IIT Madras is a public technical and research university located in Chennai, Tamil Nadu. Founded in 1959, it is recognised as an Institute of National Importance.

IIT Madras has been ranked as the top engineering institute in India for four years in a row by the National Institutional Ranking Framework of the MHRD

It currently offers undergraduate, postgraduate and research degrees across 16 disciplines in Engineering, Sciences, Humanities and Management. About 596 faculty belonging to science and engineering departments and centres of the Institute are engaged in teaching, research and industrial consultancy.

IIT Madras

A Kadec-type theorem is proved for functions belonging to the shift invariant space V(Qr) , where Qr denotes the B-spline function of even order r. It is also shown that if a non-zero function f∈ V(Qm) has infinitely many simple or double zeros on the real axis which are separated, then there exists at least one pair of consecutive zeros whose distance apart is greater than or equal to a certain number which depends on Krein–Favard constants. Further, iterative reconstruction algorithms are provided for functions in V(Qm). Finally, a sampling density theorem for irregular Gabor-type frames {MyjTxi,jg(l):l=0,1,…,k−1,i,j∈Z} is proved for g∈ V(Qm) ∩ L1(R) under certain sufficient conditions on g. © 2016, Springer Science+Business Media Dordrecht.

Department of Mathematics

Acta Applicandae Mathematicae

Sampling and Reconstruction in Shift Invariant Spaces of B -Spline Functions

Let ϕ∈W(C,ℓ1) such that {τnϕ:n∈Zd} forms a Riesz basis for V(ϕ). It is shown that Zd is a stable set of sampling for V(ϕ) if and only if Φ†(x)≠0, for every x∈Td, where Φ†(x):=∑n∈Zdϕ(n)e2πin·x,x∈Td. Sampling formulae are provided for reconstructing a function f∈V(ϕ) from uniform samples using Zak transform and complex analytic technique. The problem of sampling and reconstruction is discussed in the case of irregular samples also. The theory is illustrated with some examples, and numerical implementation for reconstruction of a function from its nonuniform samples is provided using MATLAB. © 2014, Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg.

Annali di Matematica Pura ed Applicata

Sampling and reconstruction in shift-invariant spaces on Rd

The aim of this paper is to obtain certain characterizations for the image of a Sobolev space on the Heisenberg group under the heat kernel transform. We give three types of characterizations for the image of a Sobolev space of positive order \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$H^m(\mathbb {H}^n), m\in \mathbb {N}^n,$\end{document} under the heat kernel transform on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {H}^n,$\end{document} using direct sum and direct integral of Bergmann spaces and certain unitary representations of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {H}^n$\end{document} which can be realized on the Hilbert space of Hilbert-Schmidt operators on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$L^2(\mathbb {R}^n).$\end{document} We also show that the image of Sobolev space of negative order \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$H^{-s}(\mathbb {H}^n), s(>0) \in \mathbb {R}$\end{document} is a direct sum of two weighted Bergman spaces. Finally, we try to obtain some pointwise estimates for the functions in the image of Schwartz class on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {H}^n$\end{document} under the heat kernel transform. © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Mathematische Nachrichten

On the images of Sobolev spaces under the heat kernel transform on the Heisenberg group

We study the theory of shift invariant spaces in L2(G), where G is a compact group. We define a range function and show that a relation between an H-invariant space and the range function is valid as in the case of abelian group setting. Here H is assumed to be a closed normal subgroup of G. We also obtain a decomposition for an H-invariant space in terms of principle H-invariant spaces whose generators give rise to "generalized Parseval frames" and use this result to study H-preserving operators. © 2012 Elsevier Masson SAS.

Fulltext

Bulletin des Sciences Mathematiques

Shift invariant spaces on compact groups

Let FA(Cn) denote the Fock space associated with a real linear transformation A on Cn which is symmetric and positive definite relative to the real inner product Re〈z,w〉, z,w∈Cn. Let BA denote the Bargmann transform, mapping L2(Rn) unitarily onto FA(Cn). In this note, we show that one can find a group G, whose unitary irreducible representation at its base vector coincides with BA*Kw up to a constant multiple, where BA* denotes the adjoint of BA and Kw denotes the reproducing kernel of FA(Cn). © 2010 Elsevier Masson SAS.

