This thesis discusses a few known results, and several new results, in **polynomial** in- terpolation **theory**. This chapter introduces some of the basic concepts in **interpolation** **theory**, while preparing for the material of the other chapters. Each of the next two chapters comprises a paper, one accepted for publication by Mathematics of Computa- tion and the other ready for submission, on formulas for divided diﬀerences of implicit functions. While the presentation of the ﬁrst paper was discussed in detail with my co-author Prof. Michael Floater, I developed and drafted both papers. A formula for the higher-order derivatives of implicit functions appears as a limiting case of these formulas. Insofar as the number of terms in these formulas describes a previously un- known pattern, I published these sequences in The On-Line Encyclopedia of Integer Sequences. The ﬁnal chapter shows explicitly how algebraic curves of a certain type give rise to generalized principal lattices in higher-dimensional space, yielding many new examples of such meshes.

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1 aa therefore most grateful to many persons# These include my former M#Bc# supervisor principal E.M. Wright, Southampton colleagues Professor H#B# G riffiths, Dr# P.A. Bemet and Hr# P#J* Taylor, and professor S#I# Curie of St# Andrews# I also owe much to Professor F#J# Davis, Professor of Applied Mathematics at Brown University, Rhode Island, for the stimulus from his book * **interpolation** and

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However, there is one particular case when the trapezoidal rule provides a much better accuracy of approximation. This is the case of a smooth and L-periodic function f , for which the rate of convergence of the trapezoidal quadrature will automatically adjust to the degree of regularity of f . In the literature, this result is commonly referred to as standard; a proof, for instance, can be found in [2, Section 2.9] or in [3, Section 4.1.2]. Moreover, this result has far-reaching implications in scientific computation; it facilitates construction of efficient numerical methods, for example, in the scattering **theory**, see [1]. Since one can actually come across several slightly different versions of this result, we formulate it as a theorem and provide a proof that follows [3] and is based on a simpler argument than the one used in [2].

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A non band-limited alternative approach that has been widely used since the 1990s is the spline representation. A spline of degree n is a continuous piece-wise **polynomial** function of degree n of a real variable with derivatives up to order n−1. This representation has the advantage of being equally justifiable on a theoretical and practical basis [ 19 ]. It can model the physical process of drawing a smooth curve and it is well adapted to signal processing thanks to its optimality properties. It is handy to consider the B-spline representation [ 15 ] where the continuous underlying signal is expressed as the convolution between the B-spline kernel, that is compactly supported, and the parameters of the representation, namely the B-spline coefficients. One of the strongest arguments in favor of the B-spline **interpolation** is that it approaches the Shannon-Whittaker **interpolation** as the order increases [ 2 ].

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mials cannot cancel each other out, hence f is algebraic, and so must have only finitely many poles. Therefore there is some x ∈ N such that there are no poles in the half plane Re(z) ≥ x, so f is analytic in this region. We apply Lemma 6.2.7 to f(z − x), giving that f (z − x) is a **polynomial** here. From this, we conclude that f(z) is **polynomial** in the half-plane Re(z) ≥ x, and thus by the identity theorem f(z) must be a **polynomial** on the whole plane.

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In some recent papers [9, 10, 39], a greedy algorithm has been studied, that computes (multivariate) approximate Fekete points by extracting maximum volume submatrices from rectangular Vandermonde matrices on suitable discretization meshes. It works on arbitrary geometries and uses only optimized tools of numerical linear algebra (essentially QR-like factorizations). There is a strong connection with the **theory** of admissible meshes for mul- tivariate **polynomial** approximation, recently developed by Calvi and Levenberg [15]. There are also good perspectives in the application to numerical cubature and to the numerical so- lution of PDEs by collocation and discrete least squares methods [45]. A renewed interest is indeed arising in methods based on global **polynomial** approximation; see, e.g., [29].

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rate (1.10) for the problem (1.7) can be chosen from the restricted class of mono- tone subsets of F. While we develop, as in [11], the algorithms and **theory** for (1.7), we hasten to add that all results and algorithms presented in the present paper ap- ply, without any modifications, to the adaptive numerical solution of more general parametric equations: all that is required is bounded invertibility of the parametric equation for all instances of the parameter sequence and a characterisation of the parametric solution families’ dependence on the parameters in the sequence. Such characterisations seem to hold for broad classes of parametric problems (we refer to [20–23] for details).

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polynomials of type I (deﬁned in (3)) and of type II (deﬁned in (5)). As an application, for the sake of brevity, we only mention an **interpolation** problem. The solution to this problem will be examined in detail in a future paper (Part 2), in which we will present other applications of the two **polynomial** classes, including to the operators approximation **theory**.

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For 1D and 2D signals, the Shannon-Whittaker **interpolation** with periodic extension can be formulated as a trigonometric **polynomial** **interpolation** (TPI). In this work, we describe and discuss the **theory** of TPI of images and some of its applications. First, the trigonometric **polynomial** interpolators of an image are characterized and it is shown that there is an ambiguity as soon as one size of the image is even. Three classical choices of interpolator for real-valued images are presented and cases where they coincide are pointed out. Then, TPI is applied to the geometric transformation of images, to up-sampling and to down-sampling. General results are expressed for any choice of interpolator but more details are given for the three proposed ones. It is proven that the well-known DFT-based computations have to be slightly adapted.

