Area integral functions are introduced for sectorial operators on Hilbert spaces. We establish the equivalence relationship between the square and area integral functions. This immediately extends McIntosh/Yagi＇s res...Area integral functions are introduced for sectorial operators on Hilbert spaces. We establish the equivalence relationship between the square and area integral functions. This immediately extends McIntosh/Yagi＇s results on H∞ functional calculus of sectorial operators on Hilbert spaces to the case when the square functions are replaced by the area integral functions.展开更多
Let H be a complex infinite dimensional Hilbert space and B(H)be the algebra of all bounded linear operators on H.In this paper,we mainly study the operators that satisfy both a-Weyl's theorem and property(R).Also...Let H be a complex infinite dimensional Hilbert space and B(H)be the algebra of all bounded linear operators on H.In this paper,we mainly study the operators that satisfy both a-Weyl's theorem and property(R).Also,the operators whose functional calculus satisfies the two properties are also explored.We give the features for the operator or its functional calculus for which both a-Weyl's theorem and property(R)hold.展开更多
Based on the fact that the real inductor and the real capacitor are fractional order in nature and the fractional calculus,the transfer function modeling and analysis of the open-loop Buck converter in a continuous co...Based on the fact that the real inductor and the real capacitor are fractional order in nature and the fractional calculus,the transfer function modeling and analysis of the open-loop Buck converter in a continuous conduction mode（CCM） operation are carried out in this paper.The fractional order small signal model and the corresponding equivalent circuit of the open-loop Buck converter in a CCM operation are presented.The transfer functions from the input voltage to the output voltage,from the input voltage to the inductor current,from the duty cycle to the output voltage,from the duty cycle to the inductor current,and the output impedance of the open-loop Buck converter in CCM operation are derived,and their bode diagrams and step responses are calculated,respectively.It is found that all the derived fractional order transfer functions of the system are influenced by the fractional orders of the inductor and the capacitor.Finally,the realization of the fractional order inductor and the fractional order capacitor is designed,and the corresponding PSIM circuit simulation results of the open-loop Buck converter in CCM operation are given to confirm the correctness of the derivations and the theoretical analysis.展开更多
The linear relationship between fractal dimensions of a type of generalized Weierstrass functions and the order of their fractional calculus has been proved. The graphs and numerical results given here further indicat...The linear relationship between fractal dimensions of a type of generalized Weierstrass functions and the order of their fractional calculus has been proved. The graphs and numerical results given here further indicate the corresponding relationship.展开更多
Based on the combination of fractional calculus with fractal functions, a new type of functions is introduced; the definition, graph, property and dimension of this function are discussed.
According to the necessary condition of the functional taking the extremum, that is its first variation is equal to zero, the variational problems of the functionals for the undetermined boundary in the calculus of va...According to the necessary condition of the functional taking the extremum, that is its first variation is equal to zero, the variational problems of the functionals for the undetermined boundary in the calculus of variations are researched, the functionals depend on single argument, arbitrary unknown functions and their derivatives of higher orders. A new view point is posed and demonstrated, i.e. when the first variation of the functional is equal to zero, all the variational terms are not independent to each other, and at least one of them is equal to zero. Some theorems and corollaries of the variational problems of the functionals are obtained.展开更多
In this article, we investigate the density of the solution to a class of stochastic functional differential equations by means of Malliavin calculus. Our aim is to provide upper and lower Gaussian estimates for the d...In this article, we investigate the density of the solution to a class of stochastic functional differential equations by means of Malliavin calculus. Our aim is to provide upper and lower Gaussian estimates for the density.展开更多
The object of this article is to study and develop the generalized fractional calcu- lus operators given by Saigo and Maeda in 1996. We establish generalized fractional calculus formulas involving the product of R-fun...The object of this article is to study and develop the generalized fractional calcu- lus operators given by Saigo and Maeda in 1996. We establish generalized fractional calculus formulas involving the product of R-function, Appell function F3 and a general class of poly- nomials. The results obtained provide unification and extension of the results given by Saxena et al. [13], Srivastava and Grag [17], Srivastava et al. [20], and etc. The results are obtained in compact form and are useful in preparing some tables of operators of fractional calculus. On account of the general nature of the Saigo-Maeda operators, R-function, and a general class of polynomials a large number of new and known results involving Saigo fractional calculus operators and several special functions notably H-function, /-function, Mittag-Leffier function, generalized Wright hypergeometric function, generalized Bessel-Maitland function follow as special cases of our main findings.展开更多
