Pareto interpolation is a method of estimator estimating the median and other properties of a population that follows a Pareto distribution . It is used in economics when analysing the distribution of incomes in a population, when one must base estimates on a relatively small random sample taken from the population. The family of Pareto distributions is parameterized by a positive number that is the smallest value that a random variable with a Pareto distribution can take. As applied to distribution of incomes, is the lowest income of any person in the population and a positive number the Pareto index as this increases, the tail of the distribution gets thinner. As applied to distribution of incomes, this means that the larger the value of the Pareto index the smaller the proportion of incomes many times as big as the smallest incomes. Pareto interpolation can be used when the available information includes the proportion of the sample that falls below each of two specified numbers a b . For example, it may be observed that 45 of individuals in the sample have incomes below a 35,000 per year, and 55 have incomes below b 40,000 per year. Let P sub a sub proportion of the sample that lies below a P sub b sub proportion of the sample that lies below b . Then the estimates of and are math widehat kappa left frac P b P a left 1 a widehat theta right left 1 b widehat theta right right 1 widehat theta math and math widehat theta frac log 1 P a log 1 P b log b log a . math The estimate of the median would then be math mbox estimated median widehat kappa cdot 2 1 widehat theta , , math since the actual population median is math mbox median kappa ,2 1 theta . , math References http www.sipp.census.gov sipp sourceac S&A01 20060323 Long S&A 3 .pdf U.S. Census Bureau, Memorandum on statistical techniques used in 2001 income survey PDF . See Equation 10 on p24. http mumford1.dyndns.org ... income . Gives a derivation of the equations for Pareto interpolation. Category Estimation for specific ... more details
See also Spline mathematics In the mathematics mathematical field of numerical analysis , spline interpolation is a form of interpolation where the interpolant is a special type of piecewise polynomial called a spline mathematics spline . Spline interpolation is preferred over polynomial interpolation because the interpolation error can be made small even when using low degree polynomials for the spline. Spline interpolation avoids the problem of Runge s phenomenon which occurs when interpolating between equidistant points with high degree polynomials. Introduction Flat spline Elastic rulers that were bent to pass through a number of predefined points the knots were used for making technical drawings for shipbuilding and construction by hand, as illustrated by figure 1. Image Cubic spline.svg thumb upright 1.8 right Figure 1 Interpolation with cubic splines between eight points. Hand drawn technical drawings were made for shipbuilding etc. using flexible rulers that were bent to follow pre defined points the knots The approach to mathematically model the shape of such elastic rulers fixed by n 1 knots math x i,y i quad i 0,1, cdots ,n math is to interpolate between all the pairs of knots math x i 1 , y i 1 math and math x i , y i math with polynomials math y q i x quad i 1,2, cdots ... for the spline interpolation one has that to the left of the left most knot and to the right of the right ... math Example Image Cubic splines three points.svg frame right Figure 2 Interpolation with cubic natural ... cubic interpolation NURBS Multivariate interpolation Polynomial interpolation Smoothing spline External links http jsxgraph.uni bayreuth.de wiki index.php Cubic spline interpolation Dynamic cubic ... and practice of spline interpolation http web.archive.org web 20090408054627 http online.redwoods.cc.ca.us ... spline interpolation is done. http www.akiti.ca CubicSpline.html Online Cubic Spline Interpolation ... Splines Category Interpolation ca Interpolaci per splines de Spline Interpolation it Interpolazione ... more details
