Statistics edit

In statistics, measures of central tendency and statistical dispersion, such as the mean, median, and standard deviation, can be defined in terms of ${\displaystyle L^{p}}$  metrics, and measures of central tendency can be characterized as solutions to variational problems.

In penalized regression, "L1 penalty" and "L2 penalty" refer to penalizing either the ${\displaystyle L^{1}}$  norm of a solution's vector of parameter values (i.e. the sum of its absolute values), or its squared ${\displaystyle L^{2}}$  norm (its Euclidean length). Techniques which use an L1 penalty, like LASSO, encourage sparse solutions (where the many parameters are zero).^[1] Elastic net regularization uses a penalty term that is a combination of the ${\displaystyle L^{1}}$  norm and the squared ${\displaystyle L^{2}}$  norm of the parameter vector.

Hausdorff–Young inequality edit

The Fourier transform for the real line (or, for periodic functions, see Fourier series), maps ${\displaystyle L^{p}(\mathbb {R} )}$  to ${\displaystyle L^{q}(\mathbb {R} )}$  (or ${\displaystyle L^{p}(\mathbf {T} )}$  to ${\displaystyle \ell ^{q}}$ ) respectively, where ${\displaystyle 1\leq p\leq 2}$  and ${\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}=1.}$  This is a consequence of the Riesz–Thorin interpolation theorem, and is made precise with the Hausdorff–Young inequality.

By contrast, if ${\displaystyle p>2,}$  the Fourier transform does not map into ${\displaystyle L^{q}.}$ 

Hilbert spaces edit

Hilbert spaces are central to many applications, from quantum mechanics to stochastic calculus. The spaces ${\displaystyle L^{2}}$  and ${\displaystyle \ell ^{2}}$  are both Hilbert spaces. In fact, by choosing a Hilbert basis ${\displaystyle E,}$  i.e., a maximal orthonormal subset of ${\displaystyle L^{2}}$  or any Hilbert space, one sees that every Hilbert space is isometrically isomorphic to ${\displaystyle \ell ^{2}(E)}$  (same ${\displaystyle E}$  as above), i.e., a Hilbert space of type ${\displaystyle \ell ^{2}.}$ 

The euclidean length of a vector ${\displaystyle x=(x_{1},x_{2},\dots ,x_{n})}$  in the ${\displaystyle n}$ -dimensional real vector space ${\displaystyle \mathbb {R} ^{n}}$  is given by the Euclidean norm:

The Euclidean distance between two points ${\displaystyle x}$  and ${\displaystyle y}$  is the length ${\displaystyle \|x-y\|_{2}}$  of the straight line between the two points. In many situations, the Euclidean distance is appropriate for capturing the actual distances in a given space. In contrast, consider taxi drivers in a grid street plan who should measure distance not in terms of the length of the straight line to their destination, but in terms of the rectilinear distance, which takes into account that streets are either orthogonal or parallel to each other. The class of ${\displaystyle p}$ -norms generalizes these two examples and has an abundance of applications in many parts of mathematics, physics, and computer science.

Definition edit

For a real number ${\displaystyle p\geq 1,}$  the ${\displaystyle p}$ -norm or ${\displaystyle L^{p}}$ -norm of ${\displaystyle x}$  is defined by

The Euclidean norm from above falls into this class and is the ${\displaystyle 2}$ -norm, and the ${\displaystyle 1}$ -norm is the norm that corresponds to the rectilinear distance.

The ${\displaystyle L^{\infty }}$ -norm or maximum norm (or uniform norm) is the limit of the ${\displaystyle L^{p}}$ -norms for ${\displaystyle p\to \infty .}$  It turns out that this limit is equivalent to the following definition:

See L-infinity.

For all ${\displaystyle p\geq 1,}$  the ${\displaystyle p}$ -norms and maximum norm as defined above indeed satisfy the properties of a "length function" (or norm), which are that:

• only the zero vector has zero length,
• the length of the vector is positive homogeneous with respect to multiplication by a scalar (positive homogeneity), and
• the length of the sum of two vectors is no larger than the sum of lengths of the vectors (triangle inequality).

