes to Vm(x) := Vm,2m(x) for m ≥1 and sets V0(x) = 1.Source: §3 · p. 16
• The de la Vallée Poussin operator Vm is defined on L1 using the corresponding kernel.Source: §3 · p. 16
• Theorem 2.2 states that Vm preserves polynomials from T (m) and provides bounds for all 1 ≤p ≤∞.Source: §3 · p. 16
• For every natural number N, the section asserts the existence of a polynomial of a specified form.Source: §4 · p. 17

2.2 Multivariate polynomials

• Rudin–Shapiro polynomials have Fourier coefficients of absolute value one and satisfy ∥R_N∥∞≤C(d)ϑ(N)^1/2.Their coefficient structure resembles Dirichlet kernels, while their Lp norms behave differently.Source: §2.2 · p. 19

2.3 Hyperbolic cross polynomials

• For hyperbolic-cross polynomials, the square-function operator satisfies ∥S_Qn∥p→p≤C(p, d) for 1 < p < ∞.This estimate follows from Corollary 11.8 and the Littlewood–Paley theorem.Source: §2.3 · p. 22
• Unlike parallelepiped de la Vallée Poussin kernels, the hyperbolic-cross kernels V_Qn do not have uniformly bounded L1 norms, complicating L1- and L∞-approximation.The absence of such kernels is identified as a new phenomenon for hyperbolic cross polynomials.Source: §2.3 · p. 22

2.4 The Bernstein-Nikol’skii inequalities

• The section defines the operator D_r^α for r ≥ 0 and α ∈ R on trigonometric polynomials.This begins the treatment of the Bernstein–Nikol’skii inequalities.Source: §1 · p. 22

Definitions

• The (r, α)-derivative is defined via an operator that has an inverse on each T^(n), distinguishing it from ordinary differentiation.This inverse-operator property is used for the generalized derivative framework.Source: §1 · p. 22

One-dimensional inequalities

• Theorem 2.6 establishes the Bernstein inequality for trigonometric polynomials t ∈ T^(n) for r > 0, α ∈ R, and 1 ≤ p ≤ ∞.The theorem is stated for the generalized derivative setting introduced above.Source: §1 · p. 22

Multidimensional extension

• Theorem 2.7 extends the Bernstein inequality to multidimensional trigonometric polynomials T^(N, d), using vector parameters r and α with α_j = 0 whenever r_j = 0.When all r_j are equal, the scalar r is used in place of the vector.Source: §1 · p. 23

Hyperbolic cross extension

• The multidimensional Bernstein inequalities are sharp, but extending them to hyperbolic cross polynomials is nontrivial and reveals a new phenomenon.The hyperbolic-cross extension has different forms for p = ∞ and 1 < p < ∞.Source: §1 · p. 23

Known results and limitations

• For hyperbolic cross polynomials, the p = ∞ upper bound was obtained by Babenko, its matching lower bound by Telyakovskii, and the 1 < p < ∞ case by Mityagin.The correct Bernstein inequality for p = 1 remains unknown; a result for d = 2 was proved in.Source: §1 · p. 23

The Nikol’skii inequalities

• Theorem 2.10 extends the univariate Nikol’skii inequalities to polynomials in T (N, d) for 1 ≤ q ≤ p ≤ ∞.The multivariate inequality applies for N > 0.Source: §2 · p. 23
• Theorem 2.11 gives relations for function classes defined through an array ε, with constants independent of ε.These relations are used in embedding-type inequalities, and correspond to Lemmas 3.13 and 3.14.Source: §2 · p. 24
• Theorem 2.11 was proved in and is also cited in, Ch.1, Theorem 3.3.Its relations are formulated as embeddings that support later results in Section 3.Source: §2 · p. 24
• Theorem 2.14 addresses Nikol’skii inequalities for T (N) when 1 ≤ q ≤ p < ∞, p > 1, and r ≥ 0.The q = 1 case is identified as nontrivial, and the theorem is attributed to, Ch.1, Section 2.Source: §2 · p. 24

3. The Marcinkiewicz theorem.

• Marcinkiewicz theorems establish equivalences between continuous and lattice norms for trigonometric polynomials, including the one-dimensional cases 1 < p < ∞ and 1 ≤ p ≤ ∞ and a multidimensional mixed-norm result.The multidimensional constant depends only on d, while no analogous theorem holds for polynomials from T(N).Source: §3 · p. 25

2.5 Volume estimates

• Volume estimates characterize frequency-limited trigonometric-polynomial spaces, including general finite sets, rectangular frequency sets, hyperbolic layers, and two-dimensional hyperbolic crosses.The difficult endpoint is the lower estimate for p = ∞, and hyperbolic-cross estimates differ between p = ∞ and 1 ≤ p < ∞.Source: §2.5 · pp. 26–27

2.6 Riesz products and the Small Ball Inequality

• The small-ball inequality is conjectured for coefficients {ck}; a weaker version follows from Parseval, Cauchy’s inequality, and Lp-norm monotonicity, while (2.28) remains unproved.A result proves (2.29) with exponent (d −1)/2 −δ(d), where 0 < δ(d) < 1/2, instead of (d−2)/2; inequality (2.27) plays a key role in lower bounds for entropy numbers.Source: §2.6 · pp. 29–30

2.7 Comments and open problems

• The comments identify open problems on the orders of several quantities and ask for proofs of the Small Ball Inequalities (2.28) and (2.29) for d ≥3.Theorems 2.15 and 2.16 are described as classical Marcinkiewicz variants, while Theorem 2.17 is attributed to.Source: §2.7 · pp. 30–31

3.1 Spaces of functions with bounded mixed derivative

• Bernoulli-kernel representations define W^r_{p,α} spaces, generalizing bounded mixed derivatives to fractional smoothness and multivariate settings.For integer r, these spaces admit descriptions via weak mixed partial derivatives; arbitrary real smoothness is handled using Weil fractional derivatives.Source: §3.1 · pp. 32–33

3.2 Spaces of functions with bounded mixed difference

• Mixed-difference spaces H^r_p and Besov spaces B^r_{p,θ} define dominating mixed smoothness through mixed moduli, with norms independent of the auxiliary integer m up to equivalent quasi-norms.Zero-mean subspaces yield an ANOVA-like decomposition and do not affect general multivariate approximation, although dimension dependence can matter in high-dimensional approximation.Source: §3.2 · pp. 33–35

