μ-average N-widths on the Wiener space

In this paper we investigate the asymptotic behaviour of μ-averagen-widths of integral operatorK on the Wiener space, whereK is the inverse operator of an ordinary linear differential operatorL of orderm. For 1≤p.q<∞ . and forp∈[1, ∞),q∈[2, ∞) . Supported by the Fund. of Dooctoral program of NECC.     

Riesz frames and approximation of the frame coefficients

A frame is a fmaily {f _i } _i=1 ^∞ of elements in a Hilbert space with the property that every element in can be written as a (infinite) linear combination of the frame elements. Frame theory describes how one can choose the corresponding coefficients, which are called frame coefficients. From the mathematical point of view this is gratifying, but for applications it is a problem that the calculation requires inversion of an operator on . The projection method is introduced in order to avoid this problem. The basic idea is to consider finite subfamilies {f _i } _i=1 ⁿ of the frame and the orthogonal projection P_n onto its span. For has a representation as a linear combination of f_i, i=1,2,..., n and the corresponding coefficients can be calculated using finite dimensional methods. We find conditions implying that those coefficients converge to the correct frame coefficients as n→∞, in which case we have avoided the inversion problem. In the same spirit we approximate the solution to a moment problem. It turns out, that the class of “well-behaving frames” are identical for the two problems we consider. The first named author is partially supported by NSF DMS 9201357, Danish NSRC grant 9401958, Missouri Research Board grant C-3-41743, and a Missouri Research Council Summer Fellowship.     

The convergence rate of Euler means sequence

Let denote the Euler means of the Fourier series of the 2π-perodic function f(x). For an integer q>0, setting ℰ _nq (ω)=sup‖f−ℰ _n (q,f)‖, the precise value of is obtained.     

On approximation of functions on sphere

Let f be an integrable function on the unit sphere Σ _n−1 of R ⁿ (n⩾3) and let σ _N ^δ be the Cesàro means of order σ of the Fourier-Laplace series of f. The special value λ:=n−2/2 of σ is known as the critical index. This paper proves that and where ω(f,t)_p is the 1st-order modulus of continuity of f in L^p-metric which is defined in a way different than in the classical case of n=2. In Memory of Professor M. T. Cheng Project supported by the NSF of China under the grans # 19771009.     

Strong mean approximation on HP(Tn) for the Riesz means at the critical index

In this note we extend the results in [1] to high dimensions. Let f∈H^p (Tⁿ), 0<p<1, n≥1 andσ ^δ , f denote the Riesz means of f at the critical index δ=n/p−(n+1)/2. We have the following estimate: were 0<s≤2 and , is the K-functional in H^p(Tⁿ). Supported by NSFC     

A sufficient condition for mean convergence of orthogonal series for Freud weights

Let S_n[f](x) be the n-th partial sum of the orthonormal polynomial series expansion for f corresponding to a Freud weight W(x)=e^−Q(x). We give a sufficient condition for the inequality      

Weak type inequalities for fractional maximal operator on weighted orlicz spaces

The fractional maximal operator on homogeneous space (X, d, μ) is de fined as . In this paper, the sufficient and necessary conditions for the to be of weak type and extra weak type will be given. In Memory of Professor M. T. Cheng     

On approximation of functions on sphere

Let f be an integrable function on the unit sphere Σ _n−1 of R ⁿ (n⩾3) and let σ _N ^δ be the Cesàro means of order σ of the Fourier-Laplace series of f. The special value λ:=n−2/2 of σ is known as the critical index. This paper proves that      

Numerical evaluation of a 2-D Cauchy principal value integral based on quasi-interpolating splines

In this paper the author presents a method for the numerical solution of a 2-D Cauchy principal value of the form where S is a domain with a continuous boundary. By using polar coordinates, the integral is reduced to the form where the finite-part of the integral. We construct the relative product rule based on quasi-inter polating splines. Convergence results are proved and numerical examples are given.     

A sufficient condition for mean convergence of orthogonal series for Freud weights

Let S_n[f](x) be the n-th partial sum of the orthonormal polynomial series expansion for f corresponding to a Freud weight W(x)=e^−Q(x). We give a sufficient condition for the inequality to hold. This sufficient condition is much easier to apply than that of Jha and Lubinsky [2].     

Weak type inequalities for fractional maximal operator on weighted orlicz spaces

The fractional maximal operator on homogeneous space (X, d, μ) is de fined as      

Sull’unione di classi filtrali

Riassunto In questo lavoro si cercano condizioni perchè una classe di algebre simili filtrali sia essa stessa filtrale. Si trova che se è una famiglia finita di algebre simili e filtrali, che siano indipendenti (cfr.A. L. Foster [2]), allora anche è filtrale. Introdotto poi il concetto disemi-indipendenza, si arriva a un risultato analogo per le algebre semifiltrali e a una condizione sufficiente per la semifiltralità di un'algebra finita. Infine si osserva che i risultati trovati possono essere estesi alle classi ideali.

