Siam Journal on Mathematical Analysis, Jul 1, 1982
Series of Orthogonal Polynomials as Hyperfunctions. [SIAM Journal on Mathematical Analysis 13, 66... more Series of Orthogonal Polynomials as Hyperfunctions. [SIAM Journal on Mathematical Analysis 13, 664 (1982)]. Ahmed I. Zayed, Gilbert G. Walter. Abstract. The convergence of series of orthogonal polynomials and of associated ...
Page 1. Applicable Analysis. Vol. 54. pp. 135-1 50 Reprints available directly from the publishel... more Page 1. Applicable Analysis. Vol. 54. pp. 135-1 50 Reprints available directly from the publishel Photocopying permitted by licen~e only Q 1994 Gordon and Breach Science Publishers S A. Pnnted in Malaysia New Summation Formulas for Multivariate ...
ABSTRACT Almost all the known orthonormal wavelets, except for the Haar and Shannon wavelets, can... more ABSTRACT Almost all the known orthonormal wavelets, except for the Haar and Shannon wavelets, cannot be expressed in closed form. By a closed-form expression we mean a representation of the function in terms of elementary functions, such as trigonometric, ...
Information technology: transmission, processing and storage, 2001
The aim of this chapter is to discuss the relationship between Lagrange interpolation and samplin... more The aim of this chapter is to discuss the relationship between Lagrange interpolation and sampling theorems. The first three sections can be regarded as an alternative introduction to sampling theory, avoiding the Fourier analysis of Chapter 2, at least to begin with. Because Lagrange interpolation is the central theme of the chapter, it is natural to start off with an introduction to the Lagrange interpolation method and then proceed to a more general form of it, which we call Lagrange-type interpolation. Having done that, we can then investigate the relationship between Lagrange-type interpolation and sampling theorems, in particular, the Whittaker-Shannon-Kotel’nikov (WSK) sampling theorem.
Transactions of the American Mathematical Society, Feb 1, 1988
ABSTRACT The polynomials P n (α,β) (x) which are orthogonal on (-1,1) with weight function w(x)=(... more ABSTRACT The polynomials P n (α,β) (x) which are orthogonal on (-1,1) with weight function w(x)=(1-x) α (1+x) β are known as the Jacobi polynomials and they can be represented by means of hypergeometric functions. If one changes the integer n to a complex number λ in this hypergeometric representation, then one gets Jacobi functions (of the first kind). The series expansion of a function f∈L p (w) in terms of Jacobi polynomials leads to the Fourier- Jacobi coefficients f ^ (α,β) (n)=2 -α-β-1 ∫w(x)f(x)P n (α,β) (x)dx· If one replaces n by the real number λ>-(α+β+1)/2 then one has the continuous Jacobi transform f ^ (α,β) (λ), which is well defined if f(x)(1+x) -β ∈L 1 (w). The authors obtain an inversion formula such that f(x)=∫f ^ (α,β) (λ-q)R(x,λ)dσ(λ), where σ is a measure on [0,∞), 2q=α+β+1 and R(x,λ) is a Jacobi series in x with coefficients depending on P λ-q (β,α) (n) which are, as a function of λ, orthogonal with orthogonality measure σ. The authors need to assume that (α+β+1)/2 is a positive integer. Somewhat simpler formulas are obtained for a modified transform in terms of the entire functions λ+α+ββP λ (α,β) (x). In a forthcoming paper, T. H. Koornwinder and G. G. Walter [“The finite continuous Jacobi transform and its inverse”] were able to treat the general case α>-1 and β>-1, without the restriction that (α+β+1)/2 is a positive integer.
Real Singularities of Singular SturmLiouville Expansions. [SIAM Journal on Mathematical Analysis... more Real Singularities of Singular SturmLiouville Expansions. [SIAM Journal on Mathematical Analysis 18, 219 (1987)]. Gilbert G. Walter, Ahmed J. Zayed. Abstract. Elliptic equations in polar coordinates lead to singular SturmLiouville ...
The discrete Fourier transform is considered as one of the most powerful tools in digital signal ... more The discrete Fourier transform is considered as one of the most powerful tools in digital signal processing, which enable us to find the spectrum of finite-duration signals. In this article, we introduce the notion of discrete quadratic-phase Fourier transform, which encompasses a wider class of discrete Fourier transforms, including classical discrete Fourier transform, discrete fractional Fourier transform, discrete linear canonical transform, discrete Fresnal transform, and so on. To begin with, we examine the fundamental aspects of the discrete quadratic-phase Fourier transform, including the formulation of Parseval’s and reconstruction formulae. To extend the scope of the present study, we establish weighted and non-weighted convolution and correlation structures associated with the discrete quadratic-phase Fourier transform.