In this study, the impulsive predator-prey dynamic systems on time scales calculus are studied. When the system has periodic solution is investigated, and three different conditions have been found, which are necessar...In this study, the impulsive predator-prey dynamic systems on time scales calculus are studied. When the system has periodic solution is investigated, and three different conditions have been found, which are necessary for the periodic solution of the predator-prey dynamic systems with Beddington-DeAngelis type functional response. For this study the main tools are time scales calculus and coincidence degree theory. Also the findings are beneficial for continuous case, discrete case and the unification of both these cases. Additionally, unification of continuous and discrete case is a good example for the modeling of the life cycle of insects.展开更多
The aim of this paper is to study the existence of integrable solutions of a nonlinear functional integral equation in the space of Lebesgue integrable functions on unbounded interval, L1(R+). As an application we ded...The aim of this paper is to study the existence of integrable solutions of a nonlinear functional integral equation in the space of Lebesgue integrable functions on unbounded interval, L1(R+). As an application we deduce the existence of solution of an initial value problem of fractional order that be studied only on a bounded interval. The main tools used are Schauder fixed point theorem, measure of weak noncompactness, superposition operator and fractional calculus.展开更多
In this paper, we consider a class of Sobolev-type fractional neutral stochastic differential equations driven by fractional Brownian motion with infinite delay in a Hilbert space. When α>1-H, by the ...In this paper, we consider a class of Sobolev-type fractional neutral stochastic differential equations driven by fractional Brownian motion with infinite delay in a Hilbert space. When α>1-H, by the technique of Sadovskii’s fixed point theorem, stochastic calculus and the methods adopted directly from deterministic control problems, we study the approximate controllability of the stochastic system.展开更多
Image forgery is a crucial part of the transmission of misinformation,which may be illegal in some jurisdictions.The powerful image editing software has made it nearly impossible to detect altered images with the nake...Image forgery is a crucial part of the transmission of misinformation,which may be illegal in some jurisdictions.The powerful image editing software has made it nearly impossible to detect altered images with the naked eye.Images must be protected against attempts to manipulate them.Image authentication methods have gained popularity because of their use in multimedia and multimedia networking applications.Attempts were made to address the consequences of image forgeries by creating algorithms for identifying altered images.Because image tampering detection targets processing techniques such as object removal or addition,identifying altered images remains a major challenge in research.In this study,a novel image texture feature extraction model based on the generalized k-symbolWhittaker function(GKSWF)is proposed for better image forgery detection.The proposed method is divided into two stages.The first stage involves feature extraction using the proposed GKSWF model,followed by classification using the“support vector machine”(SVM)to distinguish between authentic and manipulated images.Each extracted feature from an input image is saved in the features database for use in image splicing detection.The proposed GKSWF as a feature extraction model is intended to extract clues of tampering texture details based on the probability of image pixel.When tested on publicly available image dataset“CASIA”v2.0(ChineseAcademy of Sciences,Institute of Automation),the proposed model had a 98.60%accuracy rate on the YCbCr(luminance(Y),chroma blue(Cb)and chroma red(Cr))color spaces in image block size of 8×8 pixels.The proposed image authentication model shows great accuracy with a relatively modest dimension feature size,supporting the benefit of utilizing the k-symbol Whittaker function in image authentication algorithms.展开更多
Area integral functions are introduced for sectorial operators on L^p-spaces. We establish the equivalence between the square and area integral functions for sectorial operators on L^p spaces. This follows that the re...Area integral functions are introduced for sectorial operators on L^p-spaces. We establish the equivalence between the square and area integral functions for sectorial operators on L^p spaces. This follows that the results of Cowling, Doust, McIntosh, Yagi, and Le Merdy on H^∞ functional calculus of seetorial operators on LP-spaces hold true when the square functions are replaced by the area integral functions.展开更多
