A generalized Taylor series of a complex function was derived and some related theorems about its convergence region were given. The generalized Taylor theorem can be applied to greatly enlarge convergence regions of...A generalized Taylor series of a complex function was derived and some related theorems about its convergence region were given. The generalized Taylor theorem can be applied to greatly enlarge convergence regions of approximation series given by other traditional techniques. The rigorous proof of the generalized Taylor theorem also provides us with a rational base of the validity of a new kind of powerful analytic technique for nonlinear problems, namely the homotopy analysis method.展开更多
In this article we show that the order of the point value, in the sense of Lojasiewicz, of a tempered distribution and the order of summability of the pointwise Fourier inversion formula are closely related. Assuming ...In this article we show that the order of the point value, in the sense of Lojasiewicz, of a tempered distribution and the order of summability of the pointwise Fourier inversion formula are closely related. Assuming that the order of the point values and certain order of growth at infinity are given for a tempered distribution, we estimate the order of summability of the Fourier inversion formula. For Fourier series, and in other cases, it is shown that if the distribution has a distributional point value of order k, then its Fourier series is e.v. Cesaro summable to the distributional point value of order k＋1. Conversely, we also show that if the pointwise Fourier inversion formula is e.v. Cesaro summable of order k, then the distribution is the （k ＋ 1）-th derivative of a locally integrable function, and the distribution has a distributional point value of order k ＋ 2. We also establish connections between orders of summability and local behavior for other Fourier inversion problems.展开更多
In this article, we introduce the concept of lacunary statistical convergence of order a of real number sequences and give some inclusion relations between the sets of lacu- nary statistical convergence of order α an...In this article, we introduce the concept of lacunary statistical convergence of order a of real number sequences and give some inclusion relations between the sets of lacu- nary statistical convergence of order α and strong Nα （p）-summability. Furthermore, some relations between the spaces Nθα （p） and Sθα are examined.展开更多
In this paper,we introduce the concept of λ-statistical convergence of order α.Also some relations between the λ-statistical convergence of order α and strong（V,λ）-summability of order α are given.
In this paper the authors consider the summability of formal solutions for some first order singular PDEs with irregular singularity. They prove that in this case the formal solutions will be divergent, but except a e...In this paper the authors consider the summability of formal solutions for some first order singular PDEs with irregular singularity. They prove that in this case the formal solutions will be divergent, but except a enumerable directions, the formal solutions are Borel summable.展开更多
A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms. Under some conditions on θ, it is proved that the maximal operator of the θ-means defined in a cone ...A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms. Under some conditions on θ, it is proved that the maximal operator of the θ-means defined in a cone is bounded from the amalgam Hardy space W（hp, e∞） to W（Lp,e∞）. This implies the almost everywhere convergence of the θ-means in a cone for all f ∈ W（L1, e∞） velong to L1.展开更多
In this paper we obtain a new version of the Orlicz-Pettis theorem by using statistical convergence. To obtain this result we prove a theorem of uniform convergence on matrices related to the statistical convergence.
In a more recent paper, the second author has introduced a space ｜Cα｜k as the set of all series by absolute summable using Cesaro matrix of order α 〉 -1. In the present paper we extend it to the absolute NSrlund ...In a more recent paper, the second author has introduced a space ｜Cα｜k as the set of all series by absolute summable using Cesaro matrix of order α 〉 -1. In the present paper we extend it to the absolute NSrlund space ｜Np^θ｜k taking Norlund matrix in place of Cesaro matrix, and also examine some topological structures, α-β-γ-duals and the Schauder base of this space. Further we characterize certain matrix operators on that space and determine their operator norms, and so extend some well-known results.展开更多
In this study, as the domain of four dimensional Euler mean E(r,s) of orders r,sin the space L_p for 0 < p < 1, we examine the double sequence space ε_p^(r,s) and some properties of four dimensional Euler mean....In this study, as the domain of four dimensional Euler mean E(r,s) of orders r,sin the space L_p for 0 < p < 1, we examine the double sequence space ε_p^(r,s) and some properties of four dimensional Euler mean. We determine the α-and β(bp)-duals of the space εp r,s, and characterize the classes(ε_p^(r,s):M_u),(ε_p^(r,s):C_(bp)) and(ε_p^(r,s):L_q) of four dimensional matrix transformations, where 1 ≤q < ∞. Finally, we shortly emphasize on the Euler spaces of single and double sequences, and note some further suggestions.展开更多
The notion of μ-smoothness points of periodic functions of several variables is introduced and the rate of convergence of the Gauss-Weierstrass means of relevant Fourier series at these points is investigated.