The present study is the first of its kind, aiming to explore the interface between the ridgelet ... more The present study is the first of its kind, aiming to explore the interface between the ridgelet and linear canonical transforms. To begin with, we formulate a family of linear canonical ridgelet waveforms by suitably chirping a one-dimensional wavelet along a specific direction. The construction of novel ridgelet waveforms is demonstrated via a suitable example supported by vivid graphics. Subsequently, we introduce the notion of linear canonical ridgelet transform, which not only embodies the classical ridgelet transform but also yields another new variant of the ridgelet transform based on the fractional Fourier transform. Besides studying all the fundamental properties, we also present an illustrative example on the implementation of the linear canonical ridgelet transform on a bivariate function.
The purpose of this short paper is to show the invalidity of a Fourier series expansion of fracti... more The purpose of this short paper is to show the invalidity of a Fourier series expansion of fractional order as derived by G. Jumarie in a series of papers. In his work the exponential functions e^inω x are replaced by the Mittag-Leffler functions E_α (i (nω x)^α) , over the interval [0, M_α/ ω] where 0< ω<∞ and M_α is the period of the function E_α( ix^α), i.e., E_α( ix^α)=E_α( i(x+M_α)^α).
In this paper we consider a semigroup on trigonometric expansions that will be called the Theta s... more In this paper we consider a semigroup on trigonometric expansions that will be called the Theta semigroup since its kernel is a multiple of the third Jacobi theta function. We study properties of this semigroup and prove that it is a positive diffusion semigroup. We also obtain that its subordinated semigroup is the classical Poisson semigroup. The extensions to higher dimensions and to periodic ultra distributions are also considered.
Proceedings of the American Mathematical Society, 1996
An analog of the Whittaker-Shannon-Kotel′nikov sampling theorem is derived for functions with val... more An analog of the Whittaker-Shannon-Kotel′nikov sampling theorem is derived for functions with values in a separable Hilbert space. The proof uses the concept of frames and frame operators in a Hilbert space. One of the consequences of this theorem is that it allows us to derive sampling theorems associated with boundary-value problems and some homogeneous integral equations, which in turn gives us a generalization of another sampling theorem by Kramer.
Atlas home || Conferences | Abstracts | about Atlas 5th International ISAAC Congress July 25-30, ... more Atlas home || Conferences | Abstracts | about Atlas 5th International ISAAC Congress July 25-30, 2005 Department of Mathematics and Informatics, University of Catania Catania, Sicily, Italy. Organizers International ISAAC Board ...
Siam Journal on Mathematical Analysis, Jul 1, 1982
Series of Orthogonal Polynomials as Hyperfunctions. [SIAM Journal on Mathematical Analysis 13, 66... more Series of Orthogonal Polynomials as Hyperfunctions. [SIAM Journal on Mathematical Analysis 13, 664 (1982)]. Ahmed I. Zayed, Gilbert G. Walter. Abstract. The convergence of series of orthogonal polynomials and of associated ...
Page 1. Applicable Analysis. Vol. 54. pp. 135-1 50 Reprints available directly from the publishel... more Page 1. Applicable Analysis. Vol. 54. pp. 135-1 50 Reprints available directly from the publishel Photocopying permitted by licen~e only Q 1994 Gordon and Breach Science Publishers S A. Pnnted in Malaysia New Summation Formulas for Multivariate ...
ABSTRACT Almost all the known orthonormal wavelets, except for the Haar and Shannon wavelets, can... more ABSTRACT Almost all the known orthonormal wavelets, except for the Haar and Shannon wavelets, cannot be expressed in closed form. By a closed-form expression we mean a representation of the function in terms of elementary functions, such as trigonometric, ...
Information technology: transmission, processing and storage, 2001
The aim of this chapter is to discuss the relationship between Lagrange interpolation and samplin... more The aim of this chapter is to discuss the relationship between Lagrange interpolation and sampling theorems. The first three sections can be regarded as an alternative introduction to sampling theory, avoiding the Fourier analysis of Chapter 2, at least to begin with. Because Lagrange interpolation is the central theme of the chapter, it is natural to start off with an introduction to the Lagrange interpolation method and then proceed to a more general form of it, which we call Lagrange-type interpolation. Having done that, we can then investigate the relationship between Lagrange-type interpolation and sampling theorems, in particular, the Whittaker-Shannon-Kotel’nikov (WSK) sampling theorem.
