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| In the field of [[mathematical analysis]], an '''interpolation space''' is a space which lies "in between" two other [[Banach space]]s. The main applications are in [[Sobolev space]]s, where spaces of functions that have a noninteger number of [[derivative]]s are interpolated from the spaces of functions with integer number of derivatives.
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| ==History==
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| The theory of interpolation of vector spaces began by an observation of [[Józef Marcinkiewicz]], later generalized and now known as the [[Riesz-Thorin theorem]]. In simple terms, if a linear function is continuous on a certain [[Lp space|space ''L''<sup>''p''</sup>]] and also on a certain space ''L''<sup>''q''</sup>, then it is also continuous on the space ''L''<sup>''r''</sup>, for any intermediate ''r'' between ''p'' and ''q''. In other words, ''L''<sup>''r''</sup> is a space which is intermediate between ''L''<sup>''p''</sup> and ''L''<sup>''q''</sup>.
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| In the development of Sobolev spaces, it became clear that the trace spaces were not any of the usual function spaces (with integer number of derivatives), and [[Jacques-Louis Lions]] discovered that indeed these trace spaces were constituted of functions that have a noninteger degree of differentiability.
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| Many methods were designed to generate such spaces of functions, including the [[Fourier transform]], complex interpolation,<ref>The seminal papers in this direction are {{citation
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| | last = Lions |first = Jacques-Louis
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| | title = Une construction d'espaces d'interpolation
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| | language = French
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| | journal = C. R. Acad. Sci. Paris
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| | volume = 251
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| | year = 1960
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| | pages = 1853–1855}} and {{harvtxt|Calderón|1964}}.</ref>
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| real interpolation,<ref>first defined in {{citation
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| | last1 = Lions | first1 = Jacques-Louis
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| | last2 = Peetre | first2 = Jaak
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| | title = Propriétés d'espaces d'interpolation
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| | language = French
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| | journal = C. R. Acad. Sci. Paris
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| | volume = 253
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| | year = 1961
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| |pages = 1747–1749}}, developed in {{harvtxt|Lions|Peetre|1964}}, with notation slightly different (and more complicated, with four parameters instead of two) from today's notation. It was put later in today's form in {{citation
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| | last = Peetre | first = Jaak
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| | title = Nouvelles propriétés d'espaces d'interpolation
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| | language = French
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| | journal = C. R. Acad. Sci. Paris
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| | volume = 256
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| | year = 1963
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| | pages = 1424–1426}}, and
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| {{citation
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| | last = Peetre | first = Jaak
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| | title = A theory of interpolation of normed spaces
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| | series = Notas de Matemática,
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| | volume = 39
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| | publisher = Instituto de Matemática Pura e Aplicada, Conselho Nacional de Pesquisas
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| | location = Rio de Janeiro
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| | year = 1968
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| | pages = iii+86}}.</ref>
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| as well as other tools (see e.g. [[fractional derivative]]).
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| == The setting of interpolation ==
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| A [[Banach space]] ''X'' is said to be ''continuously embedded'' in a Hausdorff [[topological vector space]] ''Z'' when ''X'' is a linear subspace of ''Z'' such that the inclusion map from ''X'' into ''Z'' is continuous. A '''compatible couple''' {{nowrap| (''X''<sub>0</sub>, ''X''<sub>1</sub>)}} of Banach spaces consists of two Banach spaces ''X''<sub>0</sub> and ''X''<sub>1</sub> that are continuously embedded in the same Hausdorff topological vector space ''Z''.<ref>see {{harvtxt|Bennett|Sharpley|1988}}, pp. 96–105.</ref>
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| The embedding in a linear space ''Z'' allows to consider the two linear subspaces
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| :<math> X_0 \cap X_1 \ \ \text{and} \ \ X_0 + X_1 = \{ z \in Z : z = x_0 + x_1, \ x_0 \in X_0, \, x_1 \in X_1 \}.</math>
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| Interpolation does not depend only upon the isomorphic (nor isometric) equivalence classes of ''X''<sub>0</sub> and ''X''<sub>1</sub>. It depends in an essential way from the specific ''relative position'' that ''X''<sub>0</sub> and ''X''<sub>1</sub> occupy in a larger space ''Z''.
