Browsing by Author "R.S. Pathak"
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PublicationArticle A class of Hankel convolutions(2009) R.S. Pathak; S.M. TripathiHankel translations and Hankel convolutions of three different orders are defined. Their properties are investigated. An application to the Bessel differential operator is given. © 2009 Taylor & Francis.PublicationArticle PublicationArticle PublicationArticle A Distributional Hardy Transformation(1979) R.S. Pathak; J.N. PandeyThe Hardy’s F-transform [Formula Omitted] is extended to distributions. The corresponding inversion formula [Formula Omitted] is shown to be valid in the weak distributional sense. This is accomplished by transferring the inversion formula onto the testing function space for the generalized functions under consideration and then showing that the limiting process in the resulting formula converges with respect to the topology of the testing function space. © 1979, Hindawi Publishing Corporation. All rights reserved.PublicationArticle A generalization of Hμ-spaces and Hankel transforms(Kluwer Academic Publishers, 1986) R.S. Pathak; H.K. Sahoo[No abstract available]PublicationArticle A generalized pseudo-differential operator on Gel'fand-Shilov space and Sobolev space(2006) R.S. Pathak; Akhilesh PrasadA generalized pseudo-differential operator involving the symbol a(x, ξ) whose derivatives satisfy certain growth conditions is studied on Gel'fand-Shilov space Sα,Aβ,B. It is shown that the generalized pseudo-differential operator is a well defined mapping of Sα,Aβ,B(R) into another space of the same type on R2. A Sobolev space boundedness result is obtained. The product of two generalized pseudo-differential operator is shown to be a pseudo-differential operator.PublicationArticle Abelian Theorems for Whittaker Transforms(1987) Richard D. Carmichael; R.S. PathakInitial and final value Abelian theorems for the Whittaker transform of functions and of distributions are obtained. The Abelian theorems are obtained as the complex variable of the transform approaches 0 or ∞ in absolute value inside a wedge region in the right half plane. © 1989, Hindawi Publishing Corporation. All rights reserved.PublicationArticle An n-dimensional pseudo-differential operator involving the Hankel transformation(2012) R.S. Pathak; Akhilesh Prasad; Manish KumarAn n-dimensional pseudo-differential operator (p.d.o.) involving the ndimensional Hankel transformation is defined. The symbol class H m is introduced. It is shown that p.d.o.'s associated with symbols belonging to this class are continuous linear mappings of the n-dimensional Zemanian space H μ(I n) into itself. An integral representation for the p.d.o. is obtained. Using the Hankel convolution, it is shown that the p.d.o. satisfies a certain L1-norm inequality. © Indian Academy of Sciences.PublicationArticle Analytic functions having distributional boundary values in W'-spaces(1980) R.S. PathakIt is shown that the functions which are analytic in tubular radial domains and satisfy certain growth conditions have distributional boundary values in the weak topology of (WΩ)'-space. Representation of analytic functions in terms of distributional boundary values are given. Converse results are also obtained. An analytic decomposition theorem is proved. The main theorems are established by means of a number of lemmas concerning WM, WΩ spaces and their dual spaces. Several new lemmas are proved for K{Mp} spaces from which results for W,M-spaces can be easily deduced. © 1980, Cambridge Philosophical Society. All rights reserved.PublicationArticle Asymptotic behavior of the convolution transform(1993) R.S. Pathak; P.K. PandeyCarmichael and Pathak [1] have given abelian theorems for the H-transform of classical functions and generalized functions as the complex variable of the transform approaches zero or infinity in a wedge domain in the right half plane. Many classical integral transforms are special cases of H-transform. The convolution transform studied by Hirschman and Widder [3] is not a special case of the H-transform. In fact Tanno [4] has shown that certain generalized convolution transform includes the H-transform as a special case. In this paper Abelian theorems for the convolution transform