Establishing the accurate relationship between fractional calculus and fractals is an important research content of fractional calculus theory.In the present paper,we investigate the relationship between fractional ca...Establishing the accurate relationship between fractional calculus and fractals is an important research content of fractional calculus theory.In the present paper,we investigate the relationship between fractional calculus and fractal functions,based only on fractal dimension considerations.Fractal dimension of the Riemann-Liouville fractional integral of continuous functions seems no more than fractal dimension of functions themselves.Meanwhile fractal dimension of the Riemann-Liouville fractional differential of continuous functions seems no less than fractal dimension of functions themselves when they exist.After further discussion,fractal dimension of the Riemann-Liouville fractional integral is at least linearly decreasing and fractal dimension of the Riemann-Liouville fractional differential is at most linearly increasing for the Holder continuous functions.Investigation about other fractional calculus,such as the Weyl-Marchaud fractional derivative and the Weyl fractional integral has also been given elementary.This work is helpful to reveal the mechanism of fractional calculus on continuous functions.At the same time,it provides some theoretical basis for the rationality of the definition of fractional calculus.This is also helpful to reveal and explain the internal relationship between fractional calculus and fractals from the perspective of geometry.展开更多
The present paper investigates the fractal structure of fractional integrals of Weierstrass functions. The exact box dimension for such functions many important cases is established. We need to point out that, althoug...The present paper investigates the fractal structure of fractional integrals of Weierstrass functions. The exact box dimension for such functions many important cases is established. We need to point out that, although the result itself achieved in the present paper is interesting, the new technique and method should be emphasized. These novel ideas might be useful to establish the box dimension or Hausdorff dimension (especially for the lower bounds) for more general groups of functions.展开更多
Fractional integral of continuous functions has been discussed in the present paper. If the order of Riemann-Liouville fractional integral is v, fractal dimension of Riemann-Liouville fractional integral of any contin...Fractional integral of continuous functions has been discussed in the present paper. If the order of Riemann-Liouville fractional integral is v, fractal dimension of Riemann-Liouville fractional integral of any continuous functions on a closed interval is no more than 2 - v.展开更多
Recently,degenerate poly-Bernoulli polynomials are defined in terms of degenerate polyexponential functions by Kim-Kim-Kwon-Lee.The aim of this paper is to further examine some properties of the degenerate poly-Bernou...Recently,degenerate poly-Bernoulli polynomials are defined in terms of degenerate polyexponential functions by Kim-Kim-Kwon-Lee.The aim of this paper is to further examine some properties of the degenerate poly-Bernoulli polynomials by using three formulas from the recently developed‘λ-umbral calculus.’In more detail,we represent the degenerate poly-Bernoulli polynomials by Carlitz Bernoulli polynomials and degenerate Stirling numbers of the first kind,by fully degenerate Bell polynomials and degenerate Stirling numbers of the first kind,and by higherorder degenerate Bernoulli polynomials and degenerate Stirling numbers of the second kind.展开更多
Multiplicative calculus(MUC)measures the rate of change of function in terms of ratios,which makes the exponential functions significantly linear in the framework of MUC.Therefore,a generally non-linear optimization p...Multiplicative calculus(MUC)measures the rate of change of function in terms of ratios,which makes the exponential functions significantly linear in the framework of MUC.Therefore,a generally non-linear optimization problem containing exponential functions becomes a linear problem in MUC.Taking this as motivation,this paper lays mathematical foundation of well-known classical Gauss-Newton minimization(CGNM)algorithm in the framework of MUC.This paper formulates the mathematical derivation of proposed method named as multiplicative Gauss-Newton minimization(MGNM)method along with its convergence properties.The proposed method is generalized for n number of variables,and all its theoretical concepts are authenticated by simulation results.Two case studies have been conducted incorporating multiplicatively-linear and non-linear exponential functions.From simulation results,it has been observed that proposed MGNM method converges for 12972 points,out of 19600 points considered while optimizing multiplicatively-linear exponential function,whereas CGNM and multiplicative Newton minimization methods converge for only 2111 and 9922 points,respectively.Furthermore,for a given set of initial value,the proposed MGNM converges only after 2 iterations as compared to 5 iterations taken by other methods.A similar pattern is observed for multiplicatively-non-linear exponential function.Therefore,it can be said that proposed method converges faster and for large range of initial values as compared to conventional methods.展开更多