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| One can define norms on {{nowrap|''X''<sub>0</sub> ∩ ''X''<sub>1</sub>}} and {{nowrap|''X''<sub>0</sub> + ''X''<sub>1</sub>}} by
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| :<math>\|x\|_{X_0 \cap X_1} := \max ( \|x\|_{X_0}, \|x\|_{X_1} ),</math>
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| :<math>\|x\|_{X_0 + X_1} := \inf \{ \|x_0\|_{X_0} + \|x_1\|_{X_1} \ : \ x = x_0 + x_1, \; x_0 \in X_0, \; x_1 \in X_1 \}.</math>
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| Equipped with these norms, the intersection and the sum are Banach spaces. The following inclusions are all continuous:
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| :<math>X_0 \cap X_1 \subset X_0, \ X_1 \subset X_0 + X_1.</math>
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| Interpolation studies the family of spaces ''X'' that are '''intermediate spaces''' between ''X''<sub>0</sub> and ''X''<sub>1</sub> in the sense that
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| :<math>X_0 \cap X_1 \subset X \subset X_0 + X_1,</math>
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| where the two inclusions maps are continuous.
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| An example of this situation is the pair (''L''<sup>1</sup>('''R'''), ''L''<sup>∞</sup>('''R''')), where the two Banach spaces are continuously embedded in the space ''Z'' of measurable functions on the real line, equipped with the topology of convergence in measure. In this situation, the spaces ''L''<sup>''p''</sup>('''R'''), for {{nowrap| 1 ≤ ''p'' ≤ ∞}} are intermediate between ''L''<sup>1</sup>('''R''') and ''L''<sup>∞</sup>('''R'''). More generally,
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| :<math>L^{p_0}(\mathbf{R}) \cap L^{p_1}(\mathbf{R}) \subset L^p(\mathbf{R}) \subset L^{p_0}(\mathbf{R}) + L^{p_1}(\mathbf{R}), \ \ \text{when} \ \ 1 \le p_0 \le p \le p_1 \le \infty,</math>
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| with continuous injections, so that, under the given condition, ''L''<sup>''p''</sup>('''R''') is intermediate between ''L''<sup>''p''<sub>0</sub></sup>('''R''') and ''L''<sup>''p''<sub>1</sub></sup>('''R''').
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| :'''Definition.''' Given two compatible couples (''X''<sub>0</sub>, ''X''<sub>1</sub>) and (''Y''<sub>0</sub>, ''Y''<sub>1</sub>), an '''interpolation pair''' is a couple (''X'', ''Y'') of Banach spaces with the two following properties:
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| :*The space ''X'' is intermediate between ''X''<sub>0</sub> and ''X''<sub>1</sub>, and ''Y'' is intermediate between ''Y''<sub>0</sub> and ''Y''<sub>1</sub>.
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| :*If ''L'' is any linear operator from {{nowrap|''X''<sub>0</sub> + ''X''<sub>1</sub>}} to {{nowrap|''Y''<sub>0</sub> + ''Y''<sub>1</sub>}}, which maps continuously ''X''<sub>0</sub> to ''Y''<sub>0</sub> and ''X''<sub>1</sub> to ''Y''<sub>1</sub>, then it also maps continuously ''X'' to ''Y''.
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| The interpolation pair (''X'', ''Y'') is said to be of '''exponent θ''' (with {{nowrap|0 < θ < 1}}) if there exists a constant ''C'' such that
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| ::<math>\|L\|_{X,Y} \leq C \|L\|_{X_0,Y_0}^{1-\theta} \; \|L\|_{X_1,Y_1}^{\theta}</math>
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| for all operators ''L'' as above. The notation ||''L''||<sub>''X'', ''Y''</sub> is for the norm of ''L'' as a map from ''X'' to ''Y''. If ''C'' = 1, one says that (''X'', ''Y'') is an '''exact interpolation pair of exponent''' θ.
