Families of multiresolution and wavelet spaces with optimal properties

Under suitable conditions, if the scaling functions ?₁ and ?₂ generate the multiresolutions V _(j)(?₁) and V _(j)(?₂), then their convolution ?₁*?₂also generates a multiresolution V _(j)(?₁*?₂) More over, if p is an appropriate convolution operator from l ₂ into itself and if ? is a scaling function generating the multiresolution V _(j)(?),then p*?is a scaling function generating the same multiresolution V _(j)(?)=V _(j)(p*?). Using these two properties, we group the scaling and wavelet functions into equivalent classes and consider various equivalent basis functions of the associated function spaces We use the n-fold convolution product to construct sequences of multiresolution and wavelet spaces V _(j)(?ⁿ) and W _(j)(?ⁿ) with increasing regularity. We discuss the link between multiresolution analysis and Shannon's sampling theory. We then show that the interpolating and orthogonal pre- and post-filters associated with the multiresolution sequence V ₍₀₎(?ⁿ)asymptotically converge to the ideal lowpass filter of Shannon. We also prove that the filters associated with the sequence of wavelet spaces W ₍₀₎(?ⁿ)convergeto the ideal bandpass filter. Finally, we construct the basic wavelet sequences ψ ^b _nand show that they tend to Gabor functions. Thisprovides wavelets that are nearly time-frequency optimal. The theory is illustrated with the example of polynomial splines.     

The Embedding W n,1 into C 0 on Compact Riemannian Manifolds

We study the best constant in the inequality corresponding to the Sobolev embedding W ^n,1(R ⁿ ) into the space of bounded continuous functions C ₀(R ⁿ ). Then, we adapt this inequality on compact Riemannian manifolds and discuss on its optimality.     

On an integral transform with generalized associated Legendre functions

We study some new properties of generalized associated Legendre functions of first and second kind P _k ^m,n (z) and Q _k ^m,n (z). Applying these functions, we introduce an integral transform that can be used in solving boundary-value problems of mathematical physics.Translated fromVychislitel'naya i Prikladnaya Matematika, Issue 71, 1990, pp. 33–43.     

Approximate solution of differential equations with the use of asymptotic polynomials

We consider an approximate solution of differential equations with initial and boundary conditions. To find a solution, we use asymptotic polynomials Q _n ^f (x) of the first kind based on Chebyshev polynomials T _n (x) of the first kind and asymptotic polynomials G _n ^f (x) of the second kind based on Chebyshev polynomials U _n (x) of the second kind. We suggest most efficient algorithms for each of these solutions. We find classes of functions for which the approximate solution converges to the exact one. The remainder is represented as an expansion in linear functionals {L _n ^f } in the first case and {M _n ^f } in the second case, whose decay rate depends on the properties of functions describing the differential equation.     

Arithmetic of semigroups of series in multiplicative systems

We study the arithmetic of a semigroup M_P\mathcal{M}_{\mathcal{P}} of functions with operation of multiplication representable in the form f(x) = ?_{n = 0}^￥ a_nc_n(x)    ( a_n 3 0,?_{n = 0}^￥ a_n = 1 ) f(x) = \sum\nolimits_{n = 0}^\infty {{a_n}{\chi_n}(x)\quad \left( {{a_n} \ge 0,\sum\nolimits_{n = 0}^\infty {{a_n} = 1} } \right)} , where { c_n }_{n = 0}^￥ \left\{ {{\chi_n}} \right\}_{n = 0}^\infty is a system of multiplicative functions that are generalizations of the classical Walsh functions. For the semigroup M_P\mathcal{M}_{\mathcal{P}}, analogs of the well-known Khinchin theorems related to the arithmetic of a semigroup of probability measures in R ⁿ are true. We describe the class I₀(M_P)I_0(\mathcal{M}_{\mathcal{P}}) of functions without indivisible or nondegenerate idempotent divisors and construct a class of indecomposable functions that is dense in M_P\mathcal{M}_{\mathcal{P}} in the topology of uniform convergence.     

Best approximation by ridge functions in <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-spaces

We study the approximation of the classes of functions by the manifold R _n formed by all possible linear combinations of n ridge functions of the form r(a · x)): It is proved that, for any 1 ≤ q ≤ p ≤ ∞, the deviation of the Sobolev class W ^r _p from the set R _n of ridge functions in the space L _q (B ^d ) satisfies the sharp order n ^-r/(d-1).     