In numerical analysis , Hermite interpolation , named after Charles Hermite , is a method of interpolation interpolating data points as a polynomial function . The generated Hermite polynomial is closely related to the Newton polynomial , in that both are derived from the calculation of divided differences . Unlike Newton interpolation, Hermite interpolation matches an unknown function both in observed value, and the observed value of its first m derivatives. This means that n m 1 values math begin matrix x 0, y 0 , & x 1, y 1 , & ldots, & x n 1 , y n 1 , x 0, y 0 , & x 1, y 1 , & ldots, & x n 1 , y n 1 , vdots & vdots & & vdots x 0, y 0 m , & x 1, y 1 m , & ldots, & x n 1 , y n 1 m end matrix math must be known, rather than just the first n values required for Newton interpolation. The resulting polynomial may have degree at most n m 1 &minus 1, whereas the Newton polynomial has maximum degree n &minus 1. In the general case, there is no need for m to be a fixed value that is, some points may have more known derivatives than others. In this case the resulting polynomial may have degree N &minus 1, with N the number of data points. Usage Simple case When using divided differences to calculate the Hermite polynomial of a function f , the first step is to copy each point m times. Here we will consider the simplest case math m 1 math for all points. Therefore, given math n 1 math data points math x 0, x 1, x 2, ldots, x n math , and values math f x 0 , f x 1 , ldots, f x n math and math f x 0 , f x 1 , ldots, f x n math for a function f that we want to interpolate, we create a new dataset math z 0, z 1, ldots, z 2n 1 ... as finite differences Neville s schema Polynomial interpolation Lagrange polynomial Lagrange form of the interpolation polynomial Bernstein polynomial Bernstein form of the interpolation polynomial ... at Mathworld Category Interpolation Category Finite differences Category Factorial and binomial topics ... more details
Orphan date February 2011 String interpolation is a common feature in many programming language s such as Ruby programming language Ruby , PHP , Perl , etc. It means to insert a String computer science string or replace a variable with its value. It makes string formatting and specifying contents more intuitive. ref http perlmeme.org howtos using perl interpolation.html ref Examples PHP source lang PHP ?php str EOD Example of string spanning multiple lines using heredoc syntax. EOD class foo var foo var bar function foo this foo Foo this bar array Bar1 , Bar2 , Bar3 foo new foo name Jason echo EOT My name is name . I am printing some foo foo. Now, I am printing some foo bar 1 . This should print a capital A x41 EOT ? source The output will be source lang text My name is Jason . I am printing some Foo. Now, I am printing some Bar2. This should print a capital A A source Perl source lang perl usr bin perl use strict use warnings my apples 4 print I have apples apples n source The output will be source lang text I have 4 apples source Ruby source lang ruby apples 4 puts I have apples apples or puts I have s apples apples source The output will be source lang text I have 4 apples source BOO source lang boo apples 4 print I have apples apples or print I have 0 apples apples source The output will be source lang text I have 4 apples source CoffeeScript source lang coffeescript apples 4 console.log I have apples apples source The output will be source lang text I have 4 apples source Python source lang python apples 4 print I have s apples apples source The output will be source lang text I have 4 apples source Security Issues String Interpolation, like string concatenation, may lead to security problems. When failed to properly escape or filter user input data, system will expose ... XSS attacks. ref http google caja.googlecode.com svn changes mikesamuel string interpolation 29 Jan ... constructs Category String computer science Interpolation ... more details
newspaper The Times pages 7, col. 4 year date 23 January 1882 url ref The interpolation was not noticed ... interpolation of an obscene line had been discovered before the second edition was published ... the author of a gross outrage committed by the interpolation of a line in the speech of Sir William ... DEFAULTSORT Harcourt Interpolation Category Journalistic hoaxes Category Victorian era Category ... more details
Wiktionarypar CraigcraigCraig may refer to TOCright Places In the United Kingdom Craig, Angus Craigs, County Antrim , a parish and a townland in County Antrim , Northern Ireland In the United States Craig, Alaska , a city Craig, California , an unincorporated place Craig ghost town , Butte County, California , a former settlement Craig, Modoc County, California , a former settlement Craig, Colorado , a city Craig, Indiana , an unincorporated place Craig, Iowa , a city Craig, Missouri , a city Craig, Montana , an unincorporated place Craig, Nebraska , a village Craig County, Virginia Craig County, Oklahoma Craig Township disambiguation two places Geology Craig landform , a rocky hill or mountain People Craig surname