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| == Complex interpolation ==
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| If the scalars are [[complex number]]s, properties of complex [[analytic function]]s are used to define an interpolation space. Given a compatible couple (''X''<sub>0</sub>, ''X''<sub>1</sub>) of Banach spaces, the linear space <math>\scriptstyle \mathcal{F}(X_0, X_1)</math> consists of all analytic functions ''f'' with values in ''X''<sub>0</sub>+''X''<sub>1</sub>, defined on the open strip ''S'' = { {{nowrap|''z'' : 0 < Re ''z'' < 1}} } in the complex plane, continuous on the closed strip {{nowrap| 0 ≤ Re ''z'' ≤ 1}}, such that the set of values ''f''(''z''), ''z'' ∈ ''S'', is bounded in {{nowrap|''X''<sub>0</sub> + ''X''<sub>1</sub>}} and such that
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| ::the set of values { {{nowrap|''f'' (i ''t'' ) : ''t'' ∈ '''R'''}} } is bounded in ''X''<sub>0</sub> and { {{nowrap|''f'' (1+ i''t'') : ''t'' ∈ '''R'''}} } is bounded in ''X''<sub>1</sub>.
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| This space of functions is a Banach space under the norm
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| ::<math>\|f\|_{\mathcal{F}(X_0, X_1)} = \max \bigl\{ \sup_{t \in \mathbf{R}} \|f(it)\|_{X_0}, \; \sup_{t \in \mathbf{R}}\|f(1 + it)\|_{X_1} \bigr\}.</math>
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| '''Definition.'''<ref>see p. 88 in {{harvtxt|Bergh|Löfström|1976}}.</ref> For {{nowrap|0 < θ < 1}}, the '''complex interpolation space''' (''X''<sub>0</sub>, ''X''<sub>1</sub>)<sub>θ</sub> is the linear subspace of {{nowrap|''X''<sub>0</sub> + ''X''<sub>1</sub>}} consisting of all values ''f''(θ) when ''f'' varies in the preceding space of functions,
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| :<math>(X_0, X_1)_\theta = \{ x \in X_0 + X_1 : x = f(\theta), \; f \in \mathcal{F}(X_0, X_1) \}.</math>
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| The norm on the complex interpolation space (''X''<sub>0</sub>, ''X''<sub>1</sub>)<sub>θ</sub> is defined by
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| ::<math>\ \|x\|_\theta = \inf \{ \|f\|_{\mathcal{F}(X_0, X_1)} \;:\; f(\theta) = x, \; f \in \mathcal{F}(X_0, X_1) \}.</math>
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| Equipped with this norm, the complex interpolation space (''X''<sub>0</sub>, ''X''<sub>1</sub>)<sub>θ</sub> is a Banach space.
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| '''Theorem.'''<ref>see Theorem 4.1.2, p. 88 in {{harvtxt|Bergh|Löfström|1976}}.</ref> Given two compatible couples of Banach spaces (''X''<sub>0</sub>, ''X''<sub>1</sub>) and (''Y''<sub>0</sub>, ''Y''<sub>1</sub>), the pair ((''X''<sub>0</sub>, ''X''<sub>1</sub>)<sub>θ</sub>, (''Y''<sub>0</sub>, ''Y''<sub>1</sub>)<sub>θ</sub>) is an exact interpolation pair of exponent θ, ''i.e.'', if ''T'' is a linear operator from ''X''<sub>0</sub> + ''X''<sub>1</sub> to ''Y''<sub>0</sub> + ''Y''<sub>1</sub>, bounded from ''X''<sub>''j''</sub> to ''Y''<sub>''j''</sub>, ''j'' = 0, 1, then ''T'' is bounded from {{nowrap|(''X''<sub>0</sub>, ''X''<sub>1</sub>)<sub>θ</sub>}} to {{nowrap|(''Y''<sub>0</sub>, ''Y''<sub>1</sub>)<sub>θ</sub>}} and
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| ::<math> \|T\|_\theta \le \|T\|_0^{1 - \theta} \|T\|_1^\theta. </math>
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| The family of ''L''<sup>''p''</sup> spaces (consisting of complex valued functions) behaves well under complex interpolation.<ref>see Chapter 5, p. 106 in {{harvtxt|Bergh|Löfström|1976}}.</ref>
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| If {{nowrap|(''R'', Σ, μ)}} is an arbitrary [[Measure (mathematics)|measure space]], if {{nowrap| 1 ≤ ''p''<sub>0</sub>, ''p''<sub>1</sub> ≤ ∞}} and {{nowrap| 0 < θ < 1}}, then
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| :<math>\bigl( L^{p_0}(R, \Sigma, \mu), L^{p_1}(R, \Sigma, \mu) \bigr)_\theta = L^p(R, \Sigma, \mu) \ \ \text{if} \ \ \frac 1 p = \frac{1 - \theta}{p_0} + \frac{\theta}{p_1},</math>
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| with equality of norms. This fact is closely related to the [[Riesz–Thorin theorem]].
