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In mathematics, the **Tonelli–Hobson test** gives sufficient criteria for a function *ƒ* on **R**^{2} to be an integrable function. It is often used to establish that Fubini's theorem may be applied to *ƒ*. It is named for Leonida Tonelli and E. W. Hobson.

**Mathematics** includes the study of such topics as quantity, structure, space, and change.

In mathematical analysis **Fubini's theorem**, introduced by Guido Fubini in 1907, is a result that gives conditions under which it is possible to compute a double integral by using iterated integral. One may switch the order of integration if the double integral yields a finite answer when the integrand is replaced by its absolute value.

**Leonida Tonelli** was an Italian mathematician, noted for creating Tonelli's theorem, a variation of Fubini's theorem, and for introducing semicontinuity methods as a common tool for the direct method in the calculus of variations.

More precisely, the Tonelli–Hobson test states that if *ƒ* is a real-valued measurable function on **R**^{2}, and either of the two iterated integrals

In mathematics, a **real-valued function** is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain.

In mathematics and in particular measure theory, a **measurable function** is a function between two measurable spaces such that the preimage of any measurable set is measurable, analogously to the definition that a function between topological spaces is continuous if the preimage of each open set is open. In real analysis, measurable functions are used in the definition of the Lebesgue integral. In probability theory, a measurable function on a probability space is known as a random variable.

In calculus an **iterated integral** is the result of applying integrals to a function of more than one variable in a way that each of the integrals considers some of the variables as given constants. For example, the function , if is considered a given parameter, can be integrated with respect to , . The result is a function of and therefore its integral can be considered. If this is done, the result is the iterated integral

or

is finite, then *ƒ* is Lebesgue-integrable on **R**^{2}.

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In mathematics, the **mean value theorem** states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints.

In mathematics, the **convolution theorem** states that under suitable conditions the Fourier transform of a convolution of two signals is the pointwise product of their Fourier transforms. In other words, convolution in one domain equals point-wise multiplication in the other domain. Versions of the convolution theorem are true for various Fourier-related transforms. Let
and
be two functions with convolution
.

In calculus, and more generally in mathematical analysis, **integration by parts** or **partial integration** is a process that finds the integral of a product of functions in terms of the integral of their derivative and antiderivative. It is frequently used to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found. The rule can be readily derived by integrating the product rule of differentiation.

In mathematics, the **inverse** of a function
is a function that, in some fashion, "undoes" the effect of
. The inverse of
is denoted
. The statements *y* = *f*(*x*) and *x* = *f*^{ −1}(*y*) are equivalent.

In calculus, **integration by substitution**, also known as ** u-substitution**, is a method for finding integrals. Using the fundamental theorem of calculus often requires finding an antiderivative. For this and other reasons, integration by substitution is an important tool in mathematics. It is the counterpart to the chain rule for differentiation.

In mathematics, **Green's theorem** gives the relationship between a line integral around a simple closed curve *C* and a double integral over the plane region *D* bounded by *C*. It is named after George Green, though its first proof is due to Bernhard Riemann and is the two-dimensional special case of the more general Kelvin–Stokes theorem.

In linear algebra, a **one-form** on a vector space is the same as a linear functional on the space. The usage of *one-form* in this context usually distinguishes the one-forms from higher-degree multilinear functionals on the space. For details, see linear functional.

In mathematics, the **Fourier inversion theorem** says that for many types of functions it is possible to recover a function from its Fourier transform. Intuitively it may be viewed as the statement that if we know all frequency and phase information about a wave then we may reconstruct the original wave precisely.

The **Gaussian integral**, also known as the **Euler–Poisson integral**, is the integral of the Gaussian function *e*^{−x2} over the entire real line. It is named after the German mathematician Carl Friedrich Gauss. The integral is:

In mathematics—in particular, in multivariable calculus—a **volume integral** refers to an integral over a 3-dimensional domain, that is, it is a special case of multiple integrals. Volume integrals are especially important in physics for many applications, for example, to calculate flux densities.

In mathematics, the **Riemann–Lebesgue lemma**, named after Bernhard Riemann and Henri Lebesgue, is of importance in harmonic analysis and asymptotic analysis.

The **multiple integral** is a definite integral of a function of more than one real variable, for example, *f*(*x*, *y*) or *f*(*x*, *y*, *z*). Integrals of a function of two variables over a region in **R**^{2} are called double integrals, and integrals of a function of three variables over a region of **R**^{3} are called triple integrals.

In complex analysis, functional analysis and operator theory, a **Bergman space** is a function space of holomorphic functions in a domain *D* of the complex plane that are sufficiently well-behaved at the boundary that they are absolutely integrable. Specifically, for 0 < *p* < ∞, the Bergman space *A*^{p}(*D*) is the space of all holomorphic functions
in *D* for which the p-norm is finite:

In mathematical analysis, the **Schur test**, named after German mathematician Issai Schur, is a bound on the
operator norm of an integral operator in terms of its Schwartz kernel.

In calculus, interchange of the **order of integration** is a methodology that transforms iterated integrals of functions into other, hopefully simpler, integrals by changing the order in which the integrations are performed. In some cases, the order of integration can be validly interchanged; in others it cannot.

In mathematics, the **Babenko–Beckner inequality** is a sharpened form of the Hausdorff–Young inequality having applications to uncertainty principles in the Fourier analysis of L^{p} spaces. The **( q, p)-norm** of the

A **product distribution** is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Given two statistically independent random variables *X* and *Y*, the distribution of the random variable *Z* that is formed as the product

In mathematics, **integrals of inverse functions** can be computed by means of a formula that expresses the antiderivatives of the inverse
of a continuous and invertible function
, in terms of
and an antiderivative of
. This formula was published in 1905 by Charles-Ange Laisant.