Given two banach function spaces we study the pointwise product space E center dot F, especially for the case that the pointwise product of their unit balls is again convex. We then give conditions on when the pointwi...
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Given two banach function spaces we study the pointwise product space E center dot F, especially for the case that the pointwise product of their unit balls is again convex. We then give conditions on when the pointwise product E center dot M(E, F) = F, where M(E, F) denotes the space of multiplication operators from E into F.
. The paper deals with approximation of functions defined on R in spaces that are not translation invariant. The spaces under consideration are banach function spaces in which Steklov averaging operators are uniformly...
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. The paper deals with approximation of functions defined on R in spaces that are not translation invariant. The spaces under consideration are banach function spaces in which Steklov averaging operators are uniformly bounded. It is proved that operators of convolution with a kernel whose bell shaped majorant is integrable are bounded in these spaces. With the help of convolution operators, direct and inverse theorems of the theory of approximation by trigonometric polynomials and entire functions of exponential type are established. As structural characteristics, the powers of deviations of Steklov averages are used, including nonintegral powers. Theorems for periodic and nonperiodic functions are obtained in a unified way. The results of the paper generalize and refine a lot of known theorems on approximation in specific spaces such as weighted spaces, Lebesgue variable exponent spaces and others.
Consider a couple of banach function spaces X and Y over the same measure space and the space X(Y) of multiplication operators from X into Y. In this paper we develop the setting for characterizing certain summability...
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Consider a couple of banach function spaces X and Y over the same measure space and the space X(Y) of multiplication operators from X into Y. In this paper we develop the setting for characterizing certain summability properties satisfied by the elements of X(Y). At this end, using the "generalized Kothe duality" for banach function spaces, we introduce a new class of norms for spaces consisting of infinite sums of products of the type xy with x is an element of X and y is an element of Y
This paper deals with bilinear operators acting in pairs of banach function spaces that factor through the pointwise product. We find similar situations in different contexts of the functional analysis, including abst...
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This paper deals with bilinear operators acting in pairs of banach function spaces that factor through the pointwise product. We find similar situations in different contexts of the functional analysis, including abstract vector lattices-orthosymmetric maps, C-algebras-zero product preserving operators, and classical and harmonic analysis-integral bilinear operators. Bringing together the ideas of these areas, we show new factorization theorems and characterizations by means of norm inequalities. The objective of the paper is to apply these tools to provide new descriptions of some classes of bilinear integral operators, and to obtain integral representations for abstract classes of bilinear maps satisfying certain domination properties.
Let s(n)(T) denote the nth approximation, isomorphism, Gelfand, Kolmogorov or Bernstein number of the Hardy-type integral operator T given by T f(x) = v(x) integral(x)(a) u(t) f(t)dt, x is an element of (a, b) (-infin...
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Let s(n)(T) denote the nth approximation, isomorphism, Gelfand, Kolmogorov or Bernstein number of the Hardy-type integral operator T given by T f(x) = v(x) integral(x)(a) u(t) f(t)dt, x is an element of (a, b) (-infinity < a < b < +infinity) and mapping a banachfunction space E to itself. We investigate some geometrical properties of E for which C-1 integral(b)(a) u(x)v(x)dx <= lim(n ->infinity) inf ns(n) (T) <= lim(n ->infinity)sup ns(n)(T) <= C-2 integral(b)(a) u(x)v(x)dx under appropriate conditions on it and v. The constants C-1, C-2 > 0 depend only on the space E. (C) 2016 Elsevier Inc. All rights reserved.
The well-known factorization theorem of Lozanovskii may be written in the form L-1 E circle dot E', where circle dot means the pointwise product of banach ideal spaces. A natural generalization of this problem wou...
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The well-known factorization theorem of Lozanovskii may be written in the form L-1 E circle dot E', where circle dot means the pointwise product of banach ideal spaces. A natural generalization of this problem would be the question when one can factorize F through E, i.e., when F E circle dot M(E, F), where M(E, F) is the space of pointwise multipliers from E to F. Properties of M(E, F) were investigated in our earlier paper [41] and here we collect and prove some properties of the construction E circle dot F. The formulas for pointwise product of Calderon-Lozanovskii E-phi-spaces, Lorentz spaces and Marcinkiewicz spaces are proved. These results are then used to prove factorization theorems for such spaces. Finally, it is proved in Theorem 11 that under some natural assumptions, a rearrangement invariant banachfunction space may be factorized through a Marcinkiewicz space. (C) 2013 Elsevier Inc. All rights reserved.
In this paper, we give characterizations for a polynomial stability in banachspaces. This is done by using evolution cocycles and techniques of banach function spaces. Our characterizations are new versions of the th...
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In this paper, we give characterizations for a polynomial stability in banachspaces. This is done by using evolution cocycles and techniques of banach function spaces. Our characterizations are new versions of the theorems of Datko type.
It is classical that amongst all spaces L-p (G), 1 <= p <= infinity, for G = R, Z or T say, only L-2 (G) ( that is, p = 2) has the property that every bounded Borel function on the dual group Gamma determines a ...
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It is classical that amongst all spaces L-p (G), 1 <= p <= infinity, for G = R, Z or T say, only L-2 (G) ( that is, p = 2) has the property that every bounded Borel function on the dual group Gamma determines a bounded Fourier multiplier operator in L-2 (G). Stone's theorem asserts that there exists a regular, projection- valued measure (of operators on L-2 (G)), de. ned on the Borel sets of Gamma, with Fourier-Stieltjes transform equal to the group of translation operators on L-2 (G);this fails for every p not equal 2. We show that this special status of L-2 (G) amongst the spaces L-p (G), 1 <= p <= infinity, is actually more widespread;it continues to hold in a much larger class of banach function spaces de. ned over G (relative to Haar measure).
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Steyn, ClaudNWU
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We use a weighted analogue of a trace to define a weighted non-commutative decreasing rearrangement and show its relationship with the singular value function. We further show an alternative approach to constructing w...
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We use a weighted analogue of a trace to define a weighted non-commutative decreasing rearrangement and show its relationship with the singular value function. We further show an alternative approach to constructing weighted non-commutative banach function spaces using weighted non-commutative decreasing rearrangements and prove that this approach is equivalent to the original approach by Labuschagne and Majewski.
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