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| == Real interpolation ==
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| There are two ways for introducing the '''real interpolation method'''. The first and most commonly used when actually identifying examples of interpolation spaces is the K-method. The second method, the J-method, gives the same interpolation spaces as the K-method when the parameter θ is in (0, 1). That the J- and K-methods agree is important for the study of duals of interpolation spaces: basically, the dual of an interpolation space constructed by the K-method appears to be a space constructed form the dual couple by the J-method, [[Interpolation space#Discrete definitions|see below]].
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| === The K-method ===
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| The K-method of real interpolation<ref>see pp. 293–302 in {{harvtxt|Bennett|Sharpley|1988}}.</ref>
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| can be used for Banach spaces over the field '''R''' of [[real number]]s. | |
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| '''Definition.''' Let (''X''<sub>0</sub>, ''X''<sub>1</sub>) be a compatible couple of Banach spaces. For ''t'' > 0 and every {{nowrap|''x'' ∈ ''X''<sub>0</sub> + ''X''<sub>1</sub>}}, let
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| :<math>K(x, t; X_0, X_1) = \inf \{ \|x_0\|_{X_0} + t \|x_1\|_{X_1} \,:\, x = x_0 + x_1, \; x_0 \in X_0, \, x_1 \in X_1\}.</math> | |
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| Changing the order of the two spaces results<ref>see Proposition 1.2, p. 294 in {{harvtxt|Bennett|Sharpley|1988}}.</ref>
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| in
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| :<math> t^{-1} K(x, t; X_0, X_1) = K(x, t^{-1}; X_1, X_0).</math>
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| Let
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| :<math>\|x\|_{\theta,q; K} = \left( \int_0^\infty \bigl( t^{-\theta} K(x, t; X_0, X_1) \bigr)^q \, {dt \over t} \right)^{1/q}, \ 0 < \theta < 1, \ 1 \leq q < \infty,</math>
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| and
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| :<math>\|x\|_{\theta,\infty; K} = \sup_{t > 0} \; t^{-\theta} K(x, t; X_0, X_1), \ 0 \le \theta \le 1.</math>
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| The K-method of real interpolation consists in taking {{nowrap|''K''<sub>θ, ''q'' </sub>(''X''<sub>0</sub>, ''X''<sub>1</sub>) }} to be the linear subspace of {{nowrap|''X''<sub>0</sub> + ''X''<sub>1</sub>}} consisting of all ''x'' such that {{nowrap|‖''x'' ‖<sub>θ,''q'' ; ''K''</sub> < ∞}}.
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| ==== Example ====
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| An important example is that of the couple (''L''<sup>1</sup>(''R'', Σ, μ), ''L''<sup>∞</sup>(''R'', Σ, μ)), where the functional {{nowrap|''K''(''t'', ''f'' ; ''L''<sup>1</sup>, ''L''<sup>∞</sup>)}} can be computed explicitely. The measure μ is supposed [[σ-finite measure|σ-finite]]. In this context, the best way of cutting the function {{nowrap|''f'' ∈ ''L''<sup>1</sup> + ''L''<sup>∞</sup>}} as sum of two functions ''f''<sub>0</sub> in ''L''<sup>1</sup> and ''f''<sub>1</sub> in ''L''<sup>∞</sup> is, for some {{nowrap|''s'' > 0}} to be chosen as function of ''t'', to let ''f''<sub>1</sub>(''x'') be given for all {{nowrap|''x'' ∈ ''R''}} by
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|
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| :<math>f_1(x) = f(x) \ \ \text{if} \ \ |f(x)| < s, \ \ \text{and} \ \ f_1(x) = s f(x) / |f(x)| \ \ \text{otherwise}.</math>
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| The optimal choice of ''s'' leads to the formula<ref>see p. 298 in {{harvtxt|Bennett|Sharpley|1988}}.</ref>
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| :<math>K(f, t; L^1, L^\infty) = \int_0^t f^*(u) \, d u,</math>
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| where {{nowrap|''f'' *}} is the [[Lorentz space#Decreasing rearrangements|decreasing rearrangement]] of ''f''.
