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This is the second paper in a series to discuss the approximation efficacy of polynomial interpolation of functions. In particular, the approximation accuracy depends sensitively on the locations of the interpolation nodes. We address the problem of finding the optimal symmetrical polynomial interpolation schemes for the triangle. The table for the symmetrical mean minimal interpolation sets for the triangle is given in this paper. An adaptive scheme for determining the interpolation order is also presented.

We consider regular polynomial interpolation algorithms on recursively defined sets of interpolation points which approximate global solutions of arbitrary well-posed systems of linear partial differential equations. Convergence of the 'limit' of the recursively constructed family of polynomials to the solution and error estimates are obtained from a priori estimates for some standard classes of linear partial differential equations, i.e. elliptic and hyperbolic equations. Another variation of the algorithm allows to construct polynomial interpolations which preserve systems of linear partial differential equations at the interpolation points. We show how this can be applied in order to compute higher order terms of WKB-approximations of fundamental solutions of a large class of linear parabolic equations. The error estimates are sensitive to the regularity of the solution. Our method is compatible with recent developments for solution of higher dimensional partial differential equations, i.e. (adaptive) sparse grids, and weighted Monte-Carlo, and has obvious applications to mathematical finance and physics.

We consider regular polynomial interpolation algorithms on recursively defined sets of interpolation points which approximate global solutions of arbitrary well-posed systems of linear partial differential equations. Convergence of the 'limit' of the recursively constructed family of polynomials to the solution and error estimates are obtained from a priori estimates for some standard classes of linear partial differential equations, i.e. elliptic and hyperbolic equations. Another variation of the algorithm allows to construct polynomial interpolations which preserve systems of linear partial differential equations at the interpolation points. We show how this can be applied in order to compute higher order terms of WKB-approximations of fundamental solutions of a large class of linear parabolic equations. The error estimates are sensitive to the regularity of the solution. Our method is compatible with recent developments for solution of higher dimensional partial differential equations, i.e. (adaptive) sparse grids, and weighted Monte-Carlo, and has obvious applications to mathematical finance and physics.

We consider regular polynomial interpolation algorithms on recursively defined sets of interpolation points which approximate global solutions of arbitrary well-posed systems of linear partial differential equations. Convergence of the 'limit' of the recursively constructed family of polynomials to the solution and error estimates are obtained from a priori estimates for some standard classes of linear partial differential equations, i.e. elliptic and hyperbolic equations. Another variation of the algorithm allows to construct polynomial interpolations which preserve systems of linear partial differential equations at the interpolation points. We show how this can be applied in order to compute higher order terms of WKB-approximations of fundamental solutions of a large class of linear parabolic equations. The error estimates are sensitive to the regularity of the solution. Our method is compatible with recent developments for solution of higher dimensional partial differential equations, i.e. (adaptive) sparse grids, and weighted Monte-Carlo, and has obvious applications to mathematical finance and physics.

We consider regular polynomial interpolation algorithms on recursively defined sets of interpolation points which approximate global solutions of arbitrary well-posed systems of linear partial differential equations. Convergence of the 'limit' of the recursively constructed family of polynomials to the solution and error estimates are obtained from a priori estimates for some standard classes of linear partial differential equations, i.e. elliptic and hyperbolic equations. Another variation of the algorithm allows to construct polynomial interpolations which preserve systems of linear partial differential equations at the interpolation points. We show how this can be applied in order to compute higher order terms of WKB-approximations of fundamental solutions of a large class of linear parabolic equations. The error estimates are sensitive to the regularity of the solution. Our method is compatible with recent developments for solution of higher dimensional partial differential equations, i.e. (adaptive) sparse grids, and weighted Monte-Carlo, and has obvious applications to mathematical finance and physics.

We consider regular polynomial interpolation algorithms on recursively defined sets of interpolation points which approximate global solutions of arbitrary well-posed systems of linear partial differential equations. Convergence of the 'limit' of the recursively constructed family of polynomials to the solution and error estimates are obtained from a priori estimates for some standard classes of linear partial differential equations, i.e. elliptic and hyperbolic equations. Another variation of the algorithm allows to construct polynomial interpolations which preserve systems of linear partial differential equations at the interpolation points. We show how this can be applied in order to compute higher order terms of WKB-approximations of fundamental solutions of a large class of linear parabolic equations. The error estimates are sensitive to the regularity of the solution. Our method is compatible with recent developments for solution of higher dimensional partial differential equations, i.e. (adaptive) sparse grids, and weighted Monte-Carlo, and has obvious applications to mathematical finance and physics.