3.3 Characterization via Fourier transform

• Fourier-side dyadic decompositions characterize H^r_p and B^r_{p,θ}, while the difference and Fourier definitions of Besov spaces are equivalent for p, θ ≥1.For p=1 and p=∞, unbounded dyadic operators are replaced by A_s operators; for 0<p≤1, smooth dyadic decompositions extend the characterization, with Banach versus quasi-Banach status determined by min{p,q}.Source: §3.3 · pp. 35–37

3.4 Embeddings

• The embedding results distinguish non-compact embeddings from compactness when r>1/p, while the critical case r=1/p remains non-compact for θ≤1.The section also states non-trivial embeddings between Besov and W-spaces under smoothness threshold r≥1/p−1/u.Source: §3.4 · pp. 37–38
• Lemma 3.16 establishes a family of continuous embeddings for r > 0 and α ∈ R.Source: §3.4 · p. 38
• For p = 1, the characterization of the B classes uses operators A_s instead of δ_s, with the stated embeddings derived from earlier results and B_r characterizations.Source: §3.4 · p. 38
• The embedding W_r^{1,α} into B_r^{1,2} fails, and the failure persists even with B_r^{1,θ} for 2 < θ < ∞ on the right-hand side.The cited passage identifies this as a limitation of the formal counterpart of (3.13).Source: §3.4 · p. 38
• The embedding (3.15) is sharp, while the embeddings in Lemma 3.16 are strict; the tensorized and periodized hat function provides an example discussed with Figure 11.Source: §3.4 · p. 38

4.1 Introduction

• Kolmogorov-width considerations identify hyperbolic cross polynomials T(Γ(N)) as optimal L2 approximation subspaces when m=|Γ(N)|.For broader Lp settings, smooth crosses are replaced by step crosses Qn, with operators such as SQn and VQn providing important approximation methods.Source: §4.1 · p. 39
• The optimality of Fourier and linear approximation methods depends on (p,q): Vn realizes relevant orders in D1 and D2, but not in D3, while other linear operators work in D4.For most (p,q), Sn is optimal among Fourier operators, whereas better approximation in some cases sacrifices useful properties such as orthogonal projection or uniform L2 boundedness.Source: §4.1 · p. 41
• For mixed-smoothness classes, T(Qn) and SQn extend the hyperbolic-cross framework, but approximation in L1 and L∞ is substantially complicated and remains partly open.The multivariate analogue VQn is weaker than the univariate de la Vallée Poussin kernel, while SQn plays the corresponding role for 1<p<∞.Source: §4.1 · p. 42

The Smolyak algorithm

• Smolyak-type algorithms combine univariate operators through Boolean-sum constructions, and under approximation and reproduction conditions they yield ∥∆s(f)∥p ≲2−r|s|1∥f∥Hrp and ∥f−Tn(f)∥p ≲2−rnnd−1∥f∥Hrp.The framework covers SQn, VQn, and related operators, with extensions to additional mixed-smoothness classes noted.Source: The Smolyak algorithm · pp. 42–43

Approximation from the hyperbolic cross

• Best approximation from T(Qn) has parameter-dependent orders for Wr p in Lq, with complete results mainly available for 1<p,q<∞ and less complete endpoint cases.The uniform-metric bounds for Bernoulli kernels can determine the order without constructing an approximating polynomial, and endpoint formulas include 2−nr and 2−n(r−1/p) under stated conditions.Source: Approximation from the hyperbolic cross · pp. 43–44
• For Hr p classes, finding the right decay orders of EQn(Hr p)q is more difficult than the corresponding problem for Wr p classes, and even 1 < p, q < ∞ requires a new technique.Source: Approximation from the hyperbolic cross · p. 44
• In the multivariate setting, least upper bounds for best approximation in T(Qn) differ between Wr p and Hr p, unlike the univariate case where their orders coincide across all 1≤p,q≤∞.This contrast is identified as a specific feature of multidimensional hyperbolic-cross approximation.Source: Approximation from the hyperbolic cross · p. 45
• When p=q=1, the W and H classes have alike best-approximation behavior by hyperbolic-cross polynomials.This equivalence is stated through Theorems 4.5 and 4.6.Source: Approximation from the hyperbolic cross · p. 45
• For d≥2 and 1<p<∞, no linear method matches the best uniform approximation order for W^r_p classes, unlike the Lq setting where partial Fourier sums are order-optimal.The nonlinear best-approximation operator cannot be replaced by any linear operator without worsening the order in the uniform metric.Source: Approximation from the hyperbolic cross · p. 45
• For Besov classes B^r_{p,θ}, partial Fourier sums realize the upper bounds, while the asymptotic order can acquire the factor n^(1/min{2,p}−1/θ)(d−1) when q≤p and θ>min{2,p}.The cited theorems cover distinct ranges of p, q, θ, and r, with lower bounds obtained through embeddings, univariate reductions, and fooling functions.Source: Approximation from the hyperbolic cross · p. 46

4.3 The Kolmogorov widths

• Kolmogorov-width orders for W^r_p in Lq are established by Theorem 4.13 under r>r(p,q), with hyperbolic-cross subspaces optimal in R1 but not in R2.The parameter region R1 is covered by hyperbolic-cross approximation, whereas Theorem 4.13 shows nonoptimality in the complementary region R2.Source: §4.3 · pp. 47, 49
• For 2≤p≤q<∞, the main Kolmogorov-width convergence rate depends on r but neither on p nor on q, reflecting a multivariate phenomenon linked to Kashin’s discretization technique.The analysis uses finite-dimensional normed-space widths and probabilistic matrix constructions.Source: §4.3 · pp. 47–48
• The Kolmogorov-width theory is partly non-constructive because key probabilistic RIP-type matrices are known existentially, making deterministic near-optimal constructions an open problem.These matrices connect the width results to compressed sensing.Source: §4.3 · p. 48
• The width problem remains unresolved on several boundary regions, while for d≥3 the conjectured order is supported by known upper and lower bounds.Figure 4 distinguishes regions with known orders from dashed regions where the correct order is unknown; Conjecture 4.21 addresses d≥3.Source: §4.3 · pp. 49–50, Figure 4 · p. 50
• Kolmogorov-width orders are known in several parameter regions, while the P2 region and most remaining cases remain unresolved; a conjecture is proposed for d ≥3, 2 ≤p ≤∞, and r > 1/2.The subspaces T(Qn) are optimal in order in P1, while P3 and specified boundary segments are covered by the stated theorems; the general picture leaves multiple cases open.Source: §4.3 · pp. 51–52
• For Besov-type classes Br_{p,θ}, Theorems 4.28 and 4.29 give Kolmogorov-width estimates across several p,q,θ regimes, with smoothness restrictions such as r > β in the case 2 ≤p < q.The proofs combine linear approximation by SQn, embeddings, finite-dimensional width estimates, and Kashin-Gluskin, Pietsch-Stesin, and related lemmas.Source: §4.3 · p. 52