We establish the construction theory of function based upon a local field Kp as underlying space. By virture of the concept of pseudo-differential operator, we introduce ＂fractal calculus＂ （or, p-type calculus, or,...We establish the construction theory of function based upon a local field Kp as underlying space. By virture of the concept of pseudo-differential operator, we introduce ＂fractal calculus＂ （or, p-type calculus, or, Gibbs-Butzer calculus）. Then, show the Jackson direct approximation theorems, Bermstein inverse approximation theorems and the equivalent approximation theorems for compact group D（C Kp） and locally compact group Kp^＋-（= Kp）, so that the foundation of construction theory of function on local fields is established. Moreover, the Jackson type, Bernstein type, and equivalent approximation theorems on the HOlder-type space C^σ（Kp）, σ 〉0, are proved; then the equivalent approximation theorem on Sobolev-type space Wr（Kp）, σ≥0, 1≤r 〈∞, is shown.展开更多
The Lebesgue-Nikodym Theorem states that for a Lebesgue measure an additive set function ?which is -absolutely continuous is the integral of a Lebegsue integrable a measurable function;that is, for all measurable sets...The Lebesgue-Nikodym Theorem states that for a Lebesgue measure an additive set function ?which is -absolutely continuous is the integral of a Lebegsue integrable a measurable function;that is, for all measurable sets.?Such a property is not shared by vector valued set functions. We introduce a suitable definition of the integral that will extend the above property to the vector valued case in its full generality. We also discuss a further extension of the Fundamental Theorem of Calculus for additive set functions with values in an infinite dimensional normed space.展开更多
文摘Area integral functions are introduced for sectorial operators on Hilbert spaces. We establish the equivalence relationship between the square and area integral functions. This immediately extends McIntosh/Yagi＇s results on H∞ functional calculus of sectorial operators on Hilbert spaces to the case when the square functions are replaced by the area integral functions.
基金Supported by the National Natural Science Foundation of China(Grant No.11671201)。
文摘Let H be a complex infinite dimensional Hilbert space and B(H)be the algebra of all bounded linear operators on H.In this paper,we mainly study the operators that satisfy both a-Weyl's theorem and property(R).Also,the operators whose functional calculus satisfies the two properties are also explored.We give the features for the operator or its functional calculus for which both a-Weyl's theorem and property(R)hold.
基金Project supported by the National Natural Science Foundation of China (Grant No. 51007068)the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20100201120028)+2 种基金the Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2012JQ7026)the Fundamental Research Funds for the Central Universities of China (Grant No. 2012jdgz09)the State Key Laboratory of Electrical Insulation and Power Equipment of China (Grant No. EIPE12303)
文摘Based on the fact that the real inductor and the real capacitor are fractional order in nature and the fractional calculus,the transfer function modeling and analysis of the open-loop Buck converter in a continuous conduction mode（CCM） operation are carried out in this paper.The fractional order small signal model and the corresponding equivalent circuit of the open-loop Buck converter in a CCM operation are presented.The transfer functions from the input voltage to the output voltage,from the input voltage to the inductor current,from the duty cycle to the output voltage,from the duty cycle to the inductor current,and the output impedance of the open-loop Buck converter in CCM operation are derived,and their bode diagrams and step responses are calculated,respectively.It is found that all the derived fractional order transfer functions of the system are influenced by the fractional orders of the inductor and the capacitor.Finally,the realization of the fractional order inductor and the fractional order capacitor is designed,and the corresponding PSIM circuit simulation results of the open-loop Buck converter in CCM operation are given to confirm the correctness of the derivations and the theoretical analysis.
文摘The linear relationship between fractal dimensions of a type of generalized Weierstrass functions and the order of their fractional calculus has been proved. The graphs and numerical results given here further indicate the corresponding relationship.
基金National Natural Science Foundation of Zhejiang Province
文摘Based on the combination of fractional calculus with fractal functions, a new type of functions is introduced; the definition, graph, property and dimension of this function are discussed.
文摘According to the necessary condition of the functional taking the extremum, that is its first variation is equal to zero, the variational problems of the functionals for the undetermined boundary in the calculus of variations are researched, the functionals depend on single argument, arbitrary unknown functions and their derivatives of higher orders. A new view point is posed and demonstrated, i.e. when the first variation of the functional is equal to zero, all the variational terms are not independent to each other, and at least one of them is equal to zero. Some theorems and corollaries of the variational problems of the functionals are obtained.
基金supported by Viet Nam National Foundation for Science and Technology Development(NAFOSTED) under grant number 101.03-2015.15supported by the Vietnam National University,Hanoi(QG.16.09)
文摘In this article, we investigate the density of the solution to a class of stochastic functional differential equations by means of Malliavin calculus. Our aim is to provide upper and lower Gaussian estimates for the density.