We study strict inclusion relations between approximation and interpolation spaces.

This paper reports on constructive approximation methods for three classes of holomorphic functions on the unit disk which are closely connected each other: the class of starlike and spirallike functions, the class of semigroup generators, and the class of functions with positive real part. It is more-or-less known that starlike or spirallike functions can be defined as solutions of singular differential equations which, in general, are not stable under the motion of interior singular points to the boundary. At the same time, one can establish a perturbation formula which continuously transforms a starlike (or spirallike) function with respect to a boundary point to a starlike (or spirallike) function with respect to an interior point. This formula is based on an appropriate approximation method of holomorphic generators which determine the above-mentioned differential equations. In turn, the well-known Berkson--Porta parametric representation of holomorphic generators leads us to study an approximation-interpolation problem for the class of functions with positive real part. While this problem is of independent interest, the solution we present here is again based on the Berkson--Porta formula. Finally, we apply our results to solve a natural perturbation problem for one-parameter semigroups of holomorphic self-mappings, as well as the eigenvalue problem for the semigroup of composition operators.

We construct polynomial approximations of Dzjadyk type (in terms of the k-th modulus of continuity, $k \ge 1$) for analytic functions defined on a continuum E in the complex plane, which simultaneously interpolate at given points of E. Furthermore, the error in this approximation is decaying as $e^{-cn^\alpha}$ strictly inside E, where c and $\alpha$ are positive constants independent of the degree n of the approximating polynomial.

Restricted non-linear approximation is a type of N-term approximation where a measure $\nu$ on the index set (rather than the counting measure) is used to control the number of terms in the approximation. We show that embeddings for restricted non-linear approximation spaces in terms of weighted Lorentz sequence spaces are equivalent to Jackson and Bernstein type inequalities, and also to the upper and lower Temlyakov property. As applications we obtain results for wavelet bases in Triebel-Lizorkin spaces by showing the Temlyakow property in this setting. Moreover, new interpolation results for Triebel-Lizorkin and Besov spaces are obtained.

In this paper, the realization-free model approximation problem, as stated in \cite{mayo2007framework,beattie2012realization}, is revisited in the case where the interpolating model might be time-delay dependent. To this aim, the Loewner framework, initially settled for delay-free realization, is firstly generalized to the single delay case. Secondly, the (infinite) model approximation $\mathcal{H}_2$ optimality conditions are established through the use of the Lambert functions. Finally, a numerically effective iterative scheme, named \textbf{dTF-IRKA}, similar to the \textbf{TF-IRKA} \cite{beattie2012realization}, is proposed to reach a part of the aforementioned optimality conditions. The proposed method validity and interest are assessed on different numerical examples.

We update the state of the subject approximately 20 years after the publication of a previous article on this topic. This report is mostly a survey, with a sprinkling of assorted new results throughout.

We consider functions $f$ of two real variables, given as trigonometric functions over a finite set $F$ of frequencies. This set is assumed to be closed under rotations in the frequency plane of angle $\frac{2k\pi}{M}$ for some integer $M$. Firstly, we address the problem of evaluating these functions over a similar finite set $E$ in the space plane and, secondly, we address the problems of interpolating or approximating a function $g$ of two variables by such an $f$ over the grid $E.$ In particular, for this aim, we establish an abstract factorization theorem for the evaluation function, which is a key point for an efficient numerical solution to these problems. This result is based on the very special structure of the group $SE(2,N)$, subgroup of the group $SE(2)$ of motions of the plane corresponding to discrete rotations, which is a maximally almost periodic group. Although the motivation of this paper comes from our previous works on biomimetic image reconstruction and pattern recognition, where these questions appear naturally, this topic is related with several classical problems: the FFT in polar coordinates, the Non Uniform FFT, the evaluation of general trigonometric polynomials, and so on.

We consider functions $f$ of two real variables, given as trigonometric functions over a finite set $F$ of frequencies. This set is assumed to be closed under rotations in the frequency plane of angle $\frac{2k\pi}{M}$ for some integer $M$. Firstly, we address the problem of evaluating these functions over a similar finite set $E$ in the space plane and, secondly, we address the problems of interpolating or approximating a function $g$ of two variables by such an $f$ over the grid $E.$ In particular, for this aim, we establish an abstract factorization theorem for the evaluation function, which is a key point for an efficient numerical solution to these problems. This result is based on the very special structure of the group $SE(2,N)$, subgroup of the group $SE(2)$ of motions of the plane corresponding to discrete rotations, which is a maximally almost periodic group. Although the motivation of this paper comes from our previous works on biomimetic image reconstruction and pattern recognition, where these questions appear naturally, this topic is related with several classical problems: the FFT in polar coordinates, the Non Uniform FFT, the evaluation of general trigonometric polynomials, and so on.