Abstractly speaking, this means that ${\displaystyle \mathbb {R} ^{n}}$  together with the ${\displaystyle p}$ -norm is a normed vector space. Moreover, it turns out that this space is complete, thus making it a Banach space. This Banach space is the ${\displaystyle L^{p}}$ -space over ${\displaystyle \{1,2,\ldots ,n\}.}$ 

Relations between p-norms edit

The grid distance or rectilinear distance (sometimes called the "Manhattan distance") between two points is never shorter than the length of the line segment between them (the Euclidean or "as the crow flies" distance). Formally, this means that the Euclidean norm of any vector is bounded by its 1-norm:

This fact generalizes to ${\displaystyle p}$ -norms in that the ${\displaystyle p}$ -norm ${\displaystyle \|x\|_{p}}$  of any given vector ${\displaystyle x}$  does not grow with ${\displaystyle p}$ :

For the opposite direction, the following relation between the ${\displaystyle 1}$ -norm and the ${\displaystyle 2}$ -norm is known:

This inequality depends on the dimension ${\displaystyle n}$  of the underlying vector space and follows directly from the Cauchy–Schwarz inequality.

In general, for vectors in ${\displaystyle \mathbb {C} ^{n}}$  where ${\displaystyle 0<r<p:}$ 

This is a consequence of Hölder's inequality.

When 0 < p < 1 edit

In ${\displaystyle \mathbb {R} ^{n}}$  for ${\displaystyle n>1,}$  the formula

Hence, the function

Although the ${\displaystyle p}$ -unit ball ${\displaystyle B_{n}^{p}}$  around the origin in this metric is "concave", the topology defined on ${\displaystyle \mathbb {R} ^{n}}$  by the metric ${\displaystyle B_{p}}$  is the usual vector space topology of ${\displaystyle \mathbb {R} ^{n},}$  hence ${\displaystyle \ell _{n}^{p}}$  is a locally convex topological vector space. Beyond this qualitative statement, a quantitative way to measure the lack of convexity of ${\displaystyle \ell _{n}^{p}}$  is to denote by ${\displaystyle C_{p}(n)}$  the smallest constant ${\displaystyle C}$  such that the scalar multiple ${\displaystyle C\,B_{n}^{p}}$  of the ${\displaystyle p}$ -unit ball contains the convex hull of ${\displaystyle B_{n}^{p},}$  which is equal to ${\displaystyle B_{n}^{1}.}$  The fact that for fixed ${\displaystyle p<1}$  we have

When p = 0 edit

There is one ${\displaystyle \ell _{0}}$  norm and another function called the ${\displaystyle \ell _{0}}$  "norm" (with quotation marks).

The mathematical definition of the ${\displaystyle \ell _{0}}$  norm was established by Banach's Theory of Linear Operations. The space of sequences has a complete metric topology provided by the F-norm

Another function was called the ${\displaystyle \ell _{0}}$  "norm" by David Donoho—whose quotation marks warn that this function is not a proper norm—is the number of non-zero entries of the vector ${\displaystyle x.}$ ^{[citation needed]} Many authors abuse terminology by omitting the quotation marks. Defining ${\displaystyle 0^{0}=0,}$  the zero "norm" of ${\displaystyle x}$  is equal to

This is not a norm because it is not homogeneous. For example, scaling the vector ${\displaystyle x}$  by a positive constant does not change the "norm". Despite these defects as a mathematical norm, the non-zero counting "norm" has uses in scientific computing, information theory, and statistics–notably in compressed sensing in signal processing and computational harmonic analysis. Despite not being a norm, the associated metric, known as Hamming distance, is a valid distance, since homogeneity is not required for distances.

The sequence space ℓ^p edit

The ${\displaystyle p}$ -norm can be extended to vectors that have an infinite number of components (sequences), which yields the space ${\displaystyle \ell ^{p}.}$ This contains as special cases:

• ${\displaystyle \ell ^{1},}$  the space of sequences whose series are absolutely convergent,
• ${\displaystyle \ell ^{2},}$  the space of square-summable sequences, which is a Hilbert space, and
• ${\displaystyle \ell ^{\infty },}$  the space of bounded sequences.