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| === The J-method ===
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| As with the K-method, the J-method can be used for real Banach spaces.
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| '''Definition.''' Let (''X''<sub>0</sub>, ''X''<sub>1</sub>) be a compatible couple of Banach spaces. For ''t'' > 0 and for every vector {{nowrap|''x'' ∈ ''X''<sub>0</sub> ∩ ''X''<sub>1</sub>}}, let
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| :<math>J(x, t; X_0, X_1) = \max(\|x\|_{X_0}, t \|x\|_{X_1}).</math>
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| A vector ''x'' in {{nowrap|''X''<sub>0</sub> + ''X''<sub>1</sub>}} belongs to the interpolation space {{nowrap|''J''<sub>θ, ''q'' </sub>(''X''<sub>0</sub>, ''X''<sub>1</sub>) }} if and only if it can be written as
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|
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| :<math>x = \int_0^\infty v(t) \, {dt \over t},</math>
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| where ''v''(''t'') is measurable with values in {{nowrap|''X''<sub>0</sub> ∩ ''X''<sub>1</sub>}} and such that
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| ::<math>\Phi(v) = \left( \int_0^\infty \bigl( t^{-\theta} J(v(t), t; X_0, X_1) \bigr)^q \, {dt \over t} \right)^{1/q} < \infty.</math>
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| The norm of ''x'' in {{nowrap|''J''<sub>θ, ''q'' </sub>(''X''<sub>0</sub>, ''X''<sub>1</sub>) }} is given by the formula
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| :<math>\|x\|_{\theta,q;J} := \inf_v \bigl\{ \Phi(v) \;:\; x = \int_0^\infty v(t) \, dt/t \bigr\}.</math>
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| === Relations between the interpolation methods ===
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| The two real interpolation methods are equivalent when {{nowrap| 0 < θ < 1}}.<ref>see Theorem 2.8, p. 314 in {{harvtxt|Bennett|Sharpley|1988}}.</ref>
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| :'''Theorem.''' Let (''X''<sub>0</sub>, ''X''<sub>1</sub>) be a compatible couple of Banach spaces. If {{nowrap| 0 < θ < 1}} and {{nowrap|1 ≤ ''q'' ≤ ∞}}, then
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| :::<math>J_{\theta,q}(X_0, X_1) = K_{\theta,q}(X_0, X_1),</math>
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| :with [[Norm (mathematics)#Definition|equivalence of norms]].
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| The theorem covers degenerate cases that have not been excluded: if for example ''X''<sub>0</sub> and ''X''<sub>1</sub> form a direct sum, then the intersection and the J-spaces are the null space, and a simple computation shows that the K-spaces are also null.