The pointwise estimates of the deviation Tn,A,Bf（·） -- f（·） in terms of moduli of continuity w. f and w. f there are proved. Analogical results on norm approximation with remarks and corollaries are also...The pointwise estimates of the deviation Tn,A,Bf（·） -- f（·） in terms of moduli of continuity w. f and w. f there are proved. Analogical results on norm approximation with remarks and corollaries are also given. In the results there are used the essentially weaker conditions than these in [Mittal, M. L.： J. Math. Anal. Appl., 220, 434-450） （1998） Theorem 1, p. 4377.展开更多
Let V be a star shaped region. In 2006, Colzani, Meaney and Prestini proved that if function f satisfies some condition, then the multiplier transform with the characteristic function of tV as the multiplier tends to ...Let V be a star shaped region. In 2006, Colzani, Meaney and Prestini proved that if function f satisfies some condition, then the multiplier transform with the characteristic function of tV as the multiplier tends to f almost everywhere, when t goes to∞. In this paper we use a Theorem established by K.K.Chen to show that if we change their multiplier, then the condition on f can be weakened.展开更多
Various investigators such as Khan ([1-4]), Khan and Ram [5], Chandra [6,7], Leindler [8], Mishra et al. [9], Mishra [10], Mittal et al. [11], Mittal, Rhoades and Mishra [12], Mittal and Mishra [13], Rhoades et al. [1...Various investigators such as Khan ([1-4]), Khan and Ram [5], Chandra [6,7], Leindler [8], Mishra et al. [9], Mishra [10], Mittal et al. [11], Mittal, Rhoades and Mishra [12], Mittal and Mishra [13], Rhoades et al. [14] have determined the degree of approximation of 2π-periodic signals (functions) belonging to various classes Lipα, Lip(α,r), Lip(ξ(t),r) and W(Lr,ζ(t)) of functions through trigonometric Fourier approximation (TFA) using different summability matrices with monotone rows. Recently, Mittal et al. [15], Mishra and Mishra [16], Mishra [17] have obtained the degree of approximation of signals belonging to -class by general summability matrix, which generalizes the results of Leindler [8] and some of the results of Chandra [7] by dropping monotonicity on the elements of the matrix rows (that is, weakening the conditions on the filter, we improve the quality of digital filter). In this paper, a theorem concerning the degree of approximation of the conjugate of a signal (function) f belonging to Lip(ξ(t),r) class by (E,q) summability of conjugate series of its Fourier series has been established which in turn generalizes the results of Chandra [7] and Shukla [18].展开更多
文摘A generalized Taylor series of a complex function was derived and some related theorems about its convergence region were given. The generalized Taylor theorem can be applied to greatly enlarge convergence regions of approximation series given by other traditional techniques. The rigorous proof of the generalized Taylor theorem also provides us with a rational base of the validity of a new kind of powerful analytic technique for nonlinear problems, namely the homotopy analysis method.
基金support by the Louisiana State Board of Regents grant LEQSF(2005-2007)-ENH-TR-21
文摘In this article we show that the order of the point value, in the sense of Lojasiewicz, of a tempered distribution and the order of summability of the pointwise Fourier inversion formula are closely related. Assuming that the order of the point values and certain order of growth at infinity are given for a tempered distribution, we estimate the order of summability of the Fourier inversion formula. For Fourier series, and in other cases, it is shown that if the distribution has a distributional point value of order k, then its Fourier series is e.v. Cesaro summable to the distributional point value of order k＋1. Conversely, we also show that if the pointwise Fourier inversion formula is e.v. Cesaro summable of order k, then the distribution is the （k ＋ 1）-th derivative of a locally integrable function, and the distribution has a distributional point value of order k ＋ 2. We also establish connections between orders of summability and local behavior for other Fourier inversion problems.
文摘In this article, we introduce the concept of lacunary statistical convergence of order a of real number sequences and give some inclusion relations between the sets of lacu- nary statistical convergence of order α and strong Nα （p）-summability. Furthermore, some relations between the spaces Nθα （p） and Sθα are examined.
