Marcinkiewicz Interpolation Research Articles

(PVB) functions and integration

AbstractWe introduce the notion of functions of bounded proximal variation and the notion of orderly connected topology on the real line. Using these notions, we define in a novel way an integral of Perron type, including virtually all the known integrals of Perron and Denjoy types and admitting mean value theorems and integration by parts and the analog of Marcinkiewicz theorem for the ordinary Perron integral.

Stability for the Marcinkiewicz theorem. Case of a fourth-degree polynomial

A stability estimate is obtained for a special case of a theorem of Marcinkiewicz asserting that if P4(t) is a fourth-degree polynomial then P4(t) can be a characteristic function only if the degree of P4(t) is in fact not greater than 2.

The Marcinkiewicz interpolation theorem extends to weighted spaces

The Marcinkiewicz interpolation theorem extends to weighted spaces

Marcinkiewicz's theorem and approximation schemes for many-particle correlation functions: Comment on a paper by Rajagopal and Sudarshan

Recently Rajagopal and Sudarshan proposed a generalization of Marcinkiewicz's theorem to quantum systems. From it they conclude that approximation schemes, such that the irreducible equal-time correlation functions vanish for all orders larger than l but are nonvanishing for some l larger than 2, are inconsistent with basic positivity conditions on the density operator. We show that this statement is false for fermion systems. For boson systems the statement is correct and its extension to irreducible correlation functions for unequal times can be made plausible, a fully rigorous proof does not yet seem available, however. The possibility is briefly discussed that an extension of Marcinkiewicz's theorem may hold for fermion systems, provided one confines his attention to equilibrium states of sucyh systems. (auth)

Some generalizations of the Marcinkiewicz theorem and its implications to certain approximation schemes in many-particle physics

The Marcinkiewicz theorem states that the characteristic function of a probability distribution function cannot be an exponential of a polynomial of degree larger than 2. This theorem is generalized in the present paper to (i) probability distribution functions of many variables and to (ii) probability distribution functionals when the stochastic variables are both commuting (Bose) and anticommuting (Fermi). The consequences of this theorem to certain approximation schemes in many-particle physics, involving truncation of hierarchical equations, are pointed out. These follow when one observes that the hierarchical equations such as those of many-particle Green's functions can be generated from a single equation for a Green's functional whose structure is that of a characteristic functional of a probability distribution functional. The theorem implies that this would exhibit a nonpositive behavior when certain truncation schemes are employed. Specific examples illustrating our results are drawn from the theory of an electron gas, turbulence theory, and quantum optics.

A multilinear version of the Marcinkiewicz interpolation theorem

A multilinear version of the Marcinkiewicz interpolation theorem

Addendum to “An extension of the Marcinkiewicz interpolation theorem to Lorentz spaces”

Addendum to “An extension of the Marcinkiewicz interpolation theorem to Lorentz spaces”

The Marcinkiewicz interpolation theorem

If h is measurable on a measure space M with measure m, its distribution function is defined for y>O by X7,(y) =X(y) =m {xEM; f(x) >y. The nonincreasing rearrangement of h onto (0, oo) is then the function given by h*(t) =inf {y>0; X(y) O. Both h* and X are non-negative and nonincreasing functions that are continuous from the right. h* and h have the same distribution function, thus ||h*||p =-|h1HP. Moreover, supu>0y {X(y) } 1/=supt>o tllqh*(t). Consequently the theorem of Marcinkiewicz can be stated in the following way: Suppose T is quasi-linear2 and, for 1 oo, i=O, 1, with Po<pi, qo5qi,

An extension of the Marcinkiewicz interpolation theorem to Lorentz spaces

An extension of the Marcinkiewicz interpolation theorem to Lorentz spaces

Some Analogues and Extensions of Marcinkiewicz's Interpolation Theorem

Some Analogues and Extensions of Marcinkiewicz's Interpolation Theorem