In mathematics, more specifically in mathematical analysis, the Cauchy product is the discrete convolution of two infinite series. It is named after the French mathematician Augustin-Louis Cauchy.
The Cauchy product may apply to infinite series [1] [2] or power series. [3] [4] When people apply it to finite sequences [5] or finite series, that can be seen merely as a particular case of a product of series with a finite number of non-zero coefficients (see discrete convolution).
Convergence issues are discussed in the next section.
Let and be two infinite series with complex terms. The Cauchy product of these two infinite series is defined by a discrete convolution as follows:
Consider the following two power series
with complex coefficients and . The Cauchy product of these two power series is defined by a discrete convolution as follows:
Let (an)n≥0 and (bn)n≥0 be real or complex sequences. It was proved by Franz Mertens that, if the series converges to A and converges to B, and at least one of them converges absolutely, then their Cauchy product converges to AB. [6] The theorem is still valid in a Banach algebra (see first line of the following proof).
It is not sufficient for both series to be convergent; if both sequences are conditionally convergent, the Cauchy product does not have to converge towards the product of the two series, as the following example shows:
Consider the two alternating series with
which are only conditionally convergent (the divergence of the series of the absolute values follows from the direct comparison test and the divergence of the harmonic series). The terms of their Cauchy product are given by
for every integer n ≥ 0. Since for every k ∈ {0, 1, ..., n} we have the inequalities k + 1 ≤ n + 1 and n – k + 1 ≤ n + 1, it follows for the square root in the denominator that √(k + 1)(n − k + 1) ≤ n +1, hence, because there are n + 1 summands,
for every integer n ≥ 0. Therefore, cn does not converge to zero as n → ∞, hence the series of the (cn)n≥0 diverges by the term test.
For simplicity, we will prove it for complex numbers. However, the proof we are about to give is formally identical for an arbitrary Banach algebra (not even commutativity or associativity is required).
Assume without loss of generality that the series converges absolutely. Define the partial sums
with
Then
by rearrangement, hence
1 |
Fix ε > 0. Since by absolute convergence, and since Bn converges to B as n → ∞, there exists an integer N such that, for all integers n ≥ N,
2 |
(this is the only place where the absolute convergence is used). Since the series of the (an)n≥0 converges, the individual an must converge to 0 by the term test. Hence there exists an integer M such that, for all integers n ≥ M,
3 |
Also, since An converges to A as n → ∞, there exists an integer L such that, for all integers n ≥ L,
4 |
Then, for all integers n ≥ max{L, M + N}, use the representation ( 1 ) for Cn, split the sum in two parts, use the triangle inequality for the absolute value, and finally use the three estimates ( 2 ), ( 3 ) and ( 4 ) to show that
By the definition of convergence of a series, Cn → AB as required.
In cases where the two sequences are convergent but not absolutely convergent, the Cauchy product is still Cesàro summable. [7] Specifically:
If , are real sequences with and then
This can be generalised to the case where the two sequences are not convergent but just Cesàro summable:
For and , suppose the sequence is summable with sum A and is summable with sum B. Then their Cauchy product is summable with sum AB.
All of the foregoing applies to sequences in (complex numbers). The Cauchy product can be defined for series in the spaces (Euclidean spaces) where multiplication is the inner product. In this case, we have the result that if two series converge absolutely then their Cauchy product converges absolutely to the inner product of the limits.
Let such that (actually the following is also true for but the statement becomes trivial in that case) and let be infinite series with complex coefficients, from which all except the th one converge absolutely, and the th one converges. Then the limit exists and we have:
Because the statement can be proven by induction over : The case for is identical to the claim about the Cauchy product. This is our induction base.
The induction step goes as follows: Let the claim be true for an such that , and let be infinite series with complex coefficients, from which all except the th one converge absolutely, and the -th one converges. We first apply the induction hypothesis to the series . We obtain that the series converges, and hence, by the triangle inequality and the sandwich criterion, the series converges, and hence the series converges absolutely. Therefore, by the induction hypothesis, by what Mertens proved, and by renaming of variables, we have: Therefore, the formula also holds for .
A finite sequence can be viewed as an infinite sequence with only finitely many nonzero terms, or in other words as a function with finite support. For any complex-valued functions f, g on with finite support, one can take their convolution: Then is the same thing as the Cauchy product of and .
More generally, given a monoid S, one can form the semigroup algebra of S, with the multiplication given by convolution. If one takes, for example, , then the multiplication on is a generalization of the Cauchy product to higher dimension.
In mathematics, the exponential function is the unique real function which maps zero to one and has a derivative equal to its value. The exponential of a variable is denoted or , with the two notations used interchangeably. It is called exponential because its argument can be seen as an exponent to which a constant number e ≈ 2.718, the base, is raised. There are several other definitions of the exponential function, which are all equivalent although being of very different nature.
In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures in combinatorics through generating functions. The mathematical properties of infinite series make them widely applicable in other quantitative disciplines such as physics, computer science, statistics and finance.
In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
In mathematics, the branch of real analysis 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 arbitrary index set.
In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the distribution of a normalized version of the sample mean converges to a standard normal distribution. This holds even if the original variables themselves are not normally distributed. There are several versions of the CLT, each applying in the context of different conditions.
In number theory, given a prime number p, the p-adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; p-adic numbers can be written in a form similar to decimals, but with digits based on a prime number p rather than ten, and extending to the left rather than to the right.
In mathematics, a power series is an infinite series of the form where represents the coefficient of the nth term and c is a constant called the center of the series. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function.
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series.
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 A convergent series that is not absolutely convergent is called conditionally convergent.
In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Generating functions are often expressed in closed form, by some expression involving operations on the formal series.
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces. All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically not Banach spaces.
In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form and with parametric extension for arbitrary real constants a, b and non-zero c. It is named after the mathematician Carl Friedrich Gauss. The graph of a Gaussian is a characteristic symmetric "bell curve" shape. The parameter a is the height of the curve's peak, b is the position of the center of the peak, and c controls the width of the "bell".
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 and is finite, 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.
The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function over the entire real line. Named after the German mathematician Carl Friedrich Gauss, the integral is
In mathematical analysis, Cesàro summation assigns values to some infinite sums that are not necessarily convergent in the usual sense. The Cesàro sum is defined as the limit, as n tends to infinity, of the sequence of arithmetic means of the first n partial sums of the series.
In mathematics, the exponential function can be characterized in many ways. This article presents some common characterizations, discusses why each makes sense, and proves that they are all equivalent.
In mathematics, the Cauchy condensation test, named after Augustin-Louis Cauchy, is a standard convergence test for infinite series. For a non-increasing sequence of non-negative real numbers, the series converges if and only if the "condensed" series converges. Moreover, if they converge, the sum of the condensed series is no more than twice as large as the sum of the original.
In mathematics, the field of logarithmic-exponential transseries is a non-Archimedean ordered differential field which extends comparability of asymptotic growth rates of elementary nontrigonometric functions to a much broader class of objects. Each log-exp transseries represents a formal asymptotic behavior, and it can be manipulated formally, and when it converges, corresponds to actual behavior. Transseries can also be convenient for representing functions. Through their inclusion of exponentiation and logarithms, transseries are a strong generalization of the power series at infinity and other similar asymptotic expansions.