4.4 Orthowidths

• The orthowidth problem remains difficult for Hr_p when 1 ≤q < p < 2, and the corresponding right order of the Kolmogorov widths is still unknown.Even the related EQn(Hr_p) result required a new technique.Source: §4.4 · p. 55
• For approximating classes Wr_p in Lq by rank m = |Qn| orthogonal-projection operators, SQn is best in order when (p, q) differs from (1, 1) and (∞, ∞).Source: §4.4 · p. 56

4.5 The linear widths

• Linear-width orders for Wr_p are established under smoothness and parameter restrictions, with T(Qn) optimal on R1 but not in the small square (1/2, 1) × (0, 1/2); other cases remain open.Theorem 4.45 covers the stated parameter regions, while Theorem 4.46 supplies further estimates for 1 ≤p,q ≤∞ and r > (1/p −1/q)+.Source: §4.5 · pp. 57–58
• The linear-width results include additional parameter-specific theorems, but several boundary cases remain open, including 1 ≤ q ≤ p < 2 and λ_m(H^r_p,L_∞) for d ≥ 3.Theorem 4.47 applies for r > (1/p − 1/q)+ with 1 < p,q < ∞, while Theorems 4.48–4.50 cover further restrictions on p, q, θ, and r.Source: §4.5 · pp. 58–60
• Theorem 4.46 covers R1 and R2, with T(Q_n) order-optimal in R1 but not order-optimal in R2 under restrictions on r.The right orders of λ_m(H^r_p,L_q) are known in these regions under the theorem’s restrictions, while all other cases remain unresolved.Source: §4.5 · p. 59
• For p ≥ 2 or q ≤ 2, upper bounds use SQ_n and lower bounds use Kolmogorov-width estimates, whereas p ≤ 2 < q relies on finite-dimensional linear-width bounds and Gluskin’s lemma.Additional lower-bound reductions use norm inequalities such as ∥·∥_q ≥ ∥·∥_2.Source: §4.5 · p. 60

4.6 Open problems

• Open problems concern missing orders of Kolmogorov widths for W^r and H^r classes, particularly when p or q equals 1 or ∞ and for H^r_p,L_q with 1 ≤ q < p < 2.The section also highlights unresolved uniform-norm approximation by hyperbolic-cross polynomials, including EQ_n(W^r_q)_∞ for 2 < q ≤ ∞ and EQ_n(H^r_∞)_∞ for d ≥ 3.Source: §4.6 · pp. 60–61
• Open Problem 4.8 asks for the right order of λ_m(H^r_p,L_q) in the missing cases identified in Figure 7.These unresolved linear-width cases complement the section’s known results for R1 and R2.Source: §4.5 · p. 59, §4.6 · p. 61

5 Sampling recovery

• Sampling widths of hyperbolic-cross subspaces are optimal in order whenever their right orders are known, although no general relation with orthowidths is available.Sampling operators impose point-evaluation restrictions distinct from orthogonal projections, and some formal width inequalities require care because sampling operators need not act from Lp to Lp.Source: §5 · p. 62
• In one dimension, trigonometric polynomials are optimal recovery subspaces, and sampling and orthowidths have the same order under the stated parameter restrictions.For W^r_p classes, the optimal-recovery result holds for 1 ≤ p, q ≤ ∞ with r > 1/p; the orthowidth comparison excludes (p, q) = (1, 1) and (∞, ∞).Source: §5 · p. 63
• For 1 ≤ p < q < ∞, the Smolyak sampling operator with m ≍ |Qn| achieves the best approximation order for Hölder–Nikol’skii classes, while several other regimes remain unresolved.The correct order is known for 1 < p < q ≤ 2 and, for Sobolev classes, under additional parameter restrictions; sharpness and equality with linear widths remain open in important cases.Source: §5 · pp. 63–65

5.1 The univariate de la Vallée Poussin interpolation

• The de la Vallée Poussin interpolant on the equidistant grid J3m reproduces every trigonometric polynomial of degree at most m and provides the univariate building block for sparse-grid sampling.Its grid choice also gives the interpolation property, while classical trigonometric interpolation can fail to attain optimal recovery rates at endpoint exponents p = 1 and p = ∞.Source: §5.1 · pp. 65–66

5.2 Frequency-limited sampling representations - discrete Littlewood-Paley

• Discrete dyadic differences of de la Vallée Poussin sampling operators yield frequency-limited representations that characterize Besov and Sobolev classes under the stated smoothness conditions.The construction tensors the univariate differences across coordinates, and Proposition 5.3 provides the corresponding norm relations for Besov classes and for continuous functions in the Sobolev setting.Source: §5.2 · pp. 66–67
• These discrete sampling representations convert function-class descriptions into bounds for Smolyak sampling approximation, with extensions to more general Smolyak algorithms.The approach replaces continuous Littlewood–Paley convolutions by discrete convolution blocks and uses Fourier-analytic tools, including Peetre- and Hardy–Littlewood-type maximal functions for one direction.Source: §5.2 · pp. 66–68
• Frequency-limited representations characterize sampling recovery through discrete Littlewood-Paley decompositions under the stated smoothness and integrability conditions.Proposition 5.4 gives the corresponding membership and representation conclusions for the admissible sequences of trigonometric polynomials.Source: §5.2 · p. 68