Craig given name Fictional characters List of students at South Park Elementary Craig Tucker Craig Tucker , a recurring South Park character Craig Dixon EastEnders , short lived Eastenders character List of characters in Malcolm in the Middle Recurring characters Craig Feldspar , a recurring character in the U.S. Television show Malcolm in the Middle portrayed by David Anthony Higgins Harriet Craig , movie with Joan Crawford Craig Dean character in Hollyoaks Craig Manning , a character in Degrassi The Next Generation List of Tekken characters Craig Marduk Craig Marduk , a character in Tekken 4 Other uses Clan Craig , a Scottish clan Craig v. Boren , a US Supreme Court case Craig Electronics , a Chinese electronics company. disambig Category Place name disambiguation pages de Craig es Craig fr Craig it Craig nl Craig ja pl Craig pt Craig ru vo Craig ... more details
In mathematics, the Sarason interpolation theorem , introduced by harvs txt last Sarason authorlink Donald Sarason year 1967 , is a generalization of the Caratheodory interpolation theorem and the Pick interpolation theorem . References Citation last1 Sarason first1 Donald title Generalized interpolation in H sup sup url http www.jstor.org stable 1994641 id MR 0208383 year 1967 journal Transactions of the American Mathematical Society issn 0002 9947 volume 127 pages 179 203 math stub Category Mathematical analysis Category Mathematical theorems ... more details
Image Piecewise constant.svg right thumb Nearest neighbor interpolation blue lines in one dimension on a uniform dataset red points . Image Coloured Voronoi 2D.svg right thumb Example of nearest neighbor interpolation of a random set of points black dots in 2D. Each coloured cell indicates the area in which all the points have the black point in the cell as their nearest black point. Image Nearest2DInterpolExample.png right thumb Nearest neighbor interpolation on a uniform 2D grid black points . Nearest neighbor interpolation also known as proximal interpolation or, in some contexts, point sampling is a simple method of multivariate interpolation in one or more dimension s. Interpolation is the problem of approximating the value for a non given point in some space when given some colors of points around neighboring that point. The nearest neighbor algorithm selects the value of the nearest point and does not consider the values of neighboring points at all, yielding a piecewise constant interpolant. The algorithm is very simple to implement and is commonly used usually along with mipmap mipmapping in Real time computing real time 3D rendering to select color values for a texture filtering textured surface. Connection to Voronoi diagram For a given set of points in space, a Voronoi diagram is a decomposition of space into cells, one for each given point, so that anywhere in space, the closest given point is inside the cell. This is equivalent to nearest neighbour interpolation, by assigning the function value at the given point to all the points inside the cell. The figures on the right side show by colour the shape of the cells. See also Interpolation Nearest neighbor search Zero order hold Rounding Category Interpolation Category Multivariate interpolation Category Texture filtering mathapplied stub ru ... more details
In numerical analysis , inverse quadratic interpolation is a root finding algorithm , meaning that it is an algorithm for solving equations of the form f x 0. The idea is to use polynomial interpolation quadratic interpolation to approximate the inverse function inverse of f . This algorithm is rarely used on its own, but it is important because it forms part of the popular Brent s method . The method The inverse quadratic interpolation algorithm is defined by the recurrence relation math x n 1 frac f n 1 f n f n 2 f n 1 f n 2 f n x n 2 frac f n 2 f n f n 1 f n 2 f n 1 f n x n 1 math math frac f n 2 f n 1 f n f n 2 f n f n 1 x n, math where f sub k sub f x sub k sub . As can be seen from the recurrence relation, this method requires three initial values, x sub 0 sub , x sub 1 sub and x sub 2 sub . Explanation of the method We use the three preceding iterates, x sub n &minus 2 sub , x sub n &minus 1 sub and x sub n sub , with their function values, f sub n &minus 2 sub , f sub n &minus 1 sub and f sub n sub . Applying the Lagrange polynomial Lagrange interpolation formula to do quadratic interpolation on the inverse of f yields math f 1 y frac y f n 1 y f n f n 2 f n 1 f n 2 f n x n 2 frac y f n 2 y f n f n 1 f n 2 f n 1 f n x n 1 math math frac y f n 2 y f n 1 f n f n 2 f n f