文摘In this paper,we introduce the concept of λ-statistical convergence of order α.Also some relations between the λ-statistical convergence of order α and strong（V,λ）-summability of order α are given.
基金supported by the NSFC and the 973 key project of the MOST
文摘In this paper the authors consider the summability of formal solutions for some first order singular PDEs with irregular singularity. They prove that in this case the formal solutions will be divergent, but except a enumerable directions, the formal solutions are Borel summable.
基金Supported by the Hungarian Scientific Research Funds (OTKA) No. K67642
文摘A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms. Under some conditions on θ, it is proved that the maximal operator of the θ-means defined in a cone is bounded from the amalgam Hardy space W（hp, e∞） to W（Lp,e∞）. This implies the almost everywhere convergence of the θ-means in a cone for all f ∈ W（L1, e∞） velong to L1.
基金Supported by Junta de Andalucia grant FQM 257supported by MEC Project MTM-2006-15546-C02-01
文摘In this paper we obtain a new version of the Orlicz-Pettis theorem by using statistical convergence. To obtain this result we prove a theorem of uniform convergence on matrices related to the statistical convergence.
基金Supported by Pamukkale University Scientific Research Pro jects Coordinatorship(Grant No.2014FBE061)
文摘In a more recent paper, the second author has introduced a space ｜Cα｜k as the set of all series by absolute summable using Cesaro matrix of order α 〉 -1. In the present paper we extend it to the absolute NSrlund space ｜Np^θ｜k taking Norlund matrix in place of Cesaro matrix, and also examine some topological structures, α-β-γ-duals and the Schauder base of this space. Further we characterize certain matrix operators on that space and determine their operator norms, and so extend some well-known results.
文摘In this study, as the domain of four dimensional Euler mean E(r,s) of orders r,sin the space L_p for 0 < p < 1, we examine the double sequence space ε_p^(r,s) and some properties of four dimensional Euler mean. We determine the α-and β(bp)-duals of the space εp r,s, and characterize the classes(ε_p^(r,s):M_u),(ε_p^(r,s):C_(bp)) and(ε_p^(r,s):L_q) of four dimensional matrix transformations, where 1 ≤q < ∞. Finally, we shortly emphasize on the Euler spaces of single and double sequences, and note some further suggestions.
文摘The notion of μ-smoothness points of periodic functions of several variables is introduced and the rate of convergence of the Gauss-Weierstrass means of relevant Fourier series at these points is investigated.
文摘The pointwise estimates of the deviation Tn,A,Bf（·） -- f（·） in terms of moduli of continuity w. f and w. f there are proved. Analogical results on norm approximation with remarks and corollaries are also given. In the results there are used the essentially weaker conditions than these in [Mittal, M. L.： J. Math. Anal. Appl., 220, 434-450） （1998） Theorem 1, p. 4377.
基金Supported by the National Natural Science Foundation of China(11071065,11171306)
文摘Let V be a star shaped region. In 2006, Colzani, Meaney and Prestini proved that if function f satisfies some condition, then the multiplier transform with the characteristic function of tV as the multiplier tends to f almost everywhere, when t goes to∞. In this paper we use a Theorem established by K.K.Chen to show that if we change their multiplier, then the condition on f can be weakened.
文摘Various investigators such as Khan ([1-4]), Khan and Ram [5], Chandra [6,7], Leindler [8], Mishra et al. [9], Mishra [10], Mittal et al. [11], Mittal, Rhoades and Mishra [12], Mittal and Mishra [13], Rhoades et al. [14] have determined the degree of approximation of 2π-periodic signals (functions) belonging to various classes Lipα, Lip(α,r), Lip(ξ(t),r) and W(Lr,ζ(t)) of functions through trigonometric Fourier approximation (TFA) using different summability matrices with monotone rows. Recently, Mittal et al. [15], Mishra and Mishra [16], Mishra [17] have obtained the degree of approximation of signals belonging to -class by general summability matrix, which generalizes the results of Leindler [8] and some of the results of Chandra [7] by dropping monotonicity on the elements of the matrix rows (that is, weakening the conditions on the filter, we improve the quality of digital filter). In this paper, a theorem concerning the degree of approximation of the conjugate of a signal (function) f belonging to Lip(ξ(t),r) class by (E,q) summability of conjugate series of its Fourier series has been established which in turn generalizes the results of Chandra [7] and Shukla [18].