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| When {{nowrap| 0 < θ < 1}}, one can speak, up to an equivalent renorming, about ''the'' Banach space obtained by the real interpolation method with parameters θ and ''q''. The notation for this real interpolation space is (''X''<sub>0</sub>, ''X''<sub>1</sub>)<sub>θ, ''q''</sub>. One has that
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| :<math>(X_0, X_1)_{\theta, q} = (X_1, X_0)_{1 - \theta, q}, \quad 0 < \theta < 1, \ 1 \le q \le \infty.\,</math>
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| For a given value of θ, the real interpolation spaces increase with ''q'':<ref>see Proposition 1.10, p. 301 in {{harvtxt|Bennett|Sharpley|1988}}</ref>
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| if {{nowrap| 0 < θ < 1}} and {{nowrap| 1 ≤ ''q'' ≤ ''r'' ≤ ∞}}, the following continuous inclusion holds true:
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| :<math>(X_0, X_1)_{\theta, q} \subset (X_0, X_1)_{\theta, r}.\,</math>
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| '''Theorem.''' Given {{nowrap| 0 < θ < 1}}, {{nowrap|1 ≤ ''q'' ≤ ∞}} and two compatible couples (''X''<sub>0</sub>, ''X''<sub>1</sub>) and (''Y''<sub>0</sub>, ''Y''<sub>1</sub>), the pair ({{nowrap|(''X''<sub>0</sub>, ''X''<sub>1</sub>)<sub>θ, ''q'' </sub>, (''Y''<sub>0</sub>, ''Y''<sub>1</sub>)<sub>θ, ''q'' </sub>) }} is an exact interpolation pair of exponent θ.<ref>see Theorem 1.12, pp. 301–302 in {{harvtxt|Bennett|Sharpley|1988}}.</ref>
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| A complex interpolation space is usually not isomorphic to one of the spaces given by the real interpolation method. However, there is a general relationship.
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| :'''Theorem.''' Let (''X''<sub>0</sub>, ''X''<sub>1</sub>) be a compatible couple of Banach spaces. If {{nowrap|0 < ''θ'' < 1}}, then | |
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| :: <math>(X_0, X_1)_{\theta, 1} \subset (X_0, X_1)_\theta \subset (X_0, X_1)_{\theta, \infty}.\,</math>
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| ==== Examples ====
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| When ''X''<sub>0</sub> = ''C''([0, 1]) and ''X''<sub>1</sub> = ''C''<sup>1</sup>([0, 1]), the space of continuously differentiable functions on [0, 1], the (θ, ∞) interpolation method, for {{nowrap| 0 < θ < 1}}, gives the [[Hölder condition|Hölder space]] ''C''<sup>0, θ</sup> of exponent θ. This is because the K-functional {{nowrap| ''K''(''f'', ''t'' ; ''X''<sub>0</sub>, ''X''<sub>1</sub>)}} of this couple is equivalent to
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| :<math> \sup \Bigl\{ |f(u)|, \, \frac{|f(u) - f(v)|} {1 + t^{-1} |u - v|} \,:\, u, v \in [0, 1] \Bigr\}.</math>
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| Only values {{nowrap| 0 < ''t'' < 1}} are interesting here.
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| Real interpolation between ''L''<sup>''p''</sup> spaces gives<ref>see Theorem 1.9, p. 300 in {{harvtxt|Bennett|Sharpley|1988}}.</ref>
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| the family of [[Lorentz space]]s. Assuming {{nowrap| 0 < θ < 1}} and {{nowrap| 1 ≤ ''q'' ≤ ∞}}, one has that | |
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| :<math> \bigl( L^1(R, \Sigma, \mu), L^\infty(R, \Sigma, \mu) \bigr)_{\theta, q} = L^{p, q}(R, \Sigma, \mu), \ \ \text{where} \ \ 1/p = 1 - \theta,</math>
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| with equivalent norms. This follows from an [[Hardy's inequality|inequality of Hardy]] and from the value given above of the K-functional for this compatible couple. When ''q'' = ''p'', the Lorentz space ''L''<sup>''p'',''p''</sup> is equal to ''L''<sup>''p''</sup>, up to renorming. When ''q'' = ∞, the Lorentz space ''L''<sup>''p'',∞</sup> is equal to [[Lp space#Weak Lp|weak-''L''<sup>''p''</sup>]].
| |
| | |
| == The reiteration theorem ==
| |
| An intermediate space ''X'' of the compatible couple (''X''<sub>0</sub>, ''X''<sub>1</sub>) is said to be of '''class θ'''<ref>see Definition 2.2, pp. 309–310 in {{harvtxt|Bennett|Sharpley|1988}}</ref>
| |
| if
| |
| :: <math>(X_0, X_1)_{\theta,1} \subset X \subset (X_0, X_1)_{\theta,\infty},\,</math>
| |
| with continuous injections. Beside all real interpolation spaces {{nowrap|(''X''<sub>0</sub>, ''X''<sub>1</sub>)<sub>θ, ''q'' </sub>}} with parameter θ and {{nowrap| 1 ≤ ''q'' ≤ ∞}}, the complex interpolation space (''X''<sub>0</sub>, ''X''<sub>1</sub>)<sub>θ</sub> is an intermediate space of class θ of the compatible couple (''X''<sub>0</sub>, ''X''<sub>1</sub>).