n 1 x n. math We are looking for a root of f , so we substitute y f x 0 in the above equation and this results ... fails completely. Thus, inverse quadratic interpolation is seldom used as a stand alone algorithm .... Comparison with other root finding methods As noted in the introduction, inverse quadratic interpolation is used in Brent s method . Inverse quadratic interpolation is also closely related to some other root finding methods. Using linear interpolation instead of quadratic interpolation gives the secant ... parabolic interpolation is a related method that uses parabolas to find extrema rather than roots ..., 2007. ISBN 9780470049631 Category Root finding algorithms fr Interpolation quadratique inverse ... more details
Multiple issues refimprove June 2008 notability June 2008 orphan June 2008 Generalized quaternion interpolation is an interpolation method that extends the quaternion slerp Quaternion Slerp slerp algorithm. This generalized method can interpolate between more than two unit quaternion s, but is neither closed form expression closed form nor fixed time complexity time . Definition of unconstrained interpolation General interpolation of unconstrained values math left p right math with weights math left w right math is defined as the value math m math that solves the sum math sum i w i left p i m right 0 math and math sum i w i 1. math Because math m math and math p math values are unconstrained, this can be rewritten in the more familiar form of math m sum i w i p i. , math Unit quaternions, on the other hand, are constrained and the closed form interpolation solution can not be applied to them. Conversion to constrained interpolation Because the unit quaternion space is a closed Riemannian manifold, the difference between any two values on the manifold in the tangent space of the first value can be defined as math d 0,1 log left q 0 1 q 1 right math where the logarithm is the hypercomplex logarithm. This difference can be applied to the value in which it is a tangent space member as math q 0 exp left d 0,1 right q 1 math where the hypercomplex exponential is used. With these definitions in mind, the quaternion interpolation of values math left q right math with weights math left w right math can be defined nearly identically to the unconstrained mean as math sum i w i log left m 1 q i right 0 math which says that the weighted sum of all differences to math m math in math m math s tangent space is zero. Recursive formulation The quaternion mean value defined above can be found .... DEFAULTSORT Generalized Quaternion Interpolation Category Computer graphics algorithms Category Quaternions Category Interpolation Category Rotation in three dimensions Category Applied mathematics ... more details
In mathematics, Abel Goncharov interpolation determines a polynomial such that various higher derivatives are the same as those of a given function at given points. It was introduced by harvs txt last Whittaker authorlink John Macnaghten Whittaker year 1935 and rediscovered by harvs txt last Goncharov year 1954 . References Citation last1 Whittaker first1 J. M. title Interpolatory function theory url http books.google.com books?id yyPvAAAAMAAJ publisher Cambridge University Press series Cambridge Tracts in Mathematics and Mathematical Physics, No. 33 id MR 0185330 year 1935 Citation last1 Goncharov first1 V. L. title Teoriya interpolirovaniya i pribli eniya funkcii publisher Gosudarstv. Izdat. Tehn. Teor. Lit., Moscow language Russian id MR 0067947 year 1954 Category Interpolation mathematics stub ... more details
Unreferenced date December 2009 The Whittaker Shannon interpolation formula or sinc interpolation is a method to reconstruct a continuous time bandlimited signal from a set of equally spaced samples. Definition The interpolation formula , as it is commonly called, dates back to the works of E. Borel in 1898, and E. T. Whittaker in 1915, and was cited from works of J. M. Whittaker in 1935, and in the formulation of the Nyquist Shannon sampling theorem by Claude Shannon in 1949. It is also commonly called Shannon s interpolation formula and Whittaker s interpolation formula . E. T. Whittaker, who published it in 1915, called it the Cardinal series . The sampling theorem states that, under certain limiting conditions , a function x t can be recovered exactly from its samples, x n x nT , by the Whittaker Shannon interpolation formula math x t sum n infty infty x n cdot rm sinc left frac t nT T right , math where T 1 f sub s sub is the sampling interval , f sub s sub is the sampling rate , and sinc x is the normalized sinc function . Validity condition Image bandlimited.svg thumb right 240px Spectrum of a bandlimited signal as a function of frequency. The two sided bandwidth R sub N sub 2 B is known as the Nyquist rate for the signal. If the function x t is bandlimited, and sampled at a high enough rate, the interpolation formula is guaranteed to reconstruct it exactly. Formally, if there exists some B 0 such that the function x t is bandlimited to bandwidth B that is, it has ... 