基金NBHM Department of Atomic Energy,Government of India,Mumbai for the finanicai assistance under PDF sanction no.2/40(37)/2014/R&D-II/14131
文摘The object of this article is to study and develop the generalized fractional calcu- lus operators given by Saigo and Maeda in 1996. We establish generalized fractional calculus formulas involving the product of R-function, Appell function F3 and a general class of poly- nomials. The results obtained provide unification and extension of the results given by Saxena et al. [13], Srivastava and Grag [17], Srivastava et al. [20], and etc. The results are obtained in compact form and are useful in preparing some tables of operators of fractional calculus. On account of the general nature of the Saigo-Maeda operators, R-function, and a general class of polynomials a large number of new and known results involving Saigo fractional calculus operators and several special functions notably H-function, /-function, Mittag-Leffier function, generalized Wright hypergeometric function, generalized Bessel-Maitland function follow as special cases of our main findings.
文摘In this study, the impulsive predator-prey dynamic systems on time scales calculus are studied. When the system has periodic solution is investigated, and three different conditions have been found, which are necessary for the periodic solution of the predator-prey dynamic systems with Beddington-DeAngelis type functional response. For this study the main tools are time scales calculus and coincidence degree theory. Also the findings are beneficial for continuous case, discrete case and the unification of both these cases. Additionally, unification of continuous and discrete case is a good example for the modeling of the life cycle of insects.
文摘The aim of this paper is to study the existence of integrable solutions of a nonlinear functional integral equation in the space of Lebesgue integrable functions on unbounded interval, L1(R+). As an application we deduce the existence of solution of an initial value problem of fractional order that be studied only on a bounded interval. The main tools used are Schauder fixed point theorem, measure of weak noncompactness, superposition operator and fractional calculus.
文摘In this paper, we consider a class of Sobolev-type fractional neutral stochastic differential equations driven by fractional Brownian motion with infinite delay in a Hilbert space. When α>1-H, by the technique of Sadovskii’s fixed point theorem, stochastic calculus and the methods adopted directly from deterministic control problems, we study the approximate controllability of the stochastic system.
文摘Image forgery is a crucial part of the transmission of misinformation,which may be illegal in some jurisdictions.The powerful image editing software has made it nearly impossible to detect altered images with the naked eye.Images must be protected against attempts to manipulate them.Image authentication methods have gained popularity because of their use in multimedia and multimedia networking applications.Attempts were made to address the consequences of image forgeries by creating algorithms for identifying altered images.Because image tampering detection targets processing techniques such as object removal or addition,identifying altered images remains a major challenge in research.In this study,a novel image texture feature extraction model based on the generalized k-symbolWhittaker function(GKSWF)is proposed for better image forgery detection.The proposed method is divided into two stages.The first stage involves feature extraction using the proposed GKSWF model,followed by classification using the“support vector machine”(SVM)to distinguish between authentic and manipulated images.Each extracted feature from an input image is saved in the features database for use in image splicing detection.The proposed GKSWF as a feature extraction model is intended to extract clues of tampering texture details based on the probability of image pixel.When tested on publicly available image dataset“CASIA”v2.0(ChineseAcademy of Sciences,Institute of Automation),the proposed model had a 98.60%accuracy rate on the YCbCr(luminance(Y),chroma blue(Cb)and chroma red(Cr))color spaces in image block size of 8×8 pixels.The proposed image authentication model shows great accuracy with a relatively modest dimension feature size,supporting the benefit of utilizing the k-symbol Whittaker function in image authentication algorithms.
文摘Area integral functions are introduced for sectorial operators on L^p-spaces. We establish the equivalence between the square and area integral functions for sectorial operators on L^p spaces. This follows that the results of Cowling, Doust, McIntosh, Yagi, and Le Merdy on H^∞ functional calculus of seetorial operators on LP-spaces hold true when the square functions are replaced by the area integral functions.