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where V [−2] means V placed in ghost degree −2, O is the algebra of **polynomial** functions on C [2], and the cochain map is multiplication by σ. The shift by −2 is needed so that the cochain map has total degree 1. The existence of such a resolution means that we can realize the “skyscraper” line operator as a “bound state” of two Wilson lines both associated with the representation V but placed in different cohomological degrees. The corresponding bulk line operator is obtained using the formulas of Section 4.3, where the target of the gauged B-model is taken to be C [2], the vector bundle E on C [2] is trivial and of rank 2, with graded components in degrees 1 and 0, and the bundle morphism T from the former to the latter component is multiplication by σ. In accordance with the Section 4.3, we consider a superconnection on σ ∗ E of the form

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This paper offers a general formula for surface subdivision rules for quad meshes by using 2-D Lagrange inter- polating **polynomial** [1]. We also see that the result obtained is equivalent to the tensor product of (2N + 4)-point n-ary interpolating curve scheme for N ≥ 0 and n ≥ 2. The simple interpolatory subdivision scheme for quadrila- teral nets with arbitrary topology is presented by L. Kobbelt [2], which can be directly calculated from the pro- posed formula. Furthermore, some characteristics and applications of the proposed work are also discussed.

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The possibility of formulating an electrostatic problem leading to very good nodal sets for **polynomial** **interpolation** along the line inspired us to attempt a similar ap- proach in a 2-simplex with the emphasis being on an equilateral triangle. We have formulated a natural extension of the original approach and found that the solution of that electrostatic problem leads to nodal sets which are very close to the best known sets, thus confirming that the electrostatic analogy seems to carry over to the multidimensional case, provided the problem is properly posed. Indeed, there are many ways this can be done, in particular with respect to the treatment of the bounding edges, where we chose the simplest possible approach in order to minimize the computational workload required to find the sought-after steady state solutions. However, our results clearly show that there is a connection between the solution to problems of electrostatics and nodal sets well suited for **polynomial** interpola- tion in the multidimensional simplex. It is our belief that a very similar approach can be applied for finding suitable nodal sets in a higher-dimensional simplex as well.

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Both the above mentioned methods start giving erroneous results. This phenomenon is called Runge‟s phenomenon.These errors could be rectified by the use of Spline curves and Chebyshev polynomials. Spline curves are preferred over Curve fitting and Lagrange if large set of closely spaced points is to be approximated and energy consumption to be minimized with desired precision. It also enables one to move few control points to get exactly the curve one requires instead of moving a large number of curve points. This local controlled nature makes Spline Curve superior as compared to the previous **interpolation** methods.

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low resolution images, often results in visual artifacts, known as “aliasing” artifacts. These are very common in low resolution images and usually these aliasing artifacts either appear as zigzag edges called jaggies or produce blurring effects. Another type of aliasing artifacts is variation of color of pixels over a small number of pixels (termed pixel region). This type of aliasing artifacts produces noisy or flickering shading. A typical example of these artifacts is shown in Fig.1. These artifacts can be reduced by increasing the resolution of an image. This can be done using image **interpolation**, which is generally referred as a process of estimating a set of unknown pixels from a set of known pixels in an image. In this paper different **polynomial** based **interpolation** techniques are discussed which include ideal **interpolation**, nearest neighbour, bilinear, bicubic, high resolution cubic spline, Lagrange and Lanczos **interpolation**. These functions are then applied on MRI image of brain and evaluate the performance of each **interpolation** function discussed in paper. This paper, therefore, is divided into two parts. The first part presents the analytical model of various **interpolation** functions. The second part investigates the various quality measures of image like SNR, PSNR, MSE, SSIM, time taken of the zoomed images using these functions.

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From the information provided above, it is clear that each algorithm is unique. I will be studying each of these routing protocols in depth, and analyzing the efficiencies of each and I will work to find which one is best. It is well known that each of these routing protocols were built around a previously discovered graph **theory** algorithm. For example, OSPF was created knowing that Dijkstra’s algorithm would be used in finding the shortest path. There have been several revisions of this protocol since its release, but those changes were made to accomadate for new technology. The most recent release of OSPF is version 3, and this was to implement IPv6. Overall, the method used to find the shortest path has remained the same for years.

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often that (l.l) - (1.3), or their appropriate counterparts, (1.6) - (1.8), tend to zero. This is in fact so; however, unfortunately, the situation does not appear as clear cut as a subdivision of all possible behaviour into the analogues of positive and null recurrence, and transience. Moreover, we find that a sufficiently irregular initial distribution may cause the limits of the ratios to differ from those where the chain begins from an initial fixed state, again in contrast to the strictly stochastic case. In fact, with respect to a systematic classification of possible behaviour, the following treatment may be regarded only as an initial stage, in a **theory** still to be developed.

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Abstract. In the present paper a characterization of the generalized order and type of entire functions of several complex variables by means of polyno- mial approximation and **interpolation** have been obtained. Our results improve and generalize various results of S.M. Shah [5], M.N. Seremeta [4], Kapoor and Nautiyal [2], Vakarchuk [9], Vakarchuk and Zhir [10], Winiarski ([11],[12]). In this way we summarize and unify the work which has been done on this subject to-date.

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