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 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
In this paper, strong summability of Cesaro means (of critical order) of Fourier-Laplace series on unit sphere is discussed. The Pointwise convergence conditions are established. The results of this paper are anal-ogo...In this paper, strong summability of Cesaro means (of critical order) of Fourier-Laplace series on unit sphere is discussed. The Pointwise convergence conditions are established. The results of this paper are anal-ogous to those of single and multiple Fourier series.展开更多
Let f∈L p(R), 1≤p≤∞, and c j be the inner product of f and the Hermite function h j. Assume that c j′s satisfy|c j|·j r=o(1) as j→∞.If r=5/4, then the Hermite series Σc jh j converges to f almost everywhe...Let f∈L p(R), 1≤p≤∞, and c j be the inner product of f and the Hermite function h j. Assume that c j′s satisfy|c j|·j r=o(1) as j→∞.If r=5/4, then the Hermite series Σc jh j converges to f almost everywhere. If r=9/4-1/p, then Σ c jh j converges to f in L展开更多
The d-dimensional classical Hardy spaces H_p (T^d) are introduced and it is shown that the maximal operator of the Riemann sums of a distribution is bounded from H_p(T^d)to L_p(T^2) (d/(d+1)<p≤∞) and is of weak t...The d-dimensional classical Hardy spaces H_p (T^d) are introduced and it is shown that the maximal operator of the Riemann sums of a distribution is bounded from H_p(T^d)to L_p(T^2) (d/(d+1)<p≤∞) and is of weak type (1, 1) provided that the supremum in the maximal operator is taken over a positive cone. The same is proved for the conjugate Riemann sums. As a consequence we obtain that every function f∈L_1(T^d)is a.e. Riemann summable to f, provided again that the limit is taken over a positive cone.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.
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
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
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.
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
In this article we introduce the difference sequence spaces Wo [f, △m], W1 [f, △m],W∞[f ,△m] and S[f, △m], defined by a modulus function f. We obtain a relation between W1 【f, △m] ∩ l∞[f, △m] and S[f, △m]...In this article we introduce the difference sequence spaces Wo [f, △m], W1 [f, △m],W∞[f ,△m] and S[f, △m], defined by a modulus function f. We obtain a relation between W1 【f, △m] ∩ l∞[f, △m] and S[f, △m]∩ l∞[f, △m] and prove some inclusion results.展开更多
We derive necessary and sufficient conditions for the existence of a Hermitian nonnegative-definite solution to the matrix equation AXB = C. Moreover, we derive a representation of a general Hermitian nonnegative-defi...We derive necessary and sufficient conditions for the existence of a Hermitian nonnegative-definite solution to the matrix equation AXB = C. Moreover, we derive a representation of a general Hermitian nonnegative-definite solution. We then apply our solution to two examples, including a comparison of our solution to a proposed solution by Zhang in [1] using an example problem given from [1]. Our solution demonstrates that the proposed general solution from Zhang in [1] is incorrect. We also give a second example in which we derive the general covariance structure so that two matrix quadratic forms are independent.展开更多
This paper surveys some new results on the oscillation of all solutions of the linear nentral delaydifference equation of the form △(y<sub>n</sub> - p<sub>n</sub>y<sub>n-k</sub>) +...This paper surveys some new results on the oscillation of all solutions of the linear nentral delaydifference equation of the form △(y<sub>n</sub> - p<sub>n</sub>y<sub>n-k</sub>) + q<sub>n</sub>y<sub>n-l</sub> = 0, n = 0,1, 2…where { p<sub>n</sub> } and { q<sub>n</sub> } are twe real numbers sequences with q<sub>n</sub>≥0, and k and l are positive integers. These re-sults do not require the usual assumptionAlso, some interesting open problems on this topic am given.展开更多
We study the structure of Bernstein's first summable operators and show that they con- verge uniformly to continuons functions on the special real orthogonal group SO(n)in this paper.In addition,we have also discu...We study the structure of Bernstein's first summable operators and show that they con- verge uniformly to continuons functions on the special real orthogonal group SO(n)in this paper.In addition,we have also discussed the approximation degree to a class of function Lipα(0<α≤1)on SO(n).展开更多
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 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 this paper,we show that for a unicritical polynomial having a priori bounds,the unique conformal measure of minimal exponent has no atom at the critical point.For a conformal measure of higher exponent,we give a ne...In this paper,we show that for a unicritical polynomial having a priori bounds,the unique conformal measure of minimal exponent has no atom at the critical point.For a conformal measure of higher exponent,we give a necessary and sufficient condition for the critical point to be an atom,in terms of the growth rate of the derivatives at the critical value.展开更多
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.