5.3 Sampling on the Smolyak grids

• Smolyak-grid operators preserve hyperbolic-cross trigonometric polynomials, interpolate at grid points, and use grids with cardinality asymptotic to 2^n n^(d−1).The resulting frequency- and time-limited operators achieve the stated recovery bounds, while T_n and R_n are order-optimal among linear hyperbolic-cross approximations; no sampling operator using these grids improves the bounds.Source: §5.3 · pp. 69–72
• For Smolyak-grid recovery, Theorems 5.5–5.9 provide bounds across Sobolev, H-, and Besov-type classes, with rates depending on p, q, r, and θ.The q = ∞ case introduces an extra logarithmic term, and the results extend from frequency-limited T_n to quasi-interpolating time-limited R_n under appropriate B-spline smoothness restrictions.Source: §5.3 · pp. 70–72

5.4 Sampling widths

• For Smolyak sampling numbers ϱ_m, upper bounds transfer directly from earlier estimates; sharp results hold in some cases, but sharp lower bounds are unavailable in most situations, and for d > 2 the correct order is unknown.Source: §5.4 · pp. 74–75

5.5 Time-limited sampling representations–B-splines

• Time-limited sampling replaces trigonometric polynomials with tensor products of compactly supported hat functions and B-splines, supporting non-periodic mixed-smoothness spaces and fooling-function constructions.The approach is motivated by Triebel and Dinh D˜ung, with Faber-Schauder systems as a special B-spline case.Source: §5.5 · p. 76

The Faber-Schauder basis

• The Faber-Schauder construction represents continuous functions pointwise and extends componentwise to tensor products for d-variate periodic functions.The tensor system uses products v_s,k(x) = v_s1,k1(x_1) · ... · v_sd,kd(x_d) indexed by multilevel coordinates.Source: The Faber-Schauder basis · p. 76, The tensor Faber-Schauder system · p. 77

The Faber-Schauder basis for Besov spaces

• For Besov spaces, the Faber-Schauder system yields sequence-space representations and converse results under the parameter ranges stated in Propositions 5.18 and 5.19.The results discretize the spaces through coefficient mappings and establish both representation and converse directions.Source: The Faber-Schauder basis for Besov… · p. 77

The Faber-Schauder basis for Sobolev spaces

• For Sobolev spaces, tensorized Faber-Schauder representations hold in specified smoothness ranges and imply that the system is an unconditional basis in the corresponding parameter domain.The stated parameter region may be sharp, with a fundamental difference from the Besov case reflected in unresolved small-smoothness recovery problems.Source: The Faber-Schauder basis for Sobolev… · pp. 78–79

B-spline representations and general atoms

• B-spline quasi-interpolation replaces the Faber-Schauder smoothness restriction r < 2 and provides Besov and Sobolev representations over broader smoothness ranges.The coefficient functionals use finitely many function values, while the Sobolev proof relies on Fourier-analytic maximal-function tools and explicit coefficient formulas.Source: B-spline representations and general… · pp. 79–80

5.6 Open problems

• The section leaves several sampling-recovery orders open, including Besov and Sobolev cases, small smoothness, and regions where sampling numbers may differ from linear widths.In one dimension, Theorem 5.26 shows that sampling numbers and linear widths do not coincide; further conjectures concern sampling optimality and Smolyak’s algorithm.Source: §5.6 · p. 81

6.1 General notions and inequalities

• Entropy numbers quantify compact-set size through covering numbers and support inequalities relating them to Kolmogorov widths, nonlinear widths, and operator entropy numbers.The framework includes volume-based finite-dimensional bounds, best m-term approximation applications, and multiplicative inequalities for Lp spaces.Source: §6.1 · pp. 83, 84, 86

6.2 Entropy numbers for W classes in Lq

• For mixed-derivative W classes, entropy-number rates are established for broad parameter ranges, while endpoint cases remain difficult and several d > 2 rates are unknown.In the univariate case, the decay rate depends on smoothness r rather than p or q; for multivariate classes, different parameter pairs require distinct methods.Source: §6.2 · pp. 86–87

Entropy numbers in L∞

• In dimension two, W-class entropy rates differ across (p, q), whereas higher-dimensional L∞ cases retain major open problems connected to Small Ball Inequalities.For example, the d = 2 rates are n−r(log n)^r, n−r(log n)^{r+1/2}, and n−r(log n)^{r+1} in the three specified parameter regimes.Source: Entropy numbers in L∞ · pp. 87–88

6.3 Entropy numbers for H and B classes in Lq

• For H and B classes, theorems determine the right entropy-number order for nearly all Lq settings, with a remaining q = ∞ result under explicit p and r conditions.The stated results cover all 1 ≤ p, q ≤∞ except q =∞, followed by a theorem for 1 < p <∞ and r > max{1/p, 1/2}.Source: §6.3 · pp. 88–89
• For d = 2, the upper bounds for the H and B classes in Lq are sharp in order, while corresponding higher-dimensional behavior is conjectured.Theorem 6.22 covers 1 ≤ p ≤ ∞ and r > 1/p; its difficult p = ∞ lower bound was proved in.Source: §6.3 · p. 89
• The entropy-number upper bounds use discretization, finite-dimensional entropy estimates, wavelet isomorphisms, and interpolation, whereas lower bounds rely on volume estimates for Fourier coefficients and related small-ball inequalities.Theorem 6.3 and Theorem 2.18 are key tools; Theorem 6.24 was proved in the more general non-periodic setting in.Source: §6.3 · pp. 89–90
• Theorem 6.24 gives entropy-number results for B^r_p,θ classes in Lq under conditions on r, p, q, and θ, with θ influencing the logarithmic power.For small smoothness, the influence of r can be dominated by θ, and the correct order remains unknown in the stated regime.Source: §6.3 · pp. 89–90

6.4 Entropy numbers and the Small Ball Problem

• For the two-dimensional Brownian sheet, small-ball probabilities are tightly related to entropy H_ε(W^1_2, L∞), linking probability estimates with entropy-number bounds.Kuelbs and Li established the general Gaussian-measure relationship, while the difficult lower bound uses Talagrand’s Haar-polynomial inequality.Source: §6.4 · p. 91

6.5 Open problems

• Open problems concern d = 2 at low smoothness, higher-dimensional entropy orders for W^r classes, and B^r_p,θ entropy numbers when 0 < θ < p ≤ 2 and smoothness is small.The unresolved cases include W^r_∞ in L∞, W^r_1 in Lq, W^r_p in L∞, and B^r_p,θ in L2.Source: §6.5 · p. 92

