In real analysis, a branch of mathematics, a modulus of convergence is a function that tells how quickly a convergent sequence converges. These moduli are often employed in the study of computable analysis and constructive mathematics.
If a sequence of real numbers converges to a real number , then by definition, for every real there is a natural number such that if then . A modulus of convergence is essentially a function that, given , returns a corresponding value of .
Suppose that is a convergent sequence of real numbers with limit . There are two ways of defining a modulus of convergence as a function from natural numbers to natural numbers:
The latter definition is often employed in constructive settings, where the limit may actually be identified with the convergent sequence. Some authors use an alternate definition that replaces with .
In mathematics, a Cauchy sequence, named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite number of elements of the sequence are less than that given distance from each other.
In mathematics, a continuous function is a function such that a continuous variation of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is not continuous. Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity.
In mathematics, computable numbers are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. They are also known as the recursive numbers, effective numbers or the computable reals or recursive reals.
In the philosophy of mathematics, constructivism asserts that it is necessary to find a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove the existence of a mathematical object without "finding" that object explicitly, by assuming its non-existence and then deriving a contradiction from that assumption. Such a proof by contradiction might be called non-constructive, and a constructivist might reject it. The constructive viewpoint involves a verificational interpretation of the existential quantifier, which is at odds with its classical interpretation.
In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göttingen in 1854, but not published in a journal until 1868. For many functions and practical applications, the Riemann integral can be evaluated by the fundamental theorem of calculus or approximated by numerical integration.
In mathematics, real analysis is the branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability.
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members. The number of elements is called the length of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an index set that may not be numbers to another set of elements.
In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting bounds on the sequence. They can be thought of in a similar fashion for a function. For a set, they are the infimum and supremum of the set's limit points, respectively. In general, when there are multiple objects around which a sequence, function, or set accumulates, the inferior and superior limits extract the smallest and largest of them; the type of object and the measure of size is context-dependent, but the notion of extreme limits is invariant. Limit inferior is also called infimum limit, limit infimum, liminf, inferior limit, lower limit, or inner limit; limit superior is also known as supremum limit, limit supremum, limsup, superior limit, upper limit, or outer limit.
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue, although according to the Bourbaki group they were first introduced by Frigyes Riesz. Lp spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Because of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used also in the theoretical discussion of problems in physics, statistics, finance, engineering, and other disciplines.
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions converges uniformly to a limiting function on a set if, given any arbitrarily small positive number , a number can be found such that each of the functions differ from by no more than at every pointin. Described in an informal way, if converges to uniformly, then the rate at which approaches is "uniform" throughout its domain in the following sense: in order to guarantee that falls within a certain distance of , we do not need to know the value of in question — there can be found a single value of independent of , such that choosing will ensure that is within of for all . In contrast, pointwise convergence of to merely guarantees that for any given in advance, we can find so that, for that particular, falls within of whenever .
In mathematics, an infinite series of numbers is said to converge absolutely if the sum of the absolute values of the summands is finite. More precisely, a real or complex series is said to converge absolutely if for some real number Similarly, an improper integral of a function, is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if
In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the symbol. If such a limit exists, the sequence is called convergent. A sequence that does not converge is said to be divergent. The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests.
In mathematics, the Weierstrass M-test is a test for determining whether an infinite series of functions converges uniformly and absolutely. It applies to series whose terms are bounded functions with real or complex values, and is analogous to the comparison test for determining the convergence of series of real or complex numbers. It is named after the German mathematician Karl Weierstrass (1815-1897).
The Arzelà–Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence of a given family of real-valued continuous functions defined on a closed and bounded interval has a uniformly convergent subsequence. The main condition is the equicontinuity of the family of functions. The theorem is the basis of many proofs in mathematics, including that of the Peano existence theorem in the theory of ordinary differential equations, Montel's theorem in complex analysis, and the Peter–Weyl theorem in harmonic analysis and various results concerning compactness of integral operators.
In statistics, a consistent estimator or asymptotically consistent estimator is an estimator—a rule for computing estimates of a parameter θ0—having the property that as the number of data points used increases indefinitely, the resulting sequence of estimates converges in probability to θ0. This means that the distributions of the estimates become more and more concentrated near the true value of the parameter being estimated, so that the probability of the estimator being arbitrarily close to θ0 converges to one.
Convergence in measure is either of two distinct mathematical concepts both of which generalize the concept of convergence in probability.
In the mathematical areas of number theory and analysis, an infinite sequence or a function is said to eventually have a certain property, if it doesn't have the said property across all its ordered instances, but will after some instances have passed. The use of the term "eventually" can be often rephrased as "for sufficiently large numbers", and can be also extended to the class of properties that apply to elements of any ordered set.
In mathematics, the Levi-Civita field, named after Tullio Levi-Civita, is a non-Archimedean ordered field; i.e., a system of numbers containing infinite and infinitesimal quantities. Each member can be constructed as a formal series of the form
In mathematics, a limit is the value that a function approaches as the input approaches some value. Limits are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals.
In mathematics, a càdlàg, RCLL, or corlol function is a function defined on the real numbers that is everywhere right-continuous and has left limits everywhere. Càdlàg functions are important in the study of stochastic processes that admit jumps, unlike Brownian motion, which has continuous sample paths. The collection of càdlàg functions on a given domain is known as Skorokhod space.