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.
文摘In this paper, strong summability of Cesaro means (of critical order) of Fourier-Laplace series on unit sphere is discussed. The Pointwise convergence conditions are established. The results of this paper are anal-ogous to those of single and multiple Fourier series.
文摘Let f∈L p(R), 1≤p≤∞, and c j be the inner product of f and the Hermite function h j. Assume that c j′s satisfy|c j|·j r=o(1) as j→∞.If r=5/4, then the Hermite series Σc jh j converges to f almost everywhere. If r=9/4-1/p, then Σ c jh j converges to f in L
基金This research was partly supported by the Hungarian Scientific Research Funds (OTKA) No F019633.
文摘The d-dimensional classical Hardy spaces H_p (T^d) are introduced and it is shown that the maximal operator of the Riemann sums of a distribution is bounded from H_p(T^d)to L_p(T^2) (d/(d+1)<p≤∞) and is of weak type (1, 1) provided that the supremum in the maximal operator is taken over a positive cone. The same is proved for the conjugate Riemann sums. As a consequence we obtain that every function f∈L_1(T^d)is a.e. Riemann summable to f, provided again that the limit is taken over a positive cone.
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.
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].
文摘In this article we introduce the difference sequence spaces Wo [f, △m], W1 [f, △m],W∞[f ,△m] and S[f, △m], defined by a modulus function f. We obtain a relation between W1 【f, △m] ∩ l∞[f, △m] and S[f, △m]∩ l∞[f, △m] and prove some inclusion results.
文摘We derive necessary and sufficient conditions for the existence of a Hermitian nonnegative-definite solution to the matrix equation AXB = C. Moreover, we derive a representation of a general Hermitian nonnegative-definite solution. We then apply our solution to two examples, including a comparison of our solution to a proposed solution by Zhang in [1] using an example problem given from [1]. Our solution demonstrates that the proposed general solution from Zhang in [1] is incorrect. We also give a second example in which we derive the general covariance structure so that two matrix quadratic forms are independent.
基金Projects supported by the National Natural Science Foundation of China
文摘This paper surveys some new results on the oscillation of all solutions of the linear nentral delaydifference equation of the form △(y<sub>n</sub> - p<sub>n</sub>y<sub>n-k</sub>) + q<sub>n</sub>y<sub>n-l</sub> = 0, n = 0,1, 2…where { p<sub>n</sub> } and { q<sub>n</sub> } are twe real numbers sequences with q<sub>n</sub>≥0, and k and l are positive integers. These re-sults do not require the usual assumptionAlso, some interesting open problems on this topic am given.
文摘We study the structure of Bernstein's first summable operators and show that they con- verge uniformly to continuons functions on the special real orthogonal group SO(n)in this paper.In addition,we have also discussed the approximation degree to a class of function Lipα(0<α≤1)on SO(n).
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.
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.
文摘In this paper,we show that for a unicritical polynomial having a priori bounds,the unique conformal measure of minimal exponent has no atom at the critical point.For a conformal measure of higher exponent,we give a necessary and sufficient condition for the critical point to be an atom,in terms of the growth rate of the derivatives at the critical value.