Locally integrable function information

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

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

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...

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...

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

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...

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

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...

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...

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...

{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...

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...

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...

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...

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

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...

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

can be used to give sense to a derivative of such a function. Note each locally integrable function u {\displaystyle u} defines the linear functional φ...

{\displaystyle F^{*}(x)\leq C(Mf)(x)} . For a locally integrable function f on Rn, the sharp maximal function f ♯ {\displaystyle f^{\sharp }} is defined...

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

that a holomorphic function is infinitely differentiable and locally equal to its own Taylor series (is analytic). Holomorphic functions are the central...