We consider functions $f$ of two real variables, given as trigonometric functions over a finite set $F$ of frequencies. This set is assumed to be closed under rotations in the frequency plane of angle $\frac{2k\pi}{M}$ for some integer $M$. Firstly, we address the problem of evaluating these functions over a similar finite set $E$ in the space plane and, secondly, we address the problems of interpolating or approximating a function $g$ of two variables by such an $f$ over the grid $E.$ In particular, for this aim, we establish an abstract factorization theorem for the evaluation function, which is a key point for an efficient numerical solution to these problems. This result is based on the very special structure of the group $SE(2,N)$, subgroup of the group $SE(2)$ of motions of the plane corresponding to discrete rotations, which is a maximally almost periodic group. Although the motivation of this paper comes from our previous works on biomimetic image reconstruction and pattern recognition, where these questions appear naturally, this topic is related with several classical problems: the FFT in polar coordinates, the Non Uniform FFT, the evaluation of general trigonometric polynomials, and so on.

We consider functions $f$ of two real variables, given as trigonometric functions over a finite set $F$ of frequencies. This set is assumed to be closed under rotations in the frequency plane of angle $\frac{2k\pi}{M}$ for some integer $M$. Firstly, we address the problem of evaluating these functions over a similar finite set $E$ in the space plane and, secondly, we address the problems of interpolating or approximating a function $g$ of two variables by such an $f$ over the grid $E.$ In particular, for this aim, we establish an abstract factorization theorem for the evaluation function, which is a key point for an efficient numerical solution to these problems. This result is based on the very special structure of the group $SE(2,N)$, subgroup of the group $SE(2)$ of motions of the plane corresponding to discrete rotations, which is a maximally almost periodic group. Although the motivation of this paper comes from our previous works on biomimetic image reconstruction and pattern recognition, where these questions appear naturally, this topic is related with several classical problems: the FFT in polar coordinates, the Non Uniform FFT, the evaluation of general trigonometric polynomials, and so on.

This article is devoted to developing a theory for effective kernel interpolation and approximation in a general setting. For a wide class of compact, connected $C^\infty$ Riemannian manifolds, including the important cases of spheres and SO(3), we establish, using techniques involving differential geometry and Lie groups, that the kernels obtained as fundamental solutions of certain partial differential operators generate Lagrange functions that are uniformly bounded and decay away from their center at an algebraic rate, and in certain cases, an exponential rate. An immediate corollary is that the corresponding Lebesgue constants for interpolation as well as for $L_2$ minimization are uniformly bounded with a constant whose only dependence on the set of data sites is reflected in the mesh ratio, which measures the uniformity of the data. The kernels considered here include the restricted surface splines on spheres, as well as surface splines for SO(3), both of which have elementary closed-form representations that are computationally implementable. In addition to obtaining bounded Lebesgue constants in this setting, we also establish a "zeros lemma" for domains on compact Riemannian manifolds -- one that holds in as much generality as the corresponding Euclidean zeros lemma (on Lipschitz domains satisfying interior cone conditions) with constants that clearly demonstrate the influence of the geometry of the boundary (via cone parameters) as well as that of the Riemannian metric.

This article is devoted to developing a theory for effective kernel interpolation and approximation in a general setting. For a wide class of compact, connected $C^\infty$ Riemannian manifolds, including the important cases of spheres and SO(3), we establish, using techniques involving differential geometry and Lie groups, that the kernels obtained as fundamental solutions of certain partial differential operators generate Lagrange functions that are uniformly bounded and decay away from their center at an algebraic rate, and in certain cases, an exponential rate. An immediate corollary is that the corresponding Lebesgue constants for interpolation as well as for $L_2$ minimization are uniformly bounded with a constant whose only dependence on the set of data sites is reflected in the mesh ratio, which measures the uniformity of the data. The kernels considered here include the restricted surface splines on spheres, as well as surface splines for SO(3), both of which have elementary closed-form representations that are computationally implementable. In addition to obtaining bounded Lebesgue constants in this setting, we also establish a "zeros lemma" for domains on compact Riemannian manifolds -- one that holds in as much generality as the corresponding Euclidean zeros lemma (on Lipschitz domains satisfying interior cone conditions) with constants that clearly demonstrate the influence of the geometry of the boundary (via cone parameters) as well as that of the Riemannian metric.