The space of sequences has a natural vector space structure by applying addition and scalar multiplication coordinate by coordinate. Explicitly, the vector sum and the scalar action for infinite sequences of real (or complex) numbers are given by:

Define the ${\displaystyle p}$ -norm:

Here, a complication arises, namely that the series on the right is not always convergent, so for example, the sequence made up of only ones, ${\displaystyle (1,1,1,\ldots ),}$  will have an infinite ${\displaystyle p}$ -norm for ${\displaystyle 1\leq p<\infty .}$  The space ${\displaystyle \ell ^{p}}$  is then defined as the set of all infinite sequences of real (or complex) numbers such that the ${\displaystyle p}$ -norm is finite.

One can check that as ${\displaystyle p}$  increases, the set ${\displaystyle \ell ^{p}}$  grows larger. For example, the sequence

One also defines the ${\displaystyle \infty }$ -norm using the supremum:

The ${\displaystyle p}$ -norm thus defined on ${\displaystyle \ell ^{p}}$  is indeed a norm, and ${\displaystyle \ell ^{p}}$  together with this norm is a Banach space. The fully general ${\displaystyle L^{p}}$  space is obtained—as seen below—by considering vectors, not only with finitely or countably-infinitely many components, but with "arbitrarily many components"; in other words, functions. An integral instead of a sum is used to define the ${\displaystyle p}$ -norm.

General ℓ^p-space edit

In complete analogy to the preceding definition one can define the space ${\displaystyle \ell ^{p}(I)}$  over a general index set ${\displaystyle I}$  (and ${\displaystyle 1\leq p<\infty }$ ) as

For ${\displaystyle p=2,}$  the ${\displaystyle \|\,\cdot \,\|_{2}}$ -norm is even induced by a canonical inner product ${\displaystyle \langle \,\cdot ,\,\cdot \rangle ,}$  called the Euclidean inner product, which means that ${\displaystyle \|\mathbf {x} \|_{2}={\sqrt {\langle \mathbf {x} ,\mathbf {x} \rangle }}}$  holds for all vectors ${\displaystyle \mathbf {x} .}$  This inner product can expressed in terms of the norm by using the polarization identity. On ${\displaystyle \ell ^{2},}$  it can be defined by

Now consider the case ${\displaystyle p=\infty .}$  Define^{[note 1]}

The index set ${\displaystyle I}$  can be turned into a measure space by giving it the discrete σ-algebra and the counting measure. Then the space ${\displaystyle \ell ^{p}(I)}$  is just a special case of the more general ${\displaystyle L^{p}}$ -space (defined below).

An ${\displaystyle L^{p}}$  space may be defined as a space of measurable functions for which the ${\displaystyle p}$ -th power of the absolute value is Lebesgue integrable, where functions which agree almost everywhere are identified. More generally, let ${\displaystyle (S,\Sigma ,\mu )}$  be a measure space and ${\displaystyle 1\leq p\leq \infty .}$ ^{[note 3]} When ${\displaystyle p\neq \infty }$ , consider the set ${\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}$  of all measurable functions ${\displaystyle f}$  from ${\displaystyle S}$  to ${\displaystyle \mathbb {C} }$  or ${\displaystyle \mathbb {R} }$  whose absolute value raised to the ${\displaystyle p}$ -th power has a finite integral, or in symbols:

To define the set for ${\displaystyle p=\infty ,}$  recall that two functions ${\displaystyle f}$  and ${\displaystyle g}$  defined on ${\displaystyle S}$  are said to be equal almost everywhere, written ${\displaystyle f=g}$  a.e., if the set ${\displaystyle \{s\in S:f(s)\neq g(s)\}}$  is measurable and has measure zero. Similarly, a measurable function ${\displaystyle f}$  (and its absolute value) is bounded (or dominated) almost everywhere by a real number ${\displaystyle C,}$  written ${\displaystyle |f|\leq C}$  a.e., if the (necessarily) measurable set ${\displaystyle \{s\in S:|f(s)|>C\}}$  has measure zero. The space ${\displaystyle {\mathcal {L}}^{\infty }(S,\mu )}$  is the set of all measurable functions ${\displaystyle f}$  that are bounded almost everywhere (by some real ${\displaystyle C}$ ) and ${\displaystyle \|f\|_{\infty }}$  is defined as the infimum of these bounds:

For example, if ${\displaystyle f}$  is a measurable function that is equal to ${\displaystyle 0}$  almost everywhere^{[note 5]} then ${\displaystyle \|f\|_{p}=0}$  for every ${\displaystyle p}$  and thus ${\displaystyle f\in {\mathcal {L}}^{p}(S,\,\mu )}$  for all ${\displaystyle p.}$ 

