We include an application of this result to scales of interpolation spaces. Interpolation theorems in harmonic analysis mark hyunki kim bachelor of science department of mathematics rutgers, the state university of new jersey may 2012 arxiv. To downsample images, choose an interpolation methodaverage. The interpolation models a set of tabulated function values or discrete data into a continuous function. One may also notice that we sometimes avoid stating the results in the full generality.
Interpolation theorem for marcinkiewicz spaces 5 the proofs of these results appear in section 5. Interpolation of a series of data points via chebyshev. Introduction in this paper we present two main classical results of interpolation of operators. In geostatistics, it is assumed that the underlying structure of the data is a stochastic process which leads to an interpolation procedure known as kriging. An interpolation theorem for partitions which are complete. Approximation theory deals with two types of problems. In this chapter we shall prove several theorems which involve, loosely speaking, elimination of superfluous notions. By applying f to the keys, the files become uniformly distributed 2, vol. Interpolation, preservation, and pebble games barwise, jon and van benthem, johan, journal of symbolic logic, 1999 a sufficient condition for nevanlinna parametrization and an extension of heins theorem takahashi, sechiko, nagoya mathematical journal, 1999. In this section, we focus on employing the technique of atomic decomposition to establish two marcinkiewicz type interpolation theorems for weak orlicz martingale spaces. Several characterizations, carleson measures and atomic decomposition for the bloch space b are given. We prove strong type bounds for a certain range of exponents starting from weak type bounds and a distributional estimate like the one that the strong maximal function satis es.
Included in this algorithm is an interpolation of a function that is only known via some number of measurement points. In mathematics, the marcinkiewicz interpolation theorem, discovered by jozef marcinkiewicz, is a result bounding the norms of nonlinear operators acting on l p spaces marcinkiewicz theorem is similar to the rieszthorin theorem about linear operators, but also applies to nonlinear operators. The marcinkiewicz interpolation theorem and its extensions author. A introduction to interpolation theory, interpolation spaces and. The n quantities known as the roots are not related to the coefficients in a simple way.
Marcinkiewicz theorem is similar to the rieszthorin theorem about linear operators, but also applies to nonlinear operators. Polynomial approximation, interpolation, and orthogonal polynomials in the last chapter we saw that the eigenequation for a matrix was a polynomial whose roots were the eigenvalues of the matrix. The marcinkiewicz theorem is extended to function spaces that include both the lebesgueorlicz and lorentz spaces, namely the rearrangement invariant function spaces. Interpolation was the first technique for obtaining an approximation of a function. A proof of the marcinkiewicz interpolation theorem in the special case p0 q0 1,p1 q1 r, u, rn. Indeed, it is not obvious that the polynomial should be able to be written in the form of equation 3. Polynomial approximation, interpolation, and orthogonal. A rieszthorin type interpolation theorem in euclidean jordan algebras. We give a wolff theorem in the quasibanach case that improves that known before for the banach case. Average number of accesses for interpolation search. Let t be a linear operator such that the form 1 is wellde.
The rieszthorin interpolation theorem that motivates the complex interpolation and was proved by riesz in 1927 but only for the lowertriangle case, and the general case by thorin in 1938. Interpolation and extrapolation, volume 2 1st edition. An interpolation theorem for holomorphic automorphisms of c n 103 to construct f and c j, we use an inductive procedure. Our first theorem, which is an important tool in this paper, general izes that observation and it indicates that unicyclic graphs play a significant role when we investigate the interpolation character of integervalued functions. This theorem can be used for example, to prove the hausdorffyoung inequality, which establishes that the fourier transform can be extended in a unique way as a continuous linear map. Rieszthorin, marcinkiewicz matt ziemke and marcinkiewicz s theorem can also be applied to the hilbert transform, a widely used linear operator in fourier analysis. Classical theorems about interpolation of operators. Before, we do this, we recall a fact about how to compute the norm of a function in lpfrom its distribution function. The marcinkiewicz interpolation theorem and its extensions. An introduction to the approximation of functions theodore. A note on cubic splines, amath 352, march 4, 2002 h d. Numerical interpolation gregorynewton interpolation method the previous example shows that the fifthorder interpolation polynomial gives errorless interpolated values, and there was significant improvement in accuracy. In general this does not hold true in two and more variables. However, polynomials play a much larger role in numerical analysis than providing just eigenvalues.
A similar approach is possible whenever the cumulative distribution function f can be calculated. Furthermore, a comparison of the linear interpolation with a series on a nonorthogonal basis composed of equally spaced triangular pulses is provided. The continuous function curve may characterize the relation between variables x and y more. Polynomial interpolation the most common functions used for interpolation are polynomials. A note on cubic splines, amath 352, march 4, 2002 we wouldlike touse a splinetoapproximatea functionrepresented bythe points 0 0 1 0 3 2 and 4 2. The fact that a 3 polynomial approximation 57 polynomial of degree n has exactly n such roots is known as the fundamental theorem of algebra and its. The former allows us to show that a linear operator that is bounded on. Chaudhury our goal is to prove the following interpolation theorem for any subadditive and, of course, linear operator acting between lp spaces. Let1 p l p is bounded for any p 0 interpolation theorem 2 3.
