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

possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional...

]=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...

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

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...

continuously differentiable with compact support, and g is an arbitrary locally integrable function, d d x ( f ∗ g ) = d f d x ∗ g . {\displaystyle {\frac {d}{dx}}(f*g)={\frac...

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

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

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...

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

-valued functions on Ω {\displaystyle \Omega } in a number of ways. One way is to define the spaces of Bochner integrable and Pettis integrable functions, and...

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...

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...

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...

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...

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

The operator takes a locally integrable function f : Rd → C and returns another function Mf. For any point x ∈ Rd, the function Mf returns the maximum...

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...

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 ∫...

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

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)...

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...

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

every α < 1 2 {\displaystyle \alpha <{\tfrac {1}{2}}} . Functions which are locally integrable and whose integrals satisfy an appropriate growth condition...

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...

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...

Integration by parts works if u {\displaystyle u} is absolutely continuous and the function designated v ′ {\displaystyle v'} is Lebesgue integrable (but...

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