are obtained. In obtaining these results we follow closely Zemanian [5] and Carmichael and Pathak [1]. © 1993, Taylor & Francis Group, LLC. All rights reserved.PublicationArticle Asymptotic behaviours of a class of integral transforms in complex domains(1989) R.S. Pathak; S.K. Mishra[No abstract available]PublicationArticle Asymptotic expansions of the Mehler-Fock transform(Springer India, 1989) R.S. Pathak; R.N. PandeyAsymptotic expansions in the two limits x → + ∞ and x → 0+ are obtained for the Mehler-Fock transform {Mathematical expression} where P -1/2+ir m (cosh x) is the associated Legendre function. © 1989 Indian Academy of Sciences.PublicationArticle Asymptotic expansions of the wavelet transform for large and small values of b(2009) R.S. Pathak; Ashish PathakAsymptotic expansions of the wavelet transform for large and small values of the translation parameter b are obtained using asymptotic expansions of the Fourier transforms of the function and the wavelet. Asymptotic expansions of Mexican hat wavelet transform, Morlet wavelet transform, and Haar wavelet transform are obtained as special cases. Asymptotic expansion of the wavelet transform has also been obtained for small values of b when asymptotic expansions of the function and the wavelet near origin are given. Copyright © 2009 R. S. Pathak and A. Pathak.PublicationArticle Asymptotic expansions of the wavelet transform in Rn(Indian Mathematical Society, 2015) R.S. PathakMoment asymptotic expansions of the n-dimensional wavelet transform are obtained for large values of the dilation parameter a. Using representation of the n-dimensional wavelet transform as a 2n-dimensional Fourier transform asymptotic expansion of the wavelet transform with explicit error bound is also obtained. Results are illustrated by means of suitable examples. © Indian Mathematical Society, 2015.PublicationArticle Biorthogonal wavelets in a generalized Sobolev space(2006) R.S. Pathak; Gireesh PandeyA general construction of wavelets biorthogonal with respect to the generalized Sobolev space HWω (ℝ) is given. Some interesting results are discussed. Examples of biorthogonal wavelets are given. © 2006 Taylor & Francis.PublicationArticle Boundedness of the wavelet transform in certain function spaces(2007) R.S. Pathak; S.K. SinghUsing convolution transform theory boundedness results for the wavelet transform are obtained in the Triebel space-LpΩ,k, Hörmander space-Bp,q(ℝn) and general function space-L∞,k- where k denotes a weight function possessing specific properties in each case. © 2007 Victoria University All rights reserved.PublicationArticle Cauchy and Poisson integral representations for ultradistributions of compact support and distributional boundary values(1981) R.S. PathakUltradistributions of compact support are represented as the boundary values of Cauchy and Poisson integrals corresponding to tubular radial domains Tc′ = Rn + iC′, C′⊂⊂C, where C is an open, connected, convex cone. The Cauchy integral of U∈ℰ′Mp is shown to be an analytic function in Tc′ which satisfies a certain boundedness condition. Analytic functions which satisfy a specified growth condition in TC′ have a distributional boundary value which can be used to determine an ℰ′Mp distribution. © 1981, Royal Society of Edinburgh. All rights reserved.PublicationArticle Cauchy wavelet transform of ultra-distributions in tube domains(Taylor and Francis Ltd., 2015) R.S. Pathak; S.K. UpadhyayIn this paper, Cauchy wavelet transform of ultra-distributions in tube domains is defined and its various properties are studied using the theory of Cauchy integrals and Poisson integrals of ultra-distributions. © 2015 Taylor & Francis.PublicationArticle Certain pseudo-differential operator associated with the Bessel operator(2000) R.S. Pathak; S. PathakUsing inverse Hankel transform a symbol is defined, and the pseudo-differential operator (p.d.o.) G(x, D) associated with the Bessel operator d2/dx2 + (1 - 4μ2)/4x2 in terms of this symbol is defined. It is shown that the p.d.o. is bounded in a certain Sobolev type space associated with the Hankel transform.PublicationArticle Continuation of functionals on ultradifferentiable function spaces(Kluwer Academic Publishers, 1995) R.S. Pathak; A.C. Paul[No abstract available]