INTERPOLATION

We study the approximation of maps into complex manifolds along with interpolation on certain compact subsets of the plane. Results are also obtained regarding approximation and interpolation of sections of holomorphic submersions.

Book Source: Digital Library of India Item 2015.163805 dc.contributor.author: J L Walsh dc.contributor.other: Iisc Library dc.date.accessioned: 2015-07-06T23:10:42Z dc.date.available: 2015-07-06T23:10:42Z dc.date.digitalpublicationdate: 0000-00-00 dc.date.citation: 1935 dc.identifier.barcode: 1010010031983 dc.identifier.origpath: /data/upload/0031/988 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/163805 dc.description.scanningcentre: IISc, Bangalore dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 305 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Iisc dc.publisher: American Mathematical Society dc.rights: Out_of_copyright dc.source.library: Iisc Library dc.subject.classification: Mathematics dc.title: Interpolation And Approximation By The Rational Functions

This paper introduces a quasi-interpolation operator for scalar- and vector-valued finite element spaces constructed on affine, shape-regular meshes with some continuity across mesh interfaces.This operator gives optimal estimates of the best approximation error in any $L^p$-norm assuming regularity in the fractional Sobolev spaces $W^{r,p}$, where $p\in [1,\infty]$ and the smoothness index $r$ can be arbitrarily close to zero. The operator is stable in $L^1$, leaves the corresponding finite element space point-wise invariant whether homogeneous boundary conditions are imposed or not. The theory is illustrated on $H^1$-, $\mathbf{H}(\text{curl})$- and $\mathbf{H}(\text{div})$-conforming spaces.

Motivated by a recently introduced HIMMO key distribution scheme, we consider a modification of the noisy polynomial interpolation problem of recovering an unknown polynomial $f(X) \in Z[X]$ from approximate values of the residues of $f(t)$ modulo a prime $p$ at polynomially many points $t$ taken from a short interval.

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1Old-Time Makers of Medicine

By James Joseph Walsh

Dr. Walsh's <i>Old-Time Makers of Medicine</i> chronicles the history and development of modern medicine from ancient times up to the discovery of America. Throughout this historical guide, Dr. Walsh shows numerous examples of practices thought to be entirely modern that were clearly anticipated hundreds or thousands of years ago. Ancient healers sought to use the body's natural healing ability, rather than rely exclusively on external cures. Physicians even in ancient times relied on what is now recognized as the placebo effect.<br /><br /> Dr. Walsh also addresses training and certification in medicine. Medieval universities anticipate our modern medical textbooks with consolidated records of all research and independent investigations, to provide uniform training for students. Likewise, the reader will find that the ancients reacted to unsuccessful treatment in similar degrees to what might now be called medical malpractice suits.<br /><br /> The book is organized chronologically, beginning with the fall of the Roman Empire and growth of the early Christian Church. From there, Dr. Walsh details the development of medical knowledge and practice in Arabia, to Medieval and Renaissance Europe. The reader will also discover how modern cultures based much of their medical knowledge on ancient Greek teachings. The chapters on Arabian Physicians and Medieval Universities also discuss knowledge exchanged between Arabic and European cultures. Dr. Walsh exposes several misconceptions and misinterpretations of history, especially restrictions of medical research stemming from religious prohibitions. (Summary by LivelyHive)

LibriVox volunteers bring you 24 recordings of The Despairing Lover by William Walsh. This was the Fortnightly Poetry project for March 11, 2012.<br><br>William Walsh was an English poet and critic. It is not as a poet, however, but as the friend and correspondent of Alexander Pope that Walsh is remembered.<br>"Mr Walsh used to tell me," Pope says, "that there was one way left of excelling; for though we had several great poets, we never had any one great poet that was correct, and he desired me to make that my study and my aim." (Summary from Wikipedia )

3Thirteenth: Greatest of Centuries

By James Joseph Walsh

It cannot but seem a paradox to say that the Thirteenth was the greatest of centuries. To most people the idea will appear at once so preposterous that they may not even care to consider it. A certain number, of course, will have their curiosity piqued by the thought that anyone should evolve so curious a notion. Either of these attitudes of mind will yield at once to a more properly receptive mood if it is recalled that the Thirteenth is the century of the Gothic cathedrals, of the foundation of the university, of the signing of Magna Charta, and of the origin of representative government with something like constitutional guarantees throughout the west of Europe. The cathedrals represent a development in the arts that has probably never been equaled either before or since. The university was a definite creation of these generations that has lived and maintained its usefulness practically in the same form in which it was then cast for the seven centuries ever since. The foundation stones of modern liberties are to be found in the documents which for the first time declared the rights of man during this precious period. (Summary from Preface)

