In mathematics, a locally integrable function (sometimes also called locally summable function) is a function which is integrable (so its integral is...

derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used...

]=S[\varphi \circ \rho ]} for every test function φ and rotation ρ. Given any (locally integrable) function f, its radial part is given by averaging over...

d\mu .} The function is Lebesgue integrable if and only if its absolute value is Lebesgue integrable (see Absolutely integrable function). Consider the...

Square-integrable function: the square of its absolute value is integrable. Relative to measure and topology: Locally integrable function: integrable around every...

{1}{|Q|}}\int _{Q}u(y)\,\mathrm {d} y.} Definition 2. A BMO function is a locally integrable function u {\displaystyle u} whose mean oscillation supremum, taken...

Riesz–Markov–Kakutani representation theorem. If the function space of locally integrable functions, i.e. functions belonging to L loc 1 ( Ω ) {\displaystyle...

Dirichlet function is Lebesgue integrable, and convolution with test functions is used to approximate any locally integrable function by smooth functions. Whether...

is equivalent to the Riemann integral. A function is Darboux-integrable if and only if it is Riemann-integrable. Darboux integrals have the advantage of...

Chapter 1). More generally, if either function (say f) is compactly supported and the other is locally integrable, then the convolution f∗g is well-defined...

object, which, depending on the context, may be a constant, a locally integrable function or, in more general settings, a Radon measure. In the first case...

is the (n − 1)-dimensional surface measure. Conversely, all locally integrable functions satisfying the (volume) mean-value property are both infinitely...

induced locally integrable functions. The function f : U → R {\displaystyle f:U\to \mathbb {R} } is called locally integrable if it is Lebesgue integrable over...

The operator Iα associates to each integrable function f on (a,b) the function Iα f on (a,b) which is also integrable by Fubini's theorem. Thus Iα defines...

positive λ {\displaystyle \lambda } –integrable density g . {\displaystyle g.} Absolutely integrable function – function whose absolute value has finite integralPages...

of functions of interest. A necessary condition for existence of the integral is that f must be locally integrable on [0, ∞). For locally integrable functions...

value of an integrable function is the limiting average taken around the point. The theorem is named for Henri Lebesgue. For a Lebesgue integrable real or...

transform of an integrable function is continuous and the restriction of this function to any set is defined. But for a square-integrable function the Fourier...

Tau functions are an important ingredient in the modern mathematical theory of integrable systems, and have numerous applications in a variety of other...

operator takes a locally integrable function f : R d → C {\displaystyle f:\mathbb {R} ^{d}\to \mathbb {C} } and returns another function M f : R d → [ 0...

of the unit vector e ∈ Sn−1. Then for a locally integrable function f, we define the Kakeya maximal function of f to be f ∗ δ ( e ) = sup a ∈ R n 1 m...

in fact, equivalent, in the sense that a function is Darboux integrable if and only if it is Riemann integrable, and the values of the integrals are equal...

If 0 < α < n, then the Riesz potential Iαf of a locally integrable function f on Rn is the function defined by where the constant is given by c α = π...

characterizing integrable systems is the Frobenius theorem, which states that a system is Frobenius integrable (i.e., is generated by an integrable distribution)...

irreducible unitary representation on a Hilbert space, is given by a locally integrable function. Harish-Chandra (1978, 1999) proved a similar theorem for semisimple...

defined for all locally bounded and predictable integrands. More generally, it is required that Hσ be B-integrable and Hμ be Lebesgue integrable, so that ∫...

Lebesgue integration theory, if f and g are functions such that f = g almost everywhere, then f is integrable if and only if g is integrable and the integrals...

define since it is a Schwartz distribution (represented by a locally integrable function), with singularities. The character is given on the maximal torus...

Henstock–Kurzweil integrable, f is Lebesgue integrable, f is Lebesgue measurable. In general, every Henstock–Kurzweil integrable function is measurable,...

b\right]} , then f {\displaystyle f} is Riemann integrable. An important application of monotonic functions is in probability theory. If X {\displaystyle...