7.1 Introduction

• Best m-term approximation selects function-dependent basis elements or dictionary atoms, providing efficient sparse representations for applications including image, signal, data processing, and neural networks.The section distinguishes basis-based nonlinearity from the more complicated use of redundant dictionaries and defines optimization over admissible approximation methods.Source: §7.1 · p. 93
• For every 1 ≤ p < 2, an orthonormal system can satisfy σ_1(f, Φ)_L_p = 0 for every f ∈ L_p, so nontrivial lower bounds require additional restrictions on the dictionary.This limitation contrasts with the useful optimality and greedy-approximation results obtained for structured systems such as U_d.Source: §7.1 · p. 94

7.2 Orthogonal bases

• The tensor-product orthogonal dictionary U_d is optimal in order among orthogonal dictionaries for approximating H^r_p classes in L_q under the stated parameter ranges, and near-best approximants can be produced by G_q(·, U_d).The results cover 1 < p, q < ∞, with the optimality comparison explicitly stated for 2 ≤ q < ∞; wavelet and Faber-Schauder systems provide related sharp approximation results under additional conditions.Source: §7.2 · pp. 96–98

7.3 Some related problems

• Best m-term trigonometric approximation yields bilinear approximation rates of optimal order for some function classes, while tensor-product approximation remains largely unresolved in dimensions d > 2.In the two-variable L_2 case, bilinear approximation is closely related to singular value decomposition of the associated integral operator.Source: §7.3 · pp. 98–99

7.4 A comment on Stechkin’s lemma

• Stechkin’s lemma bounds coefficient tails through monotone sequences for 0 < p ≤ q ≤ ∞, with the case p < 1 highlighted as yielding better convergence rates.The lemma is presented as a useful tool in nonlinear approximation and extends earlier uses and proofs beyond the range 1 ≤ p ≤ 2, q = 2.Source: §7.4 · pp. 99–100
• Convex optimization improves Lemma 7.8 to constant c_p,q = 1, with c_1,2 = 1/2 enabling p-selection to optimize the bound.Choosing p depending on m can balance the quantities in Lemma 7.9.Source: §7.4 · p. 100
• The resulting two-sided estimate characterizes absolute convergence of multivariate Fourier series through trigonometric best m-term L2 approximation and identifies Wiener algebra A(T^d) with approximation spaces A_1/2.Source: §7.4 · p. 100

7.5 Sparse trigonometric approximation

• For W^r_p classes, trigonometric m-term approximation has rates m^(-r+η)(log m)^((d−1)(r−2η)) and m^(-r)(log m)^(r(d−1)) in the stated parameter regimes, with optimal order in key ranges.Greedy algorithms provide constructive upper bounds, improving conditions on r in some cases; uniform-norm bounds lack matching lower bounds.Source: §7.5 · pp. 101–103
• For W^r_p, a multivariate improvement occurs when 1 < p < q ≤ 2 through the logarithmic power, whereas small-smoothness rates can become dimension-independent asymptotically.The dimension dependence is hidden in constants, and the improvement is absent in the univariate case d = 1.Source: §7.5 · p. 102
• Trigonometric systems match wavelet-type systems for selected parameter ranges, but wavelet systems yield better m-term approximation for other 1 < p, q < ∞ regimes and for certain H classes.The comparison is attributed as possibly related to trigonometric systems being orthonormal and uniformly bounded.Source: §7.5 · p. 102
• Greedy-type methods construct explicit m-term approximants through feasible one-parameter optimization, although the infinite-dimensional procedures are theoretical rather than directly computational algorithms.The methods are built using thresholding and weak Chebyshev greedy algorithms and rely on the dictionary A-norm.Source: §7.5 · pp. 102–103

Consider the class

• For F^r classes, constructive greedy methods attain m^(-r+1−1/q)(log m)^((d−1)(r−1+2/q)) for 1 < q ≤ 2 and m^(-r+1/2)(log m)^(r(d−1)) for 2 ≤ q < ∞.Source: Consider the class · p. 105
• For H^r_p,θ and B^r_p,θ classes, the stated rates depend on p, q, r, and θ; B-class logarithms can disappear at small smoothness, while dimension affects only constants.For H^r_p,θ, trigonometric systems are optimal in the stated 2 ≤ p, q < ∞ range, whereas the B-class extension is constructively attained by greedy methods.Source: Consider the class · pp. 105–106
• Theorems 7.25 and 7.26 establish best m-term approximation bounds for several smoothness and integrability regimes, including 1 ≤ p ≤ q ≤ 2 and 1 ≤ p ≤ 2 < q.For 1 < p ≤ 2 < q, Temlyakov obtained constructive greedy-type upper bounds in a special case [377], while another proof gives non-constructive bounds [273].Source: Consider the class · pp. 106–107

The case q ≤p

• For q ≤ p, sparse trigonometric approximation is optimal in some regimes, but wavelet dictionaries can improve the logarithmic convergence factor, including when p = q < 2.The improvement is a multivariate effect; for B-classes, sparse trigonometric approximation can also outperform hyperbolic-cross projections when q ≤ p.Source: The case q ≤p · pp. 107–108

7.6 Different types of Greedy Algorithms

• WCGA provides Lebesgue-type inequalities for normalized real multivariate trigonometric systems, giving progress toward constructive sparse approximation comparable to best m-term approximation.The result is the first such WCGA inequality for the trigonometric system, but an extra ln(m + 1) factor remains unresolved; analogous bounds hold for uniformly bounded orthogonal systems.Source: §7.6 · pp. 108–109

7.7 Open problems

• Open problems concern constructive rates and approximation orders for W-classes across endpoint parameter values, small smoothness, and related H- and B-classes.The section emphasizes that extreme p or q values and small smoothness remain difficult, with several stated orders of σ_m still unknown.Source: §7.7 · p. 109

8.1 The problem setting

• Numerical integration approximates the integral by weighted function values, with optimal error κ_m(F) minimized over nodes and weights for classes embedded in continuous functions.The section focuses on optimal-order dependence on the number of nodes m for bounded mixed-smoothness classes.Source: §8.1 · p. 110