2B math then the interpolation formula will exactly reconstruct the original x t from its samples. Otherwise .... See Aliasing for further discussion on this point. Interpolation as convolution sum The interpolation ... . Convergence The interpolation formula always converges absolute convergence absolutely and uniform ... sum of samples raised to a power p does not have a finite expected value. Nevertheless, the interpolation ... Whittaker Shannon Interpolation Formula Category Digital signal processing Category Signal processing ... more details
Context date October 2009 In mathematics , Thiele s interpolation formula is an interpolation formula for a function f of a single variable, due to the Denmark Danish mathematician Thorvald N. Thiele Thorvald Nicolai Thiele . It is expressed as a continued fraction , where represents the reciprocal difference math f x f x 1 cfrac x x 1 rho x 1,x 2 cfrac x x 2 rho 2 x 1,x 2,x 3 f x 1 cfrac x x 3 rho 3 x 1,x 2,x 3,x 4 rho x 1,x 2 cdots math Here is a version of Thiele interpolation in Algol 68 The MODE of lx and ly here should really be a UNION of something REAL , something COMPLex , and something SYMBOLIC ... PROC thiele REAL lx,ly, REAL x REAL BEGIN REAL xx lx 1 ,yy ly 1 INT n UPB xx IF UPB yy n THEN Assuming that the values of xx are distinct ... 0 n 1,1 n REAL p p 0, yy FOR i TO n 1 DO p 1,i xx i xx 1 i p 0,i p 0,1 i OD FOR i FROM 2 TO n 1 DO FOR j TO n i DO p i,j xx j xx j i p i 1,j p i 1,j 1 p i 2,j 1 OD OD REAL a 0 FOR i FROM n 1 BY 1 TO 2 DO a x xx i p i,1 p i 2,1 a OD yy 1 x xx 1 p 1,1 a ELSE error Unequal length arrays supplied FI END Note that, although it works in most cases, it is sensitive to input values, especially those due to equally spaced absciss . Essentially, in such a case, it may reduce to the ratio of two polynomials, which may have factors in common thus yielding a 0 0 form. References mathworld urlname ThielesInterpolationFormula title Thiele s Interpolation Formula Category Finite differences Category Articles with example ALGOL 68 code mathanalysis stub ... more details
Unreferenced date May 2007 In the context of computer animation , interpolation refers to inbetweening , or filling in frames between the key frames. It typically calculates the inbetween frames through use of usually piecewise polynomial interpolation to draw images semi automatically. For all applications of this type, a set of key points is defined by the graphic artist. These are values that are rather widely separated in space or time, and represent the desired result, but only in very coarse steps. The computed interpolation process is then used to insert many new values in between these key points to give a smoother result. In its simplest form, this is the drawing of two dimensional curves. The key points, placed by the artist, are used by the computer algorithm to form a smooth curve either through, or near these points. For a typical example of 2 D interpolation through key points see cardinal spline . For examples which go near key points see nonuniform rational B spline , or B zier curve . This is extended to the forming of three dimensional curves, shapes and complex, dynamic artistic patterns such as used in laser light shows. The process can be extended to motions. The path of an object can be interpolated by providing some key locations, then calculating many in between locations for a smooth motion. In addition to position, the speed or velocity, as well as accelerations along a path, can be calculated to mimic real life motion dynamics. Where the subjects are too large or complex to move, the camera position and orientation can be moved by this process. This last is commonly called motion control . Going further, orientations rotations of objects and parts of objects can be interpolated as well as parts of complete characters. This process mimics that used ... is called key frame animation . To make these motions appear realistic, interpolation algorithms have ... Morphing Category Splines Category Interpolation de Tweening ... more details