Invariants of an augmentation of Γ0(n)

Let G _n ^× be the 2-group of primary factors of a positive integer n and fix a direct product decomposition of this group. We define an augmentation of Γ₀(n) based on G _n ^×, paralleling augmentations used by Fricke, Cohn and Knopp, and others. Using the decomposition of G _n ^×, we then define a family of functions based on η-functions and use these functions to construct invariants of the augmented group. Along with proving results analogous to those of Cohn and Knopp, we make a complete determination of the multiplier systems for these new functions.     

Interpolation problems on the spectrum of H∞

We characterize those sequences (x _n ) in the spectrum of H ^∞ whose Nevanlinna–Pick interpolation problems admit thin Blaschke products as solutions. We also study under which conditions there is a Blaschke product B with prescribed zero-set distribution and solving problems of the form B(x) = f _n (x) for every x ∈ P(x _n ), where P(x _n ) is the Gleason part associated with the point x _n and where (f _n ) is an arbitrary sequence of functions in the unit ball of H ^∞. As a corollary we get a new characterization of Carleson–Newman Blaschke products in terms of bounded universal functions, a result first proved by Gallardo and Gorkin.     

Plain Bases for Classes of Primitive Recursive Functions

Abasisforaset C of functions on natural numbers is a set F of functions such that C is the closure with respect to substitution of the projection functions and the functions in F. This paper introduces three new bases, comprehending only common functions, for the Grzegorczyk classes ℰⁿ with n ≥ 3. Such results are then applied in order to show that ℰⁿ⁺¹ = K_n for n ≥ 2, where {K_n}_n∈ℕ is the Axt hierarchy.     

Weak*- convergence of Monge-Ampère measures

We study the convergence of sequences of Monge-Ampère measures (dd ^c u _j ) ⁿ where (u _j ) is a given sequence of plurisubharmonic functions. Our main theorem is about approximation by multipole pluricomplex Green functions. Partially supported by the Swedish Research Council contract no 621-2002-5308     

Weakly Outer Inner Functions

An inner function I in the unit ball B_nB_nⁿ is said to be weakly outer if the closed subspace I H ^p(B ⁿ) is weakly dense in the Hardy space H^p(B ⁿ), 0n for all n1. We also investigate inner functions I such that the subspace IH^p(B ⁿ) is not weakly dense in H^p(B ⁿ).     

A note on the Σ1 collection scheme and fragments of bounded arithmetic

We show that for each n ≥ 1, if T₂ⁿ does not prove the weak pigeonhole principle for Σ^b_n functions, then the collection scheme B Σ₁ is not finitely axiomatizable over T₂ⁿ. The same result holds with Sⁿ₂ in place of T 2ⁿ (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Interpolation problems on the spectrum of <Emphasis Type="Italic">H</Emphasis><Superscript>∞</Superscript>

We characterize those sequences (x _n ) in the spectrum of H ^∞ whose Nevanlinna–Pick interpolation problems admit thin Blaschke products as solutions. We also study under which conditions there is a Blaschke product B with prescribed zero-set distribution and solving problems of the form B(x) = f _n (x) for every x ∈ P(x _n ), where P(x _n ) is the Gleason part associated with the point x _n and where (f _n ) is an arbitrary sequence of functions in the unit ball of H ^∞. As a corollary we get a new characterization of Carleson–Newman Blaschke products in terms of bounded universal functions, a result first proved by Gallardo and Gorkin.      

Limits of functions and elliptic operators

We show that a subspaceS of the space of real analytical functions on a manifold that satisfies certain regularity properties is contained in the set of solutions of a linear elliptic differential equation. The regularity properties are thatS is closed inL ² (M) and that if a sequence of functions fn in ƒ_n converges inL ²(M), then so do the partial derivatives of the functions ƒ_n.     