Transactions of the American Mathematical Society, Feb 1, 1988
ABSTRACT The polynomials P n (α,β) (x) which are orthogonal on (-1,1) with weight function w(x)=(... more ABSTRACT The polynomials P n (α,β) (x) which are orthogonal on (-1,1) with weight function w(x)=(1-x) α (1+x) β are known as the Jacobi polynomials and they can be represented by means of hypergeometric functions. If one changes the integer n to a complex number λ in this hypergeometric representation, then one gets Jacobi functions (of the first kind). The series expansion of a function f∈L p (w) in terms of Jacobi polynomials leads to the Fourier- Jacobi coefficients f ^ (α,β) (n)=2 -α-β-1 ∫w(x)f(x)P n (α,β) (x)dx· If one replaces n by the real number λ&amp;gt;-(α+β+1)/2 then one has the continuous Jacobi transform f ^ (α,β) (λ), which is well defined if f(x)(1+x) -β ∈L 1 (w). The authors obtain an inversion formula such that f(x)=∫f ^ (α,β) (λ-q)R(x,λ)dσ(λ), where σ is a measure on [0,∞), 2q=α+β+1 and R(x,λ) is a Jacobi series in x with coefficients depending on P λ-q (β,α) (n) which are, as a function of λ, orthogonal with orthogonality measure σ. The authors need to assume that (α+β+1)/2 is a positive integer. Somewhat simpler formulas are obtained for a modified transform in terms of the entire functions λ+α+ββP λ (α,β) (x). In a forthcoming paper, T. H. Koornwinder and G. G. Walter [“The finite continuous Jacobi transform and its inverse”] were able to treat the general case α&amp;gt;-1 and β&amp;gt;-1, without the restriction that (α+β+1)/2 is a positive integer.
Real Singularities of Singular SturmLiouville Expansions. [SIAM Journal on Mathematical Analysis... more Real Singularities of Singular SturmLiouville Expansions. [SIAM Journal on Mathematical Analysis 18, 219 (1987)]. Gilbert G. Walter, Ahmed J. Zayed. Abstract. Elliptic equations in polar coordinates lead to singular SturmLiouville ...
The discrete Fourier transform is considered as one of the most powerful tools in digital signal ... more The discrete Fourier transform is considered as one of the most powerful tools in digital signal processing, which enable us to find the spectrum of finite-duration signals. In this article, we introduce the notion of discrete quadratic-phase Fourier transform, which encompasses a wider class of discrete Fourier transforms, including classical discrete Fourier transform, discrete fractional Fourier transform, discrete linear canonical transform, discrete Fresnal transform, and so on. To begin with, we examine the fundamental aspects of the discrete quadratic-phase Fourier transform, including the formulation of Parseval’s and reconstruction formulae. To extend the scope of the present study, we establish weighted and non-weighted convolution and correlation structures associated with the discrete quadratic-phase Fourier transform.
The present study is the first of its kind, aiming to explore the interface between the ridgelet ... more The present study is the first of its kind, aiming to explore the interface between the ridgelet and linear canonical transforms. To begin with, we formulate a family of linear canonical ridgelet waveforms by suitably chirping a one-dimensional wavelet along a specific direction. The construction of novel ridgelet waveforms is demonstrated via a suitable example supported by vivid graphics. Subsequently, we introduce the notion of linear canonical ridgelet transform, which not only embodies the classical ridgelet transform but also yields another new variant of the ridgelet transform based on the fractional Fourier transform. Besides studying all the fundamental properties, we also present an illustrative example on the implementation of the linear canonical ridgelet transform on a bivariate function.
The purpose of this short paper is to show the invalidity of a Fourier series expansion of fracti... more The purpose of this short paper is to show the invalidity of a Fourier series expansion of fractional order as derived by G. Jumarie in a series of papers. In his work the exponential functions e^inω x are replaced by the Mittag-Leffler functions E_α (i (nω x)^α) , over the interval [0, M_α/ ω] where 0< ω<∞ and M_α is the period of the function E_α( ix^α), i.e., E_α( ix^α)=E_α( i(x+M_α)^α).
In this paper we consider a semigroup on trigonometric expansions that will be called the Theta s... more In this paper we consider a semigroup on trigonometric expansions that will be called the Theta semigroup since its kernel is a multiple of the third Jacobi theta function. We study properties of this semigroup and prove that it is a positive diffusion semigroup. We also obtain that its subordinated semigroup is the classical Poisson semigroup. The extensions to higher dimensions and to periodic ultra distributions are also considered.
Proceedings of the American Mathematical Society, 1996
An analog of the Whittaker-Shannon-Kotel′nikov sampling theorem is derived for functions with val... more An analog of the Whittaker-Shannon-Kotel′nikov sampling theorem is derived for functions with values in a separable Hilbert space. The proof uses the concept of frames and frame operators in a Hilbert space. One of the consequences of this theorem is that it allows us to derive sampling theorems associated with boundary-value problems and some homogeneous integral equations, which in turn gives us a generalization of another sampling theorem by Kramer.
Atlas home || Conferences | Abstracts | about Atlas 5th International ISAAC Congress July 25-30, ... more Atlas home || Conferences | Abstracts | about Atlas 5th International ISAAC Congress July 25-30, 2005 Department of Mathematics and Informatics, University of Catania Catania, Sicily, Italy. Organizers International ISAAC Board ...
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