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Summary This paper refers to the concept offiltrale class of similar algebras, which has been introduced byR. Magari in [6]. First I observe that a class of similarfiltrale algebras is not generallyfiltrale, not even if is a finite family of finite algebras. However I find that if is a finite family of similarfiltrale algebras, which are also independent (using this term in the sense ofA. L. Foster [2]), then also isfiltrale. Then introducing the notion of independent varieties, one arrives at a generalization of the above mentioned result. Given then a family of similar algebras , we say that these aresemi-indipendenti if for everyJ⊂I there exists an algebraic functionf _J defined in II _i so thatf _J(x, y)=z wherez _i=x_i ifi∈J andz _i=y_i ifi∉J for every . I prove that if is a finite family ofsemifiltrali andsemi-indipendenti algebras, then also issemifiltrale. I also find that a finite algebra, which is simple andsemi-indipendente of itself (in the sense that there exists in a binary algebraic functionf withf(x, y)=<x ₀,y ₁>) issemifiltrale. Then I give the notion ofsemi-indipendente variety, which leads to some further results. Finally I observe that all these results can be extended to ideal classes of similar algebras.

     

Minimal presentations for certain group extensions

IfG is a finite group thend(G) denotes the minimal number of generators ofG. IfH andK are groups then the extension, 1 →H →G →K → 1, is called an outer extension ofK byH ifd(G)=d(H)+d(K). Let be the class of groups containing all finitep-groupsG which has a presentation withd(G) = dimH ¹(G,z _p ) generators andr(G)=dimH ² (G,Z _p ) relations: in this article it is shown that ifK is a non cyclic group belonging to andH is a finite abelian p-group then any outer extension ofK byH belongs to .     

On the singularities of composition of entire functions

Let f be an entire function. A point zo is called a critical point of f if f'(zo) = 0, and f(zo) is called a critical value (or an algebraic singularity) of f. Next a ∈ (C^) is said to be an asymptotic value (or a transcendental singularity) of f if there exists a curve Γ:[0,1]→(C^) such that limt→1Γ(t)=∞ and limt→1(foΓ)(t)=a. In this paper we find relations between the asymptotic values of f, g and f o g, relations between critical points of f, g and f o g and also in the case when the two functions f and g are semi-conjugated with another entire function.     

Operator analogue of the Krein-Milman theorem in the generalized state spaces

We discuss the Krein-Milman-type problems in the C* -convexity theory for the generalized state space of C*-algebraA. The main results are that every BW-compact, C*-convex subset of possesses a C*-extreme point and every BW-compact, C* -convex subset of is the C^*-convex hull of its C*-extreme points.     

Una dimostrazione del fatto che ogni varietà ammette algebre semplici

Riassunto Si dimostra che se è una struttura algebrica priva di quozienti semplici, allora un’opportuna sottoalgebra di ammette quozienti semplici. Ne segue il risultato del titolo.

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Summary Let be an algebraic structure with no simple quotients. Then there is a subalgebra of which possess simple quotients. Therefore every variety of algebras (=algebraic structures) has simple algebras.

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Lavoro eseguito nell’ambito dell’attività del Comitato Nazionale per le Scienze Matematiche del C.N.R. (anno ’68–’69, programma 8).     

On the numerical inversion of the Laplace transform by the use of an optimized Legendre polynomials

A method for inverting the Laplace transform is presented, using a finite series of the classical Legendre polynomials. The method recovers a real valued function f(t) in a finite interval of the positive real axis when f(t) belongs to a certain class      

Sistemi di assiomi che definiscono un piano affine di ordinen

Riassunto Un insieme di proprietà soddisfatte da una struttura è detto essere un sistema completo di assiomi se è sufficiente a definire come un piano affine. Un sistema completo è minimo se non possiede alcun sottosistema proprio completo. In questo lavoro sono determinati i possibili sistemi minimi completi di assiomi che definiscono un piano affine di ordinen.

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Summary A set of properties which are satisfied by an incidence structure is said to be a complete system of axioms if such a system is sufficient to define as an affine plane. A complete system is minimal if it does not possess any proper complete subsystem. In this work we find all possible minimal complete systems of axioms which define an affine plane of ordern.

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Lavoro eseguito nell’ambito dell’attività dei contratti di ricerca del Comitato per le Scienze Matematiche del C.N.R.     

Best one-sided approximation of some classes of smooth functions on the whole real axis by cardinal splines of higher order

The objective of this paper is to study the quantity . In this paper we establish a duality theorem for the quantity for m≥r. By virtue of the duality theroem we obtain the exact estimation for for all m≥r.     

Inductive definitions,models of comprehension and invariant definability

The connections between inductive definability and models of comprehension are studied. Let = 〈A, R _l, ...,R _n 〉 be an infinite structure and letI _φ be a set inductively defined by a formulaφ of the second order language . We prove that if is a model of Δ ₁ ¹ -Comprehension relativized toφ, andφ is -absolute, then for everyη smaller than the height of (h( )),I _φ is in . If is aβ-structure which satisfies Σ ₁ ¹ -Comprehension relativized toφ and WF(X), and φ is -absolute, thenI _φ is in and ‖φ| <h ( ). These results imply that Barwise-Grilliot theorem is false in the case of uncountable acceptable structures. We also study the notion of invariant definability over models₁ of Δ ₁ ¹ -Comprehension. This paper is registered as Report ZW 69/76 of the Mathematical Centre.