The Lebesgue constant bounds the interpolation error math f X f le Lambda n T 1 f p . math We ... as a special case of Lebesgue s lemma . In other words, the interpolation polynomial is at most a factor ... for a set of interpolation nodes with a small Lebesgue constant. The Lebesgue constant can be expressed ... conclude again that Chebyshev nodes are a very good choice for polynomial interpolation. However, there is an easy ... constant. Though if we assume that we always take &minus 1 and 1 as nodes for interpolation, then such a set ... 3 interpolation nodes in which case the property is not trivial . One can check that each set ... that we take &minus 1 and 1 as nodes for interpolation, then as shown by H. J. Rack, for the case ... Citation last Brutman first L. title Lebesgue functions for polynomial interpolation &mdash ... Citation last Smith first Simon J. author link title Lebesgue constants in polynomial interpolation ... An example of optimal nodes for interpolation journal International Journal of Mathematical Education ... Interpolation Category Polynomials ca Constant de Lebesgue ... more details
In the mathematics mathematical subfield of numerical analysis , monotone cubic interpolation is a variant of cubic interpolation that preserves Monotone function monotonicity of the data set being interpolated. Monotonicity is preserved by linear interpolation but not guaranteed by cubic interpolation . Monotone cubic Hermite interpolation Image MonotCubInt.png thumb 300px right Example showing cubic and monotone cubic interpolation of a monotone data set. Note that the red line is not monotone. Monotone interpolation can be accomplished using cubic Hermite spline with the tangents math m i math modified to ensure the monotonicity of the resulting Hermite spline. Interpolant selection There are several ways of selecting interpolating tangents for each data point. This section will outline the use of the Fritsch Carlson method. Let the data points be math x k,y k math for math k 1,...,n math Compute the slopes of the secant line s between successive points blockquote math Delta k frac y k 1 y k x k 1 x k math blockquote for math k 1, dots,n 1 math . Initialize the tangents at every data point as the average of the secants, blockquote math m k frac Delta k 1 Delta k 2 math blockquote for math k 2, dots,n 1 math these may be updated in further steps. For the endpoints, use one sided differences blockquote math m 1 Delta 1 quad text and quad m n Delta n 1 math blockquote For math k 1, dots,n 1 math , if math Delta k 0 math if two successive math y k y k 1 math are equal , then set math m k m k 1 0, math as the spline connecting these points must be flat to preserve monotonicity. Ignore step 4 and 5 for those math k math . Let math alpha k m k Delta k math and math beta k m k .... Cubic interpolation After the preprocessing, evaluation of the interpolated spline is equivalent ... cite journal last Fritsch first F. N. coauthors Carlson, R. E. title Monotone Piecewise Cubic Interpolation ... SIAM date 1980 doi 10.1137 0717021 Category Interpolation Category Splines ... more details
Successive parabolic interpolation is a technique for finding the extremum minimum or maximum of a continuous unimodal function by successively fitting parabola s polynomials of degree two to the function at three unique points, and at each iteration replacing the oldest point with the extremum of the fitted parabola. Advantages Only function values are used, and when this method converges to an extremum, it does so with a rate of convergence of approximately 1.324 . The superlinear rate of convergence is superior to that of other methods with only linear convergence such as line search . Moreover, not requiring the computation or approximation of function derivative s makes successive parabolic interpolation a popular alternative to other methods that do require them such as gradient descent and Newton s method in optimization Newton s method . Disadvantages On the other hand, convergence even to a local extremum is not guaranteed when using this method in isolation. For example, if the three points are line mathematics collinear , the resulting parabola is degeneracy mathematics degenerate and thus does not provide a new candidate point. Furthermore, if function derivatives are available, Newton s method is applicable and exhibits quadratic convergence. Improvements Alternating the parabolic iterations with a more robust method golden section search is a popular choice to choose candidates can greatly increase the probability of convergence without hampering the convergence rate. See also Inverse quadratic interpolation is a related method that uses parabolas to find root of a function root s rather than extrema. Simpson s rule uses parabolas to approximate definite integrals. External links http www.cse.uiuc.edu iem optimization SuccessiveParabolic An interactive demonstration References cite book author Michael Heath computer scientist Michael Heath title Scientific Computing An Introductory Survey publisher McGraw Hill location New York year 2002 pages edition 2nd ... more details