For every positive ${\displaystyle p,}$  the value under ${\displaystyle \|\,\cdot \,\|_{p}}$  of a measurable function ${\displaystyle f}$  and its absolute value ${\displaystyle |f|:S\to [0,\infty ]}$  are always the same (that is, ${\displaystyle \|f\|_{p}=\||f|\|_{p}}$  for all ${\displaystyle p}$ ) and so a measurable function belongs to ${\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}$  if and only if its absolute value does. Because of this, many formulas involving ${\displaystyle p}$ -norms are stated only for non-negative real-valued functions. Consider for example the identity ${\displaystyle \|f\|_{p}^{r}=\|f^{r}\|_{p/r},}$  which holds whenever ${\displaystyle f\geq 0}$  is measurable, ${\displaystyle r>0}$  is real, and ${\displaystyle 0<p\leq \infty }$  (here ${\displaystyle \infty /r\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;\infty }$  when ${\displaystyle p=\infty }$ ). The non-negativity requirement ${\displaystyle f\geq 0}$  can be removed by substituting ${\displaystyle |f|}$  in for ${\displaystyle f,}$  which gives ${\displaystyle \|\,|f|\,\|_{p}^{r}=\|\,|f|^{r}\,\|_{p/r}.}$  Note in particular that when ${\displaystyle p=r}$  is finite then the formula ${\displaystyle \|f\|_{p}^{p}=\||f|^{p}\|_{1}}$  relates the ${\displaystyle p}$ -norm to the ${\displaystyle 1}$ -norm.

Seminormed space of ${\displaystyle p}$ -th power integrable functions

Each set of functions ${\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}$  forms a vector space when addition and scalar multiplication are defined pointwise.^{[note 6]} That the sum of two ${\displaystyle p}$ -th power integrable functions ${\displaystyle f}$  and ${\displaystyle g}$  is again ${\displaystyle p}$ -th power integrable follows from ${\textstyle \|f+g\|_{p}^{p}\leq 2^{p-1}\left(\|f\|_{p}^{p}+\|g\|_{p}^{p}\right),}$ ^{[proof 1]} although it is also a consequence of Minkowski's inequality

Absolute homogeneity, the triangle inequality, and non-negativity are the defining properties of a seminorm. Thus ${\displaystyle \|\cdot \|_{p}}$  is a seminorm and the set ${\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}$  of ${\displaystyle p}$ -th power integrable functions together with the function ${\displaystyle \|\cdot \|_{p}}$  defines a seminormed vector space. In general, the seminorm ${\displaystyle \|\cdot \|_{p}}$  is not a norm because there might exist measurable functions ${\displaystyle f}$  that satisfy ${\displaystyle \|f\|_{p}=0}$  but are not identically equal to ${\displaystyle 0}$ ^{[note 5]} (${\displaystyle \|\cdot \|_{p}}$  is a norm if and only if no such ${\displaystyle f}$  exists).

Zero sets of ${\displaystyle p}$ -seminorms

If ${\displaystyle f}$  is measurable and equals ${\displaystyle 0}$  a.e. then ${\displaystyle \|f\|_{p}=0}$  for all positive ${\displaystyle p\leq \infty .}$  On the other hand, if ${\displaystyle f}$  is a measurable function for which there exists some ${\displaystyle 0<p\leq \infty }$  such that ${\displaystyle \|f\|_{p}=0}$  then ${\displaystyle f=0}$  almost everywhere. When ${\displaystyle p}$  is finite then this follows from the ${\displaystyle p=1}$  case and the formula ${\displaystyle \|f\|_{p}^{p}=\||f|^{p}\|_{1}}$  mentioned above.

Thus if ${\displaystyle p\leq \infty }$  is positive and ${\displaystyle f}$  is any measurable function, then ${\displaystyle \|f\|_{p}=0}$  if and only if ${\displaystyle f=0}$  almost everywhere. Since the right hand side (${\displaystyle f=0}$  a.e.) does not mention ${\displaystyle p,}$  it follows that all ${\displaystyle \|\cdot \|_{p}}$  have the same zero set (it does not depend on ${\displaystyle p}$ ). So denote this common set by

Quotient vector space

Like every seminorm, the seminorm ${\displaystyle \|\cdot \|_{p}}$  induces a norm (defined shortly) on the canonical quotient vector space of ${\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}$  by its vector subspace ${\textstyle {\mathcal {N}}=\{f\in {\mathcal {L}}^{p}(S,\,\mu ):\|f\|_{p}=0\}.}$  This normed quotient space is called Lebesgue space and it is the subject of this article. We begin by defining the quotient vector space.

