ISSN:0003-6811    

DOI:10.1080/00036817708839152

出版商:Gordon and Breach Science Publishers Ltd.    

年代:1977

数据来源: Taylor

摘要:

In the foregoing Note (this Journal Vol.I.p. 75-99) the space of n-dimensional Bessel potentials Lpxwas deseribed in terms of generalized Lipschitz conditions of f or its Riesz transform for 0<∝≦2 The still open case ∝>1 is treated in the first half of this paper, firstly by introducing appropriate iterates of the cited conditions, secondly by using derivatives of f and its Riesz transform, in particular the Laplacian △ and the gradient of the Riesz transformation(▽,Rand by applying the former results In Section 6 a definition of a Riesz derivative of order ∝ is given and based upon the concept: Integrate f(m-α)-times in the sense of Riesz and then differentiate [d]m-times (by considering the limit of suitable difference quotients of f). Necessary and sufficient conditions for the existence of these Riesz derivatives are obtained All results also hold in the non-reflexive spaces[d]

ISSN:0003-6811    

DOI:10.1080/00036817708839143

出版商:Gordon and Breach Science Publishers Ltd.    

年代:1977

数据来源: Taylor

摘要:

Let m and n be positive integers, and z1,z2,...,zmbe m points in real n dimensional Euclidean space En, For eachj=1,2,...,m,let fjbe a real valued continuously differentiate function of the positive real variable r, with fj(r)gt;0 for r>0, Consider the real valued function F, which is defined, for w in En, with w=zjforj= 1,2,..., m, by the equation

ISSN:0003-6811    

DOI:10.1080/00036817708839144

出版商:Gordon and Breach Science Publishers Ltd.    

年代:1977

数据来源: Taylor

摘要:

In a well-known paper R.J. Duffin [J. Math. Anal. Appl.,5 (1962) 200-215] extends the notion of extremal length to electrical networks, by allowing variable specific resistivity along the arcs of the network, and shows extremal length and extremal width are reciprocals for planar networks yielding Rayleigh's Theorem that conjugate* conductors have reciprocal resistances, Similar results are obtained by F. W. Gehring [Mich. Math. J.,9 (1962), 137-150] for continuous conductors with constant specific resistivity in 3-space, This paper unifies the continuous and discrete cases by obtaining the Reciprocity Theorem for p-capacities in n-space provided the specific resistivity satisfies certain realistic smoothness conditions, It follows that Rayleigh's Theorem holds for nonptanar networks confirming a conjecture of Duffin

ISSN:0003-6811    

DOI:10.1080/00036817708839145

出版商:Gordon and Breach Science Publishers Ltd.    

年代:1977

数据来源: Taylor

摘要:

Copson has shown that, in the two dimensional case, the Cauchy problem for the damped wave equation can be solved, using the original Riesz kernel, without introducing a new kernel. Copson stated that his approach could be used in any number of independent variables; this statement is verified in the present paper

ISSN:0003-6811    

DOI:10.1080/00036817708839146

出版商:Gordon and Breach Science Publishers Ltd.    

年代:1977

数据来源: Taylor

摘要:

This paper deals with the concent of stability and in variance of dynamical polysystems. In Section 2 the relevant definitions ate given. In Section 3 the different types of stability and invariance are discussed and several theorems are proved. The most important is Theorem 3 which gives sufficient conditions for global weak asymptotic stability. It is pointed out that weak stability is closely related to the notion of controllability for control systems

ISSN:0003-6811    

DOI:10.1080/00036817708839147

出版商:Gordon and Breach Science Publishers Ltd.    

年代:1977

数据来源: Taylor

摘要:

An existence and uniqueness theory is established for the classical triple trigonometric series having the kernels {cos(n-i)x}, {sinnx}, or {cosnx} when the right hand sides of the equations are given functions of bounded variation. It is shown that these series do not, in general, converge in the ordinary sense on any set of positive measure but do converge at all points in the sense of Abel-Poisson The key to the proof is use of a generalized reflection principle to establish uniqueness and thus prove the equivalence of different representations of the solution. Best possible asymptotic estimates for growth and uniqueness of the coefficients are derived

ISSN:0003-6811    

DOI:10.1080/00036817708839148

出版商:Gordon and Breach Science Publishers Ltd.    

年代:1977

数据来源: Taylor

摘要:

The search for a "missing link" reflection principle for solutions of the biharmonic equation, is shown to lead to the class of functions described in the title.

ISSN:0003-6811    

DOI:10.1080/00036817708839149

出版商:Gordon and Breach Science Publishers Ltd.    

年代:1977

数据来源: Taylor

ISSN:0003-6811    

DOI:10.1080/00036817708839150

出版商:Gordon and Breach Science Publishers Ltd.    

年代:1977

数据来源: Taylor

ISSN:0003-6811    

DOI:10.1080/00036817708839151

出版商:Gordon and Breach Science Publishers Ltd.    

年代:1977

数据来源: Taylor