In telecommunication , a time assignment speech interpolation TASI was an analog technique used on certain long transmission telecommunications transmission links to increase voice transmission capacity. TASI worked by switching additional users onto any channel communications channel temporarily idled because an original User telecommunications user has stopped speaking. When the original user resumes speaking, that user would, in turn, be switched to any channel that happened to be idle. The speech detector function is called voice activity detection . One of the issues with using this type of technology was that the users listening on an idled channel can sometimes hear the conversation that has been switched onto it. Generally the sound heard was of very low volume and individual words are not distinguishable. See also crosstalk for a similar phenomenon in telecommunications. TASI was invented by Bell Labs in the 1960s to increase the capacity of transatlantic telephone cable s. It was one of their first applications requiring Electronic switching system electronic switching of voice circuits. Later Digital Circuit Multiplication Equipment included TASI as a feature, not as distinct hardware. References Patent US 5016247 Multihop time assigned speech interpolation, 1991 TASI FS1037C MS188 See also Talkspurt Category Multiplexing telecom stub es Sistema TASI de Zeitzugeordnete Sprachinterpolation ... more details
In complex analysis , Nevanlinna Pick interpolation is the problem of finding a holomorphic function from the unit disc to the unit disc denoted math mathbb D math , which takes specified points to specified points. Equivalently, it is the problem of finding a holomorphic function f which interpolate s a data set, subject to the upper bound math left vert f z right vert le 1 math for all math z in mathbb D math . More formally, if z sub 1 sub , ..., z sub N sub and w sub 1 sub , ..., w sub N sub are collections of points in the unit disc, the Nevanlinna Pick problem is the problem of finding a holomorphic function math f mathbb D to mathbb D math such that f z sub i sub w sub i sub for all i between 1 and N . The problem was independently solved by Georg Alexander Pick G. Pick and Rolf Nevanlinna R. Nevanlinna in 1916 and 1919 respectively. It was shown that such an f exists if and only if the Pick matrix math left frac 1 w i overline w j 1 z i overline z j right i,j 1 N math is positive definite matrix positive semi definite . Also, the function f is unique if and only if the Pick matrix has zero determinant . Pick s original proof was based on Blaschke product s. Generalisation It can be shown that the Hardy space H sup 2 sup is a reproducing kernel Hilbert space , and that its reproducing kernel known as the G bor Szeg Szeg kernel is math K a,b left 1 b bar a right 1 . , math Because of this, the Pick matrix can be rewritten as math left 1 w i overline w j K z j,z i right i,j 1 N. , math This description of the solution has motivated various attempts to generalise Nevanlinna and Pick s result. The Nevanlinna Pick problem can be generalised to that of finding a holomorphic .... References cite book last Agler first Jim coauthors John E. McCarthy title Pick Interpolation and Hilbert ... year 2002 isbn 0 8218 2898 3 cite journal first M. B. last Abrahamse title The Pick interpolation ... pages 195 203 doi 10.1307 mmj 1029002212 Category Interpolation ... more details