Given any ${\displaystyle f\in {\mathcal {L}}^{p}(S,\,\mu ),}$  the coset ${\displaystyle f+{\mathcal {N}}\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;\{f+h:h\in {\mathcal {N}}\}}$  consists of all measurable functions ${\displaystyle g}$  that are equal to ${\displaystyle f}$  almost everywhere. The set of all cosets, typically denoted by

Two cosets are equal ${\displaystyle f+{\mathcal {N}}=g+{\mathcal {N}}}$  if and only if ${\displaystyle g\in f+{\mathcal {N}}}$  (or equivalently, ${\displaystyle f-g\in {\mathcal {N}}}$ ), which happens if and only if ${\displaystyle f=g}$  almost everywhere; if this is the case then ${\displaystyle f}$  and ${\displaystyle g}$  are identified in the quotient space.

The ${\displaystyle p}$ -norm on the quotient vector space

Given any ${\displaystyle f\in {\mathcal {L}}^{p}(S,\,\mu ),}$  the value of the seminorm ${\displaystyle \|\cdot \|_{p}}$  on the coset ${\displaystyle f+{\mathcal {N}}=\{f+h:h\in {\mathcal {N}}\}}$  is constant and equal to ${\displaystyle \|f\|_{p};}$  denote this unique value by ${\displaystyle \|f+{\mathcal {N}}\|_{p},}$  so that:

The Lebesgue ${\displaystyle L^{p}}$  space

The normed vector space ${\displaystyle \left(L^{p}(S,\mu ),\|\cdot \|_{p}\right)}$  is called ${\displaystyle L^{p}}$  space or the Lebesgue space of ${\displaystyle p}$ -th power integrable functions and it is a Banach space for every ${\displaystyle 1\leq p\leq \infty }$  (meaning that it is a complete metric space, a result that is sometimes called the Riesz–Fischer theorem). When the underlying measure space ${\displaystyle S}$  is understood then ${\displaystyle L^{p}(S,\mu )}$  is often abbreviated ${\displaystyle L^{p}(\mu ),}$  or even just ${\displaystyle L^{p}.}$  Depending on the author, the subscript notation ${\displaystyle L_{p}}$  might denote either ${\displaystyle L^{p}(S,\mu )}$  or ${\displaystyle L^{1/p}(S,\mu ).}$ 

If the seminorm ${\displaystyle \|\cdot \|_{p}}$  on ${\displaystyle {\mathcal {L}}^{p}(S,\,\mu )}$  happens to be a norm (which happens if and only if ${\displaystyle {\mathcal {N}}=\{0\}}$ ) then the normed space ${\displaystyle \left({\mathcal {L}}^{p}(S,\,\mu ),\|\cdot \|_{p}\right)}$  will be linearly isometrically isomorphic to the normed quotient space ${\displaystyle \left(L^{p}(S,\mu ),\|\cdot \|_{p}\right)}$  via the canonical map ${\displaystyle g\in {\mathcal {L}}^{p}(S,\,\mu )\mapsto \{g\}}$  (since ${\displaystyle g+{\mathcal {N}}=\{g\}}$ ); in other words, they will be, up to a linear isometry, the same normed space and so they may both be called "${\displaystyle L^{p}}$  space".

The above definitions generalize to Bochner spaces.

In general, this process cannot be reversed: there is no consistent way to define a "canonical" representative of each coset of ${\displaystyle {\mathcal {N}}}$  in ${\displaystyle L^{p}.}$  For ${\displaystyle L^{\infty },}$  however, there is a theory of lifts enabling such recovery.