Journal of the american mathematical society volume 20, number 2, april 2007, pages 385439 s 0894034706005339 article electronically published on may 18, 2006 noncommutativ. A wolff theorem for interpolation methods associated to. Interpolation theorems for domination numbers of a graph. We have presented interpolation search only for uni formly distributed files. An interpolation theorem for holomorphic automorphisms of c n. Such operators are important, for instance, in proving carlesons theorem on the almost everywhere convergence of fourier series of lp functions. Preface the present thesis contains an exposition of. Interpolation of spatial data is a very general mathematical problem with various applications. The interpolation theory was aimed in the two classical theorems. In section 4 we give applications of the interpolation theorems of. Weiss, an extension of a theorem of marcinkiewicz and some of its applications, j.
A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. This method is mathematically equivalent to kernel interpolation, a method used in numer. Girardi rieszthorin interpolation theorem notatationbasics. Complex interpolation ucla department of mathematics. In mathematical logic, craigs interpolation theorem is a result about the relationship between different logical theories. Introductionerrors in polynomial interpolation finite differences forward difference, backward differences, central differences, symbolic relations and separation of symbolsdifference equations differences of a polynomial newtons formulae for interpolation central difference interpolation. An introduction to the approximation of functions blaisdell book in numerical analysis and computer science dover books on mathematics dover books on advanced mathematics. As an application, we prove some martingale inequalities with weak orlicz space norm. Interpolation is an old subject since, as noticed in the paper by m. Rieszthorin interpolation theorem was then generalized by elias m. Stein, allowing for interpolation of families of operators. We take cl 3el and use the results of section 2 to find an automorphism f1 with prescribed jet p1 at cl and which is near the identity on kl 2b, where b is the unit ball in c n.
Interpolation theory and applications to the boundedness. This result, known as the stein interpolation theorem, was included in his 1955 doctoral dissertation and was subsequently published in ste56. Now, having taken the first few weeks of this, it is not at all about fourier analysis but about the hardylittlewoodmaximaloperator, interpolation theorems, steins theorem lemma and a lot of constants which we try to improve constantly in some bounds. Marcinkiewicz type interpolation theorems for weak orlicz. An interpolation function is also called interpolant. Learn how to save and export pdf files in adobe photoshop, and find. The idea of hermite interpolation is clear in the univariate case, namely, when some of the interpolation points coalesce, the interpolating polynomials converge to the hermite interpolation polynomial which interpolates function values and derivatives. This chapter is devoted to the proofs and some of the applications of the theorems of rieszthorin and of marcinkiewicz, each of which is concerned with operators t defined on subsets of lebesgue spaces constructed over fairly general measure spaces and taking values in similar such spaces.
Now we need to take t to be linear rather than sublinear. A rieszthorin type interpolation theorem in euclidean jordan. Marcinkiewicz interpolation theorem with initial restricted weaktype conditions and multiplicative bounds for the intermediate spaces. Thus, it is usually tempting to try an interpolation polynomial of higher order. For their applications, free interpolations from b are also discussed.
The rieszthorin interpolation theorem in this notes we give a proof of the rieszthorin interpolation theorem. Carleson measure, atomic decomposition and free interpolation from bloch space jie xiao peking university, department of mathematics beijing 100871, p. The rst two sections of the chapters are devoted to developing the nec. As a consequence, we obtain boundedness results for several classical operators as well as a sobolev inequality in this setting. This result generalizes the homomorphism interpolation theorem of harary, heditniemi and prins, and its proof supplies an algorithm for the. In mathematics, the marcinkiewicz interpolation theorem, discovered by jozef marcinkiewicz 1939, is a result bounding the norms of nonlinear operators. The interpolation technique in proof complexity cav 20. Polynomial interpolation was then used in quadrature methods and methods for the numerical solution of ordinary differential. Marcinkiewicz interpolation we now turn to proving a general theorem about interpolation of sublinear operators. Rieszthorin interpolation theorem and its extension by stein, both of which are presented in the fourth section. Interpolation theorems and applications contents 1. This thesis is \dedicated to the rst rutgersnyu segway polo champion.
In mathematics, the marcinkiewicz interpolation theorem, discovered by jozef marcinkiewicz, is a result bounding the norms of nonlinear operators acting on l p spaces. The fourier transform and convolution 12 acknowledgments 14 references 14 1. The authors claim to approximate the unknown function via chebyshev polynomial approximation, namely by usage of the chebyshev polynomial of degree 16. Interpolation interpolation is a process of finding a formula.
The portable document format pdf is a file format developed by adobe in the 1990s to present documents, including text formatting and images, in a manner. The interpolation uses the sample values of artigo submetido em 24052001 1a. Marcinkiewicz interpolation theorems for orlicz and lorentz gamma spaces ron kerman, colin phipps, and lubo s pick abstract. P has monotone feasible interpolation, if whenever ax,y is monotone in x then a x,y and bx,z have an interpolant of monotone circuit size. Marcinkiewicz type interpolation theorems for weak orlicz martingale spaces. This thesis is primarily devoted to the study of the marcinkiewicz interpolation theorem and its applications. Let1 p pdf on jun 11, 2012, lech maligranda and others published marcinkiewicz interpolation theorem and marcinkiewicz spaces find, read and cite all the research you need on researchgate. Interpolation theory and applications to the boundedness of. Interpolation, approximation and their applications.
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