In mathematics , the Marcinkiewicz interpolation theorem , discovered by harvs txt authorlink J zef Marcinkiewicz first J zef last Marcinkiewicz year 1939 , is a result bounding the norms of non linear operators acting on lp space L sup p sup spaces . Marcinkiewicz theorem is similar to the Riesz Thorin theorem about linear operators, but also applies to non linear operators. Preliminaries Let f be a measurable function with real or complex values, defined on a measure space X , F , . The cumulative distribution function distribution function of f is defined by math lambda f t omega left x in X mid f x t right . math Then f is called weak math L 1 math if there exists a constant C such that the distribution of f satisfies the following inequality for all t 0 math lambda f t leq frac C t . math The smallest constant C in the inequality above is called the weak math L 1 math norm and is usually denoted by f sub 1, w sub or f sub 1, sub . Similarly the space is usually denoted by L sup 1, w sup or L sup 1, sup . Note This terminology is a bit misleading since the weak norm does not satisfy the triangle inequality as one can see by considering the sum of the functions on math 0,1 math given by math 1 x math and math 1 1 x math , which has norm 4 not 2. Any math L 1 math function belongs to L sup 1, w sup and in addition one has the inequality math f 1,w leq f ... is at most N sub q sup . Then the following interpolation inequality holds for all r between p and q ... formulation of the interpolation theorem is as follows If T is a quasilinear operator of weak type ... to weak math L 1 math estimate by a clever change of variables, Marcinkiewicz interpolation is a more ... the weak 1,1 estimate and interpolation. History The theorem was first announced by harvtxt Marcinkiewicz ... Marcinkiewicz first1 J. title Sur l interpolation d operations year 1939 journal C. R. Acad. des Sciences ... of Marcinkiewicz concerning interpolation of operations id MathSciNet id 0080887 year 1956 journal ... more details
Elizabeth Craig may refer to Elizabeth Craig writer Elizabeth Craig rower Elizabeth A. Craig , biochemist Liz Craig , fictional character in Dallas Elizabeth Tudor Craig hndis Craig, Elizabeth Category Article Feedback 5 de Elizabeth Craig ... more details
Craig Conway may refer to Craig Conway footballer Craig Conway actor , who starred in the film Doomsday hndis name Conway, Craig DEFAULTSORT Conway, Craig fr Craig Conway ... more details
Craig County is the name of two counties in the United States Craig County, Oklahoma Craig County, Virginia geodis uscounty de Craig County es Condado de Craig fr Comt de Craig it Contea di Craig nl Craig County no Craig nds Craig County pl Hrabstwo Craig pt Condado de Craig sv Craig County uk zh ... more details
Craig Stevens may refer to Craig Stevens actor Craig Stevens reporter , reporter on WSVN Craig Stevens photographer Craig Stevens presenter , British TV and radio presenter Craig Stevens swimmer , Australian swimmer Craig Stevens American football , American football player hndis name Stevens, Craig DEFAULTSORT Stevens, Craig ... more details
In mathematical logic , Craig s theorem states that any recursively enumerable set of well formed formula s of a first order language is primitively recursively axiomatizable. This result is not related to the well known Craiginterpolation theorem. Recursive axiomatization Let math A 1,A 2, dots math be an enumeration of the axioms of a recursively enumerable set T of first order formulas. Construct another set T consisting of math underbrace A i land dots land A i i math for each positive integer i . The deductive closures of T and T are thus equivalent the proof will show that T is a decidable set. A decision procedure for T lends itself according to the following informal reasoning. Each member of T is either math A 1 math or of the form math underbrace B j land dots land B j j. math Since each formula has finite length, it is checkable whether or not it is math A 1 math or of the said form. If it is of the said form and consists of j conjuncts, it is in T if it is the expression math A j math otherwise it is not in T . Again, it is checkable whether it is in fact math A n math by going through the enumeration of the axioms of T and then checking symbol for symbol whether the expressions are identical. Primitive recursive axiomatizations The proof above shows that for each recursively enumerable set of axioms there is a recursive set of axioms with the same deductive closure. A set of axioms is primitive recursive if there is a primitive recursive function that decides membership in the set. To obtain a primitive recursive aximatization, instead of replacing a formula math A i math with math underbrace A i land dots land A i i math one instead replaces it with math underbrace A i land dots land A i f i math where f x is a function that, given i , returns a computation ... function to verify that j is indeed a computation history as required. References William Craig logician William Craig . On Axiomatizability Within a System , The Journal of Symbolic Logic , Vol ... more details
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