基金Supported by National Natural Science Foundation of China(Grant No.12071218)the Fundamental Research Funds for the Central Universities(Grant No.30917011340)。
文摘Establishing the accurate relationship between fractional calculus and fractals is an important research content of fractional calculus theory.In the present paper,we investigate the relationship between fractional calculus and fractal functions,based only on fractal dimension considerations.Fractal dimension of the Riemann-Liouville fractional integral of continuous functions seems no more than fractal dimension of functions themselves.Meanwhile fractal dimension of the Riemann-Liouville fractional differential of continuous functions seems no less than fractal dimension of functions themselves when they exist.After further discussion,fractal dimension of the Riemann-Liouville fractional integral is at least linearly decreasing and fractal dimension of the Riemann-Liouville fractional differential is at most linearly increasing for the Holder continuous functions.Investigation about other fractional calculus,such as the Weyl-Marchaud fractional derivative and the Weyl fractional integral has also been given elementary.This work is helpful to reveal the mechanism of fractional calculus on continuous functions.At the same time,it provides some theoretical basis for the rationality of the definition of fractional calculus.This is also helpful to reveal and explain the internal relationship between fractional calculus and fractals from the perspective of geometry.
文摘The present paper investigates the fractal structure of fractional integrals of Weierstrass functions. The exact box dimension for such functions many important cases is established. We need to point out that, although the result itself achieved in the present paper is interesting, the new technique and method should be emphasized. These novel ideas might be useful to establish the box dimension or Hausdorff dimension (especially for the lower bounds) for more general groups of functions.
文摘Fractional integral of continuous functions has been discussed in the present paper. If the order of Riemann-Liouville fractional integral is v, fractal dimension of Riemann-Liouville fractional integral of any continuous functions on a closed interval is no more than 2 - v.
文摘Recently,degenerate poly-Bernoulli polynomials are defined in terms of degenerate polyexponential functions by Kim-Kim-Kwon-Lee.The aim of this paper is to further examine some properties of the degenerate poly-Bernoulli polynomials by using three formulas from the recently developed‘λ-umbral calculus.’In more detail,we represent the degenerate poly-Bernoulli polynomials by Carlitz Bernoulli polynomials and degenerate Stirling numbers of the first kind,by fully degenerate Bell polynomials and degenerate Stirling numbers of the first kind,and by higherorder degenerate Bernoulli polynomials and degenerate Stirling numbers of the second kind.
文摘Multiplicative calculus(MUC)measures the rate of change of function in terms of ratios,which makes the exponential functions significantly linear in the framework of MUC.Therefore,a generally non-linear optimization problem containing exponential functions becomes a linear problem in MUC.Taking this as motivation,this paper lays mathematical foundation of well-known classical Gauss-Newton minimization(CGNM)algorithm in the framework of MUC.This paper formulates the mathematical derivation of proposed method named as multiplicative Gauss-Newton minimization(MGNM)method along with its convergence properties.The proposed method is generalized for n number of variables,and all its theoretical concepts are authenticated by simulation results.Two case studies have been conducted incorporating multiplicatively-linear and non-linear exponential functions.From simulation results,it has been observed that proposed MGNM method converges for 12972 points,out of 19600 points considered while optimizing multiplicatively-linear exponential function,whereas CGNM and multiplicative Newton minimization methods converge for only 2111 and 9922 points,respectively.Furthermore,for a given set of initial value,the proposed MGNM converges only after 2 iterations as compared to 5 iterations taken by other methods.A similar pattern is observed for multiplicatively-non-linear exponential function.Therefore,it can be said that proposed method converges faster and for large range of initial values as compared to conventional methods.
文摘We establish the construction theory of function based upon a local field Kp as underlying space. By virture of the concept of pseudo-differential operator, we introduce ＂fractal calculus＂ （or, p-type calculus, or, Gibbs-Butzer calculus）. Then, show the Jackson direct approximation theorems, Bermstein inverse approximation theorems and the equivalent approximation theorems for compact group D（C Kp） and locally compact group Kp^＋-（= Kp）, so that the foundation of construction theory of function on local fields is established. Moreover, the Jackson type, Bernstein type, and equivalent approximation theorems on the HOlder-type space C^σ（Kp）, σ 〉0, are proved; then the equivalent approximation theorem on Sobolev-type space Wr（Kp）, σ≥0, 1≤r 〈∞, is shown.
文摘The Lebesgue-Nikodym Theorem states that for a Lebesgue measure an additive set function ?which is -absolutely continuous is the integral of a Lebegsue integrable a measurable function;that is, for all measurable sets.?Such a property is not shared by vector valued set functions. We introduce a suitable definition of the integral that will extend the above property to the vector valued case in its full generality. We also discuss a further extension of the Fundamental Theorem of Calculus for additive set functions with values in an infinite dimensional normed space.