Some Hierarchies of Primitive Recursive Functions on Term Algebras

We compare two different Grzegorczyk hierarchies {H_n^σ}n≥0 and {L_n^σ}n≥1 on term algebras, which grow according to the height and length of terms, respectively. The solution of almost all inclusion problems among the Grzegorczyk classes and the (simultaneous) recursion number classes R_n^σ and S_n^σ on term algebras shows {H_n^σ}n≥0 to generalize Weihrauch's Grzegorczyk hierarchy on words {E_n^k}n≥0 to arbitrary term algebras. However, by regarding terms as words, {L_n^σ}n≥1 turns out to be computationally equivalent to Weihrauch's hierarchy {E_n^σ}n≥0 on the whole. Especially, L₂^σ} is equivalent to polynomial time computability and contains several natural term algebra functions. This establishes a notion of feasible term algebra functions and predicates.     

Adaption allows efficient integration of functions with unknown singularities

We study numerical integration for functions f with singularities. Nonadaptive methods are inefficient in this case, and we show that the problem can be efficiently solved by adaptive quadratures at cost similar to that for functions with no singularities. Consider first a class of functions whose derivatives of order up to r are continuous and uniformly bounded for any but one singular point. We propose adaptive quadratures Q*_n, each using at most n function values, whose worst case errors are proportional to n^−r. On the other hand, the worst case error of nonadaptive methods does not converge faster than n⁻¹. These worst case results do not extend to the case of functions with two or more singularities; however, adaption shows its power even for such functions in the asymptotic setting. That is, let F^∞_r be the class of r-smooth functions with arbitrary (but finite) number of singularities. Then a generalization of Q*_n yields adaptive quadratures Q**_n such that |I(f)−Q**_n(f)|=O(n^−r) for any f ∈ F^∞_r. In addition, we show that for any sequence of nonadaptive methods there are `many' functions in F^∞_r for which the errors converge no faster than n⁻¹. Results of numerical experiments are also presented. The authors were partially supported, respectively, by the State Committee for Scientific Research of Poland under Project 1 P03A 03928 and by the National Science Foundation under Grant CCR-0095709.     

Trace Formulas for Some Singular Differential Operators and Applications

We characterize the convergence of the series ∑ λ^–1_n, where λⁿ are the non‐zero eigenvalues of some boundary value problems for degenerate second order ordinary differential operators and we prove a formula for the above sum when the coefficient of the zero‐order term vanishes. We study these operators both in weighted Hilbert spaces and in spaces of continuous functions. After investigating the boundary behaviour of the eigenfunctions, we give applications to the regularity of the generated semigroups.     

On the Rate of Convergence of Two Bernstein–Bézier Type Operators for Bounded Variation Functions

In this paper we study the rate of convergence of two Bernstein–Bézier type operatorsB^(α)_nandL^(α)_nfor bounded variation functions. By means of construction of suitable functions and the method of Bojanic and Vuillemier (J. Approx. Theory31(1981), 67–79), using some results of probability theory, we obtain asymptotically optimal estimations ofB^(α)_nandL^(α)_nfor bounded variation functions at points of continuity and points of discontinuity.     

Local Change of Coordinates for Grid-Functions on Regions with Curved Boundary

Local change of coordinates, providing an appropriate transformation of functions on a domain Ω ∩ R ⁿ into functions on R ⁿ and R ⁺_n, is a well-known and frequently used technical tool in the theory of Sobolev-spaces W^m,p (Ω) and partial differential equations. In this paper we propose a corresponding transformation mapping grid-functions on regular grids Ω_h into functions on R ⁿ_h and R ⁿ_h,+ which as in the continuous case can be used to remedy various difficulties arising the curved boundary of Ω.     

On the L2n 1– Extension Properties

We have proved that for all compact linear operator u from R into an L_p ([0,1], ν) (0 < p < 1) extends to L ₁ ([0,1], ν), where R denotes the closed linear subspace in L ₁ ([0,1], ν) of the Rademacher functions {r_n }_{n ? N}. In this paper, we study this type of extension for E_n ? L_2n ¹ where E_n is the n–dimensional subspace which appears in Kasin's theorem such that L_2n ¹ = E_n ⊕ E ^⊥ _n and the L_2n ¹ , L_2n ² norms are universally equivalent on both E_n , E ^⊥ _n. We show that, the precedent extension fails for the pair (E_n , L_2n ¹ ) and we generalize this to any E in an L ₁(Ω, A, P) by giving some conditions on E.