Special cases edit

Similar to the ${\displaystyle \ell ^{p}}$  spaces, ${\displaystyle L^{2}}$  is the only Hilbert space among ${\displaystyle L^{p}}$  spaces. In the complex case, the inner product on ${\displaystyle L^{2}}$  is defined by

The additional inner product structure allows for a richer theory, with applications to, for instance, Fourier series and quantum mechanics. Functions in ${\displaystyle L^{2}}$  are sometimes called square-integrable functions, quadratically integrable functions or square-summable functions, but sometimes these terms are reserved for functions that are square-integrable in some other sense, such as in the sense of a Riemann integral (Titchmarsh 1976).

If we use complex-valued functions, the space ${\displaystyle L^{\infty }}$  is a commutative C*-algebra with pointwise multiplication and conjugation. For many measure spaces, including all sigma-finite ones, it is in fact a commutative von Neumann algebra. An element of ${\displaystyle L^{\infty }}$  defines a bounded operator on any ${\displaystyle L^{p}}$  space by multiplication.

For ${\displaystyle 1\leq p\leq \infty }$  the ${\displaystyle \ell ^{p}}$  spaces are a special case of ${\displaystyle L^{p}}$  spaces, when ${\displaystyle S=\mathbb {N} )}$  consists of the natural numbers and ${\displaystyle \mu }$  is the counting measure on ${\displaystyle \mathbb {N} .}$ More generally, if one considers any set ${\displaystyle S}$  with the counting measure, the resulting ${\displaystyle L^{p}}$  space is denoted ${\displaystyle \ell ^{p}(S).}$  For example, the space ${\displaystyle \ell ^{p}(\mathbb {Z} )}$ is the space of all sequences indexed by the integers, and when defining the ${\displaystyle p}$ -norm on such a space, one sums over all the integers. The space ${\displaystyle \ell ^{p}(n),}$  where ${\displaystyle n}$  is the set with ${\displaystyle n}$  elements, is ${\displaystyle \mathbb {R} ^{n}}$  with its ${\displaystyle p}$ -norm as defined above. As any Hilbert space, every space ${\displaystyle L^{2}}$  is linearly isometric to a suitable ${\displaystyle \ell ^{2}(I),}$  where the cardinality of the set ${\displaystyle I}$  is the cardinality of an arbitrary Hilbertian basis for this particular ${\displaystyle L^{2}.}$ 

As in the discrete case, if there exists ${\displaystyle q<\infty }$  such that ${\displaystyle f\in L^{\infty }(S,\mu )\cap L^{q}(S,\mu ),}$  then

Hölder's inequality

Suppose ${\displaystyle p,q,r\in [1,\infty ]}$  satisfy ${\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}={\tfrac {1}{r}}}$  (where ${\displaystyle {\tfrac {1}{\infty }}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~0}$ ). If ${\displaystyle f\in L^{p}(S,\mu )}$  and ${\displaystyle g\in L^{q}(S,\mu )}$  then ${\displaystyle fg\in L^{r}(S,\mu )}$  and^[6]

This inequality, called Hölder's inequality, is in some sense optimal^[6] since if ${\displaystyle r=1}$  (so ${\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}=1}$ ) and ${\displaystyle f}$  is a measurable function such that

Minkowski inequality

Minkowski inequality, which states that ${\displaystyle \|\cdot \|_{p}}$  satisfies the triangle inequality, can be generalized: If the measurable function ${\displaystyle F:M\times N\to \mathbb {R} }$  is non-negative (where ${\displaystyle (M,\mu )}$  and ${\displaystyle (N,\nu )}$  are measure spaces) then for all ${\displaystyle 1\leq p\leq q\leq \infty ,}$ ^[7]

Atomic decomposition edit

If ${\displaystyle 1\leq p<\infty }$  then every non-negative ${\displaystyle f\in L^{p}(\mu )}$  has an atomic decomposition,^[8] meaning that there exist a sequence ${\displaystyle (r_{n})_{n\in \mathbb {Z} }}$  of non-negative real numbers and a sequence of non-negative functions ${\displaystyle (f_{n})_{n\in \mathbb {Z} },}$  called the atoms, whose supports ${\displaystyle \left(\operatorname {supp} f_{n}\right)_{n\in \mathbb {Z} }}$  are pairwise disjoint sets of measure ${\displaystyle \mu \left(\operatorname {supp} f_{n}\right)\leq 2^{n+1},}$  such that

An atomic decomposition can be explicitly given by first defining for every integer ${\displaystyle n\in \mathbb {Z} ,}$ ^[8]