8.2 Lower bounds

• Lower bounds apply to arbitrary cubature formulas and are obtained by constructing functions that vanish at the sampling nodes while retaining controlled norms and nonzero integrals.The survey presents both trigonometric-polynomial and local-bump-function fooling constructions.Source: §8.2 · p. 110, Local fooling functions · p. 112

Fooling polynomials

• Fooling polynomials yield lower bounds for Sobolev and Besov classes, with extensions to structured node sets such as (n,n−1)-nets and restricted-weight formulas.The Besov estimate has order m−r(log m)^(d−1)(1−1/θ), while net-based bounds are expressed in terms of 2^n and n.Source: Fooling polynomials · pp. 110–111

Local fooling functions

• Local fooling functions use atomic decompositions and disjoint cubes avoiding the nodes, controlling Besov norms and extending the lower-bound approach to quasi-Banach parameters, though this survey restricts proofs to Banach spaces.For p, θ ≥ 1, the resulting test functions have norms uniformly bounded in the scale parameter and produce zero cubature output.Source: Local fooling functions · p. 112

8.3 Cubature on Smolyak grids

• Smolyak-grid cubature is asymptotically worse than optimal rules, but its lower bounds are sharp because matching upper bounds exist; extra arbitrary nodes do not improve a stated bound.A level-ℓ sparse grid contains m ≍ 2^ℓℓ^(d−1) points, and its integration difficulty matches Smolyak sampling in the described setting.Source: §8.3 · pp. 113–114

8.4 The Fibonacci cubature formulas

• Fibonacci cubature formulas are order-optimal in the stated bivariate settings, providing upper bounds that match the lower estimates for the corresponding mixed-smoothness classes.They are rank-1-lattice rules exact on trigonometric polynomials supported in a hyperbolic cross.Source: §8.4 · pp. 114–115

Small smoothness and limiting cases

• Small smoothness introduces a distinct regime, including an additional log log factor at r = 1/2; sharp multivariate lower bounds for the analogous Frolov situation remain unavailable.The small-smoothness range is 2 < p < ∞ with 1/p < r ≤ 1/2, outside the earlier Fibonacci results.Source: Small smoothness and limiting cases · pp. 115–116

8.5 The Frolov cubature formulas

• Frolov cubature formulas provide upper bounds for several mixed-smoothness spaces, using a lattice construction whose axis-parallel parallelepipeds contain at most |P| + 1 lattice points.Unlike Fibonacci and Korobov rules, Frolov formulas require a two-step treatment because they are not directly designed for periodic functions.Source: §8.5 · pp. 116–117
• Chebyshev lattices are orthogonal, while Chebyshev-Frolov lattices are orthogonal and admissible exactly when d = 2^ℓ.For d = 2^ℓ, both the lattice and its dual admit orthogonal representations involving a diagonal matrix and its inverse.Source: §8.5 · p. 118
• Frolov cubature achieves the rates a−rd(log a)(d−1)/2 and a−rd(log a)(d−1)(1−1/θ) under the stated smoothness conditions, with the logarithm absent when θ=1.The first rate holds for 1 < p < ∞ and r > max{1/p, 1/2}; the second holds for 1 ≤ p, θ ≤ ∞ and r > 1/p.Source: §8.5 · p. 120

The case of small smoothness

• For 1/p < r < 1/2, modified Frolov cubature gives a−rd(log a)(d−1)(1−r), while at r = 1/2 it gives a−rd(log a)(d−1)/2√log log a.The corresponding lower bounds remain open in this small-smoothness regime.Source: The case of small smoothness · p. 120

8.6 Modifications of Frolov’s method

• A compactly supported partition function modifies Frolov’s method for periodic functions, and boundedness of the associated operator yields the corresponding integration result.The construction assumes ψ(x + k) = 1 for x ∈ R^d and integer shifts k.Source: §8.6 · pp. 120–121

Change of variable

• The Bykovskii change-of-variable modification recovers the convergence rate of Theorem 8.14 for W^r_p,θ when the smoothing parameter ℓ is sufficiently large.The required size of ℓ depends on p and r.Source: Change of variable · p. 121

Random Frolov

• Randomized Frolov cubature obtains the typical min{1/2, 1−1/p} gain in the main convergence rate, while the rooted mean-square error loses the logarithmic factor.The latter phenomenon was observed for 1 < p < ∞ and r ≥ (1/p − 1/2)+.Source: Random Frolov · p. 122

8.7 Quasi-Monte Carlo cubature

• Quasi-Monte Carlo rules use equal weights summing to one, with Korobov formulas integrating constants exactly; optimal constructions remain unresolved in higher dimensions and higher smoothness.Known results include asymptotically optimal two-dimensional van der Corput and higher-order digital-net rates, but d ≥ 2 generally retains the restriction r < 2.Source: §8.7 · pp. 122, 124

Classical discrepancy

• Discrepancy measures the distribution of point sets in the unit cube, with L∞ error representing a worst case and finite-Lp error an average case.Digital (0,n,2)-nets place exactly one point in every dyadic interval of volume 2−n.Source: §8.7 · p. 123, Classical discrepancy · p. 125
• Generalized r-discrepancy extends classical discrepancy by allowing arbitrary positive integer smoothness r and nonuniform weights, linking its decay to numerical integration error.For equal weights and r = 1, it reduces to classical Lq discrepancy; optimizing weights gives minimal weighted discrepancy no larger than minimal r-discrepancy.Source: Classical discrepancy · p. 126, General setting · p. 128

General setting

• Kernel-defined function classes and kernel discrepancies form a general numerical-integration/discrepancy duality, extending from Lp spaces to arbitrary Banach spaces and connecting integration error with m-term approximation.The approximation connection uses a special function and dictionary generated by kernel sections, motivating greedy-approximation methods.Source: General setting · pp. 127–128

Historical remarks on discrepancy

• Historical discrepancy theory established successive lower and upper bounds, while Levin’s results [218] [219] [220] show that digital nets, Halton points, and Frolov lattices satisfy the conjectured lower bound.The section also records weighted r-discrepancy lower estimates such as C(r, d, q)m^-r(log m)^(d−1)/2 for 1 < q ≤∞.Source: Historical remarks on discrepancy · pp. 128–129
• Random points achieve star-discrepancy upper bounds with nonzero probability, but classical bounds become nontrivial only beyond m > ed−1, and Smolyak nets are poor for discrepancy.The expected star-discrepancy of random point sets is reported as being of order d/m, while constructive greedy sets provide effective absolute constants.Source: Historical remarks on discrepancy · p. 130

