Ram Shankar Pathak

Orthogonal series representations for generalized functions

Abstract A test function space A L wp of smooth weighted L p -functions is constructed whose elements possess orthonormal series expansion converging in the space. Conditions for convergence in A L wp -space of any orthogonal series are obtained. The space of continuous linear functionals on A L wp is investigated and orthogonal series representations for its elements are given. Various special cases of the expansion are discussed, and their applications are pointed out.

ON CONVOLUTION FOR WAVELET TRANSFORM

A basic function D(x,y,z) associated with the general wavelet transform is defined and its properties are investigated. Using D(x,y,z) associated with the wavelet transform, translation and convolution for this transform are defined and certain existence theorems are proved. An approximation theorem involving wavelet convolution is also proved.

On Hankel transforms of ultradistributions

On developpe une theorie des espaces de fonctions test Hμ ,ad k, Hμ bq et Hμ ,ad k bq . Ce sont des generalisations des espaces fonctions test de type Hμ. Les elements des espaces duaux sont appeles ultradistributions. On montre que la transformation de Hankel hμ pour μ≥-1/2 est une application lineaire continue pour chacun de ces espaces dans certains espaces du type ci-dessus

TheWavelet Transform on Spaces of Type S

The spaces of W-type were studied by I.M. Gel’fand and G.E. Shilov [22]. They investigated the behaviour of Fourier transformation on theW-spaces. AlsoW-spaces are applied to the theory of partial differential equations.

Holomorphlc functions in tubes associated with ultradistributions

Let C be a regular cone in and in some instances a more general proper open connected subset of . We study holomorphic functions in tubes which satisfy a norm growth in L r with the bound involving the associated function M∗(ρ) corresponding to sequences M p p = 0,1,2,…, which are used to define ultradistributions. We show that these holomorphic functions haveFourier–Laplace integral representations and obtain boundary values in the ultradistribution spaces of Beurling type D′((M P ),L r ) which are generalizations of the Schwartz distributions D′ L r. The L r norm growth which we study here is motivated by the norm growth proved here for the Cauchy integral of elements in D′(∗,L r ), where ∗ is either (M p ) or {M p }, ultradistributions of Beurling type or Roumieu type, respectively.

Recent developments on the Stieltjes transform of generalized functions.

This paper is concerned with recent developments on the Stieltjes transform of generalized functions. Sections 1 and 2 give a very brief introduction to the subject and the Stieltjes transform of ordinary functions with an emphasis to the inversion theorems. The Stieltjes transform of generalized functions is described in section 3 with a special attention to the inversion theorems of this transform. Sections 4 and 5 deal with the adjoint and kernel methods used for the development of the Stieltjes transform of generalized functions. The real and complex inversion theorems are discussed in sections 6 and 7. The Poisson transform of generalized functions, the iteration of the Laplace transform and the iterated Stieltjes transfrom are included in sections 8, 9 and 10. The Stieltjes transforms of different orders and the fractional order integration and further generalizations of the Stieltjes transform are discussed in sections 11 and 12. Sections 13, 14 and 15 are devoted to Abelian theorems, initial-value and final-value results. Some applications of the Stieltjes transforms are discussed in section 16. The final section deals with some open questions and unsolved problems. Many important and recent references are listed at the end.

Asymptotic Expansions of the Wavelet Transform for Large and Small Values of b

Asymptotic 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 bwhen asymptotic expansions of the function and the wavelet near origin are given.

Continuity of Pseudo-differential Operators Associated with the Bessel Operator in Some Gevrey Spaces

The pseudo-differential operator (p.d.o) h w , a associated with the Bessel operator d 2 / dx 2 + (1 m 4 w 2 )/4 x 2 involving the symbol a ( x , y ) whose derivatives satisfy certain growth conditions depending on some increasing sequences, is studied on certain Gevrey spaces (ultradifferentiable function spaces). It is shown that the operator h w , a is a continuous linear map of one Gevrey space into another Gevrey space. A special p.d.o. called the Gevrey-Hankel potential is defined and some of its properties are investigated. A variant H w , a of h w , a is also discussed.

Mexican hat wavelet transform of distributions

ABSTRACT Theory of Weierstrass transform is exploited to derive many interesting new properties of the Mexican hat wavelet transform. A real inversion formula in the differential operator form for the Mexican hat wavelet transform is established. Mexican hat wavelet transform of distributions is defined and its properties are studied. An approximation property of the distributional wavelet transform is investigated which is supported by a nice example.

Pseudo-differential operators involving Watson transform

A class of pseudo-differential operators (p.d.o.s) associated with the general Fourier kernel studied by Hardy and Titchmarsh is defined. A symbol class T m is introduced. It is shown that the p.d.o.s associated with the symbol are continuous linear mappings of the Braaksma and Schuitman space T(λ,μ) into itself. An integral representation of p.d.o. is obtained. Some special forms of the symbol are considered. It is shown that these p.d.o.s and their products are bounded in certain Sobolev type space.