Generalized Bargmann transform and a group representation

Let A denote a real linear transformation on Cn which is symmetric and positive-definite relative to the real inner product Re〈z,w〉, z,w∈Cn. Let FA(Cn) denote the Fock space consisting of holomorphic functions on Cn which are square integrable with respect to the Gaussian measure dμA(z)=detRAΠne-Re〈Az,z〉. For w∈Cn, let ewA(z)=eA(z,w)=e-12Re〈Aw,w〉KA(z,w), z∈Cn, where KA is the reproducing kernel for FA(Cn). The main aim of this paper is to show that there exist a,b>0 such that the set of functions {ema+inbA:m,n∈Zn} forms a frame in FA. © 2010 Elsevier Inc.

Journal of Mathematical Analysis and Applications

Frames in generalized Fock spaces

In this paper, it is shown that the class of right Fourier multipliers for the Sobolev space Wk,p(Hn) coincides with the class of right Fourier multipliers for Lp(Hn) for k ∈ ℕ, 1 &lt; p &lt; ∞. Towards this end, it is shown that the operators RjR̄jℒ-1 and RjR̄jRjℒ-1 are bounded on Lp(Hn), 1 &lt; p &lt; ∞, ℒ is the sublaplacian on Hn. This proof is based on the Calderon-Zygmund theory on the Heisenberg group. It is also shown that when p = 1, the class of right multipliers for the Sobolev space Wk,1(Hn) coincides with the dual space of the projective tensor product of two function spaces. © 2010 Akadémiai Kiadó, Budapest, Hungary.

Analysis Mathematica

Fourier multipliers for Sobolev spaces on the Heisenberg group

It is well known that if m is an Lp-multiplier for the Fourier transform on ℝn, (1 &lt; p &lt; ∞) then there exists a pseudomeasure σ such that Tm f = σ * f. A similar problem is discussed for the Lp-Fourier multipliers for H-valued functions on the Heisenberg group, where H is a separable Hilbert space. © Springer-Verlag 2009.

Monatshefte fur Mathematik

Lp-multipliers for the Hilbert space valued functions on the Heisenberg group

The images of Hermite and Laguerre-Sobolev spaces under the Hermite and special Hermite semigroups (respectively) are characterized. These are used to characterize the image of Schwartz class of rapidly decreasing functions f on Rn and Cn under these semigroups. The image of the space of tempered distributions is also considered and a Paley-Wiener theorem for the windowed (short-time) Fourier transform is proved. © 2009 Elsevier Inc.

Holomorphic Sobolev spaces, Hermite and special Hermite semigroups and a Paley-Wiener theorem for the windowed Fourier transform

A local reconstruction method is discussed for functions belonging to a shift invariant space arising out of a function which has polynomial decay in time domain and moderate decay in frequency domain such that its Fourier transform is non-vanishing on a unit interval. This method is useful in solving the scaling identity (associated wavelet basis) locally. © 2008 IMACS.

Applied Numerical Mathematics

Local reconstruction of a function from a non-uniform sampled data

Let Hn denote the Heisenberg group. In this paper, it is shown that if A is a commutative Banach algebra with a bounded approximate identity such that A* and A** have wide Radon Nikodym property, then there is an isometric isomorphism of the class of right translation invariant operators from Lp(Hn, A) into Lq(Hn, A**) onto the dual space of a certain function space where 1 &lt; p, q &lt; ∞. © SAS International Publications.

World University root node

Journal of Analysis and Applications

A characterization of right translation invariant operators from L P(Hn, A) into Lq(Hn, A)

It is shown that a local reconstruction method from a nonuniform sampled data along with discrete wavelet transform and a simple statistical method is applicable in a voice system. © 2009 Elsevier Ltd. All rights reserved.