8.9 Open problems and historical comments

• Open problems concern the optimal numerical-integration orders for large and small smoothness, the star-discrepancy order, and missing lower bounds in the small-smoothness Sobolev case.Conjecture 8.30 specifies proposed rates for 2 < p < ∞ and 1/p < r ≤1/2, including a √log log m factor at r = 1/2.Source: §8.9 · pp. 130–131
• The lower estimate in Conjecture 8.28 is best possible.This establishes sharpness of the conjectured lower bound.Source: §8.9 · p. 132
• Skriganov’s estimate gives an m^-r(log m)^(d−1)/2 rate for 1 < p ≤∞ and r ∈N.The passage states the estimate with constant dependence suppressed as C(r, d, p).Source: §8.9 · p. 132
• Combining Skriganov’s estimate with (8.35) yields the corresponding m^-r(log m)^(d−1)/2 upper-rate bound.The resulting bound holds for 1 < p ≤∞ and r ∈N.Source: §8.9 · p. 132
• Recent work [393], extended in [242], proves Theorems 8.18 and 8.19 and, with Theorems 8.1 and 8.3, implies Theorem 8.14.The extension covers periodic and non-periodic spaces on the cube [0, 1]^d; lower bounds in Theorem 8.8 were proved in and can also follow from [376].Source: §8.9 · p. 132
• Upper bounds for Smolyak cubature and further numerical-integration results are available in Triebel and related works,.These references concern Smolyak-type methods and related cubature results.Source: §8.9 · p. 132

9.1 Why classes with mixed smoothness

• Mixed-smoothness approximation connects hyperbolic cross methods to numerical integration, discrepancy, entropy, probability, sampling, and difficult open problems.Integral-equation kernels provide an a priori motivation, while cubature, entropy, and sampling problems require specialized techniques and remain partly unresolved.Source: §9.1 · pp. 133–134

9.2 Universality

• Fibonacci cubature formulas are universal and order-optimal for anisotropic Hölder-Nikol’skii classes under the stated condition g(r) > 1/p.The universal formula does not depend on the anisotropy vector r or p, while higher-dimensional universal formulas have a different character and include Frolov-type constructions.Source: §9.2 · pp. 135–136

NHr

• The orthonormal bases Ud are order-optimal for m-term approximation of mixed-smoothness classes and universal for anisotropic-smoothness classes.The universal-dictionary formulation addresses unknown smoothness classes; when universality is unavailable, accuracy and universality involve a trade-off.Source: NHr · pp. 136–137

9.3 Further generalizations

• Further generalizations unify Sobolev, Hölder-Nikol’skii, and mixed-smoothness classes, while extending approximation from scalar Lp and Lq settings to vector and Lorentz-type spaces.The unified approach groups variables and combines within-group Sobolev smoothness with mixed smoothness across groups; related work studies embeddings, approximation, and widths.Source: §9.3 · p. 137

9.4 Direct and inverse theorems

• Hyperbolic cross approximation supports Jackson-type direct theorems, but unlike the univariate case it does not generally characterize multivariate classes from mixed-modulus decay.Characterization by best-approximation decay was eventually obtained through newly introduced mixed moduli of smoothness.Source: §9.4 · p. 138

9.5 Kolmogorov widths of the intersection of function classes

• For intersections of smoothness classes, Kolmogorov-width behavior is governed by quantities such as r(A) and ν(A), with general asymptotic orders addressed under non-degeneracy assumptions.These classes support the study of embeddings, hyperbolic cross approximation, widths, entropy numbers, and m-term approximation; the general L2 width problem had remained open before recent progress.Source: §9.5 · pp. 138–139
• Kolmogorov widths of function-class intersections have asymptotic orders attained by trigonometric polynomials whose frequencies lie in corresponding hyperbolic-cross intersections.The section also points to surveys and bibliographies on embeddings, widths, entropy numbers, and m-term approximation.Source: §9.5 · p. 139

9.6 Further s-numbers

• Approximation and Kolmogorov numbers form s-number sequences with s_m(T) ≤ a_m(T), whereas orthowidths need not be s-numbers and can be asymptotically larger than linear widths.Gelfand numbers are dual to Kolmogorov numbers, can model reconstruction from linear measurements, and may permit nonlinear algorithms to outperform linear widths; Bernstein numbers provide lower bounds for Kolmogorov and Gelfand numbers and Monte-Carlo error analysis.Source: §9.6 · pp. 140–142

Complements to Section 3: Mixed differences

• Replacing moduli of continuity by integral means extends mixed-difference characterizations to 0 < p ≤∞ and r > (1/p −1)+, with Theorem 9.5 covering 0 < p, θ ≤∞ under s > σ_p and m > s.For θ = ∞, the relevant sum becomes a supremum.Source: Complements to Section 3: Mixed… · p. 142

Complements to Section 5: Sampling representations

• Sampling representations for small p require kernels balancing time-side decay and Fourier smoothness, while B-splines can achieve higher-order convergence by adapting their order to p.For p < 1 in L2, classical Dirichlet-Smolyak operators suffice, whereas piecewise-linear Faber-Schauder functions impose restrictions such as 1/p < r < 2.Source: Complements to Section 5: Sampling… · p. 143

Complements to Section 5: Sampling widths

• Sampling-width methods extend to p, q, θ < 1 whenever the corresponding sampling representations are available, yielding Theorem 9.6 for 0 < p, q, θ ≤∞ and r > 1/p.Source: Complements to Section 5: Sampling… · p. 143

Complements to Section 8: Numerical integration

• Numerical-integration complements give asymptotic estimates for broad parameter ranges, with analogous estimates for Fibonacci-cubature formulas.The supplied passages state separate rates for the cases 0 < p, θ ≤∞, r > 1/p and 0 < p < ∞, 0 < θ ≤1.Source: Complements to Section 8: Numerical… · pp. 143–144