4Bumper the White Rabbit

By George Ethelbert Walsh

All little boys and girls who love animals should become acquainted with Bumper the white rabbit. Bumper had many friends, such as Mr. Blind Rabbit, Fuzzy Wuzz and Goggle Eyes, his country cousins; and Bobby Gray Squirrel had his near cousins, Stripe the chipmunk and Webb the flying squirrel; while Buster and White Tail were favored with an endless number of friends and relatives. These books of Twilight Animal Stories are dedicated to all little boys and girls who love wild animals. So come out into the woods with me, and let us listen and watch, and I promise you it will be worth while. - Summary by Introduction of the book

In this second volume of the Twilight Animal series, we'll find out what happens to our friends Bumper, Fuzzy Wuzz, Goggle Eyes, and all the rest, after the events in the first book of the series, "Bumper the White Rabbit".

6Bumper the White Rabbit and His Friends

By George Ethelbert Walsh

In this fourth book of the Twilight Animal series, we find 16 stories of Bumper the white rabbit and the adventures he has with his friends in the forest.

In this sixth volume of the Twilight Animal series, we read about the adventures of Bobby the Gray Squirrel, who (together with his friends from the woods!) brings joy and happiness to a very special new friend. - Summary by Foon

In the seventh volume of the Twilight Animal series, we meet Buster the Bear, a cub who lives in a cave the woods with his mother. After an encounter with an unfriendly forest animal, he gets lost and captured, and has some marvelous adventures with humans. Will he make it back to the forest, or will he end up somewhere else? (Summary by Foon)

White Tail is the son of Father Buck, who is the leader of the herd. He and his rival Young Black Buck get into all kinds of adventures, but they have to be wary of Puma the Mountain Lion and Timber Wolf! Will White Tail be able to keep away from them and follow in his father's hoofsteps? - Summary by Foon

10Religion and Health

By James Joseph Walsh

In the introductory chapter and the one following we find the strong presentation of his thesis on the everlasting reality of religion. The chapter on Prayer, marked by absence of psychological speculation, treats practically of the naturalness and good sense of the constant habit of prayer, and of the value of prayer in all psychoneurosis....The chapter on the Bible and Health is of especial interest, with its argument that the sanitary laws of the Jews could have been no outcome of human development, but rather of Divine origin....The wide reading, extended experience, and specialized scholarship of the writer certify to the value of anything from his pen, and when we find a work of this kind as simple as a primer and as attractive as a story, we may well offer thanks for the boon. Nobody who values knowledge concerning the mysterious relation between holy living and bodily health should be without this book. - Summary by New Catholic World

11Story of Santa Klaus

By William S. Walsh

Who was Santa Klaus? Did he exist? In this study of custom and folklore, the author looks at the real and the legendary man, according to various traditions around the world. - Summary by LynneT

12Tea: Its History and Mystery

By Joseph M. Walsh

Utility, not originality, has been aimed at in the compilation of this work. The obstacles and difficulties its author had met with in his endeavors to learn something of the article he was commissioned to sell when he first entered the Tea trade, the almost total lack of knowledge displayed by the average dealer in the commodity, allied to the numerous inquiries for a work containing “all about tea,” first prompted the undertaking. The material was collated at intervals, in a fragmentary manner, covering a period of over twenty years, and arranged amid the many interruptions incident to an active business life, subjected to constant revisions, repeated prunings and innumerable corrections, due mainly to the varying statements and conflicting opinions of admitted authorities in every branch of the subject. Still, as careful and judicious an arrangement of the data has been given as possible, a faithful effort being made to omit nothing that may prove useful, instructive or profitable to the expert, the dealer or general reader. Aware that many facts have been omitted, and many errors committed in its preparation, he still trusts that the pains he has taken to avoid both have not been in vain, that the former may be few, and the latter of no great importance. The work was compiled under impulse, not under inducement, a single line not being intended originally for the market, and is now being published solely for the benefit of those “whom it may concern.” Philadelphia, December, 1892. - Summary by From the Preface

The present volume is published at the solicitation of many friends who have read the articles contained in it as they appeared at various times in magazines and who deemed that they were worth preservation in a more permanent form. The only possible claim for its filling a want lies in the fact that it presents these workers in medicine not only as scientists but also and especially as men, in relation to their environment, social, religious and educational. - Summary by Preface