Computers and Mathematics with Applications

Local reconstruction method and voice system

It is shown that the Hermite 'polynomial' semigroup {e-tℍ: t &gt; 0} maps Lp(ℝn, ρ into the space of holomorphic functions in Lr(ℂn, Vt,p/2(r+e)/2) for each ε &gt; 0, where ρ is the Gaussian measure, Vt,p/2r+e/2 is a scaled version of Gaussian measure with r = p if 1 &lt; p &lt; 2 and r = p′ if 2 &lt; p &lt; ∞ with 1/p + 1/p′ = 1. Conversely if F is a holomorphic function which is in a "slightly" smaller space, namely Lrℂn, Vt,p/2r/2), then it is shown that there is a function f ∈ Lpℝn, ρ) such that e-tH f = F. However, a single necessary and sufficient condition is obtained for the image of L2(ℝn,Pp/2 under e-tH, 1 &lt; p &lt; ∞. Further it is shown that if F is a holomorphic function such that F ε L1 (ℂn, Vt,p/21/2 or F ε Lm1,p ℝ2n, then there exists a function f ε Lp (ℝn,p) such that e-tℍf = F, where m(x,y), e-x2/ (p-1)e4t+1e-y2/e4t-1 and 1 &lt; p &lt; ∞.

International Journal of Mathematics and Mathematical Sciences

Image of Lp(ℝn) under the Hermite semigroup

International Conference on Theoretical and Mathematical Foundations of Computer Science 2008, TMFCS 2008

Foreword

The sampling problem for a shift-invariant space in L2(Rn) arising out of certain subclass of Schwarz class of rapidly decreasing functions is discussed. This is proved to be useful in reconstructing an image from its nonuniform sample values. An algorithm is provided along with illustrations using MatLab.

International Conference on Artificial Intelligence and Pattern Recognition 2007, AIPR 2007

Reconstruction of image in shift-invariant space

Let H be a separable Hilbert space with an orthonormal basis {en/n∈N}, T be a bounded tridiagonal operator on H and Tn be its truncation on span ({e1, e2, ... , en}). We study the operator equation Tx = y through its finite dimensional truncations T nxn = yn. It is shown that if {∥Tn-1en∥} and {∥Tn*-1en∥} are bounded, then T is invertible and the solution of Tx = y can be obtained as a limit in the norm topology of the solutions of its finite dimensional truncations. This leads to uniform boundedness of the sequence {Tn-1}. We also give sufficient conditions for the boundedness of {∥Tn-1en∥} and {∥Tn*-1en∥} in terms of the entries of the matrix of T. © 2005 Elsevier Inc. All rights reserved.

Linear Algebra and Its Applications

Solution of a tridiagonal operator equation

Journal of Inequalities in Pure and Applied Mathematics

Hardy-type inequalities for Hermite expansions

Hardy's inequalities are proved for higher-dimensional Hermite and special Hermite expansions of functions in Hardy spaces. Inequalities for multiple Laguerre expansions are also deduced.

Proceedings of the American Mathematical Society

Hardy's inequalities for hermite and laguerre expansions

Chemoselective reductions of alkenes, α,β-unsaturated carbonyl compounds, nitro and nitroso compounds, N,N-hydrogenolysis of azo and hydrazo functions as well as simultaneous reduction and hydrodehalogenation of substituted aryl halides, including bulkier substrates, were achieved by catalytic transfer hydrogenation (CTH) using mesoporous PdMCM-41 catalyst. The yields were practically unaffected upon recycling of the catalyst. Further, the CTH process is accomplished without affecting the reduction of any other reducible functional group.Palladium-incorporated mesoporous silicate (PdMCM-41) molecular sieves were found to be a highly efficient solid acid catalyst under transfer hydrogen conditions.

Department Of Chemistry

Selective reduction of alkenes, α,β-unsaturated carbonyl compounds, nitroarenes, nitroso compounds, N,N-hydrogenolysis of azo and hydrazo functions as well as simultaneous hydrodehalogenation and reduction of substituted aryl halides over PdMCM-41 catalyst under transfer hydrogen conditions

Catalytic transfer hydrogenation (CTH) of aromatic nitro and carbonyl compounds over novel Fe(III) substituted hexagonal mesoporous aluminophosphate (FeHMA) molecular sieve catalyst showed excellent regioselectivity/chemoselectivity as well as superior recycling capability. Furthermore, the reduction occurs without affecting other functional groups such as –CN, –CHO, –Cl, –CH3, –OCH3 and –NH2.