Nonlinear approximation of non-compact embeddings

• Non-compact embeddings hold at r ≥ 1/p −1/q and are compact exactly when r > 1/p −1/q, while critical embeddings support nonlinear approximation results under specified p and q regimes.For spaces on R^d with p ≤ q, the embeddings hold under the same threshold but are never compact; the q = 2 case yields an additional nonlinear approximation-space identification for 0 < p < 2.Source: Nonlinear approximation of non-compact… · p. 144

9.8 Sampling along lattices

• Rank-1 lattice sampling offers FFT acceleration and improved numerical stability through oversampling, but its information-complexity rate is only half that of sparse grids and is never optimal.For rank-M tensor classes, Halton-point recovery has L∞-error ≲ C(r,d)N^-r with dimension-dependent constants, while randomized recovery avoids the curse of dimensionality exactly when M < 2^r r!.Source: §9.8 · p. 145

9.10 Continuous algorithms in m-term approximation and nonlinear widths

• A continuous m-term method is a continuous mapping from X into Σm(D), with dictionaries restricted so finite-dimensional subspaces intersect them finitely.This restriction defines the class used to formulate the corresponding nonlinear width.Source: §9.10 · p. 147
• Theorem 9.14 establishes a result for 1 < p, q < ∞, 2 ≤ θ ≤ ∞, and r > max(0, 1/p − 1/q, 1/p − 1/2).The supplied passage states the parameter conditions but does not include the theorem’s conclusion.Source: §9.10 · p. 147
• Alternative nonlinear widths based on pseudo-dimension avoid continuity assumptions and have been studied for mixed-smoothness function classes.Related results and surveys are cited across nonlinear-width theory.Source: §9.10 · p. 147

10 High-dimensional approximation

• When d is very large, the logarithmic factor can dominate m^−r, making finite-smoothness models impractical and motivating sparsity-based structural assumptions.The survey identifies dictionary sparsity, including tensor-product dictionaries, as a promising way to address this high-dimensional difficulty.Source: §10 · p. 148

10.1 Anisotropic mixed smoothness

• Anisotropic mixed smoothness replaces (log m)^(d−1)ξ with (log m)^(ν−1)ξ, and under moderate conditions the approximation rate can be independent of d, even when d is infinite.An anisotropic hyperbolic cross allocates more samples to directions with smaller smoothness; Fibonacci and Frolov cubature formulas can detect anisotropy without adaptation.Source: §10.1 · p. 149

10.2 Explicit constants and preasymptotics

• For large d, logarithmic factors can delay error decay exponentially, so controlling dimension-dependent constants and establishing preasymptotic bounds is essential; refined estimates yield quasi-polynomial tractability.The survey reports explicit and super-exponentially decaying constants, while preasymptotic estimates apply for ranges such as m less than 2^d and have no hidden constant in the stated upper bound.Source: §10.2 · pp. 150–151

10.3 Approximation in the energy norm

• Energy-norm approximation can remove the usual logarithmic factor, with dimension-explicit upper and lower bounds whose dimension-dependent terms decay exponentially for 1 < r < 2.The construction uses an energy hyperbolic cross, and for m > λ^d the bounds hold with explicit constants depending only on r.Source: §10.3 · p. 152

10.4 ε-dimension and approximation in infinite dimensions

• Infinite-dimensional approximation can be tractable through decaying weights or increasing smoothness, and linear hyperbolic cross methods achieve bounds independent of parametric or stochastic dimension.For suitable mixed Sobolev and Sobolev-analytic smoothness, tight ε-dimension bounds are obtained; in an energy norm, a rank-n method converges at rate n^−(α−β)/m.Source: §10.4 · pp. 153–154

11.1 General notation

• The appendix defines duality exponents for scalar and vector p, normalized integration on T^d, periodic L^q spaces, and the distinction between norms and quasi-norms.For 0 < p < 1, the L^p quantity is a quasi-norm; C(T^d) is used as a continuous periodic replacement for L^∞(T^d).Source: §11.1 · p. 155

11.2 Inequalities; 1.1. The Hölder inequality.; 1.4. The monotonicity of Lp-norms.

• The scalar and vector Hölder inequalities provide integrability of products under conjugate-exponent conditions, including the multilinear case.The appendix also records Hölder’s inequality for sums and allows the index range to include N = ∞.Source: §1.1 · p. 155, §1.4 · p. 156

1.4. The monotonicity of Lp-norms.; 1.7. The Minkowski inequality.

• The appendix summarizes monotonicity relations for L^p norms and the Minkowski and generalized Minkowski inequalities under their stated exponent conditions.These results extend the basic product estimates to norm comparisons and iterated or summed expressions.Source: §1.4 · p. 155, §1.7 · p. 156, §1.10 · p. 156

1.10. The Young inequality.

• Young’s inequality controls convolution of periodic one-variable functions in L^p and L^a, with a corresponding vector-exponent version also stated.The appendix introduces convolution before presenting these estimates.Source: §1.10 · p. 156

1.12. The Abel inequality.

• The Abel section derives an inequality through Abel transformation and introduces sequence spaces l^p, which are Banach spaces for 1 ≤ p ≤ ∞.The same section connects the transformation with estimates for nonnegative non-increasing sequences.Source: §1.12 · pp. 156–157

2.1. Let f ∈Lp, g ∈Lp′. We denote

• Duality results pair L^p and L^p′, extend the relevant statement to sequence spaces, characterize dual normed-space functionals, and state Nikol’skii duality.The section also gives a theorem involving a fixed system of functionals in the conjugate space.Source: §2.1 · p. 157

3.1. For a function f ∈L1(Td) we define Fourier coefficients

• The Fourier-analysis section defines Fourier coefficients on T^d and records Parseval equality, the Riesz–Fischer theorem, Hausdorff–Young, Riesz–Thorin interpolation, Littlewood–Paley, and multiplier results.It also introduces dyadic frequency blocks ρ(s), partial-sum operators, Marcinkiewicz multipliers bounded on L^p for 1 < p < ∞, and related corollaries.Source: §3.1 · pp. 158–159

Then the inequality

• The final inequality section states a Hardy–Littlewood–Sobolev result for 1 < q < p < ∞ and concludes that the operator A_β is bounded from L^q to L^p.Here β = 1/q − 1/p.Source: §3.1 · p. 159, Then the inequality · p. 159