| |
| | |
| The reiteration theorems says, in essence, that interpolating with a parameter θ behaves, in some way, like forming a [[convex combination]] ''a'' = {{nowrap| (1 - θ) ''x''<sub>0</sub> + θ ''x''<sub>1</sub> }}: taking a further convex combination of two convex combinations gives another convex combination.
| |
| | |
| '''Theorem.'''<ref>see Theorem 2.4, p. 311 in {{harvtxt|Bennett|Sharpley|1988}}</ref>
| |
| Let ''A''<sub>0</sub>, ''A''<sub>1</sub> be intermediate spaces of the compatible couple (''X''<sub>0</sub>, ''X''<sub>1</sub>), of class θ<sub>0</sub> and θ<sub>1</sub> respectively, with {{nowrap| 0 < θ<sub>0</sub>, θ<sub>1</sub> < 1}} and {{nowrap| θ<sub>0</sub> ≠ θ<sub>1</sub>}}. When {{nowrap| 0 < θ < 1}} and {{nowrap| 1 ≤ ''q'' ≤ ∞}}, one has
| |
| | |
| :<math>(A_0, A_1)_{\theta, q} = (X_0, X_1)_{\eta, q}, \ \ \text{where} \ \ \eta = (1 - \theta) \theta_0 + \theta \, \theta_1.</math>
| |
| | |
| It is notable that when interpolating with the real method between ''A''<sub>0</sub> = {{nowrap|(''X''<sub>0</sub>, ''X''<sub>1</sub>)<sub>θ<sub>0</sub>,''q''<sub> 0</sub></sub>}} and ''A''<sub>1</sub> = {{nowrap|(''X''<sub>0</sub>, ''X''<sub>1</sub>)<sub>θ<sub>1</sub>,''q''<sub> 1</sub></sub>}}, only the values of θ<sub>0</sub> and θ<sub>1</sub> matter. Also, ''A''<sub>0</sub> and ''A''<sub>1</sub> can be complex interpolation spaces between ''X''<sub>0</sub> and ''X''<sub>1</sub>, with parameters θ<sub>0</sub> and θ<sub>1</sub> respectively.
| |
| | |
| There is also a reiteration theorem for the complex method.
| |
| | |
| '''Theorem.''' <ref>see 12.3, p. 121 in {{harvtxt|Calderón|1964}}.</ref>
| |
| Let (''X''<sub>0</sub>, ''X''<sub>1</sub>) be a compatible couple of complex Banach spaces, and assume that {{nowrap|''X''<sub>0</sub> ∩ ''X''<sub>1</sub>}} is dense in ''X''<sub>0</sub> and in ''X''<sub>1</sub>. Let {{nowrap|''A''<sub>0</sub> {{=}} (''X''<sub>0</sub>, ''X''<sub>1</sub>)<sub>θ<sub>0</sub></sub>}} and {{nowrap|''A''<sub>1</sub> {{=}} (''X''<sub>0</sub>, ''X''<sub>1</sub>)<sub>θ<sub>1</sub></sub>}}, where {{nowrap| 0 ≤ θ<sub>0</sub> ≤ θ<sub>1</sub> ≤ 1}}. Assume further that {{nowrap|''X''<sub>0</sub> ∩ ''X''<sub>1</sub>}} is dense in {{nowrap|''A''<sub>0</sub> ∩ ''A''<sub>1</sub>}}. Then, for every {{nowrap|θ ∈ [0, 1]}},
| |
| | |
| :<math> \bigl( (X_0, X_1)_{\theta_0}, (X_0, X_1)_{\theta_1} \bigr)_\theta = (X_0, X_1)_\eta, \ \ \text{to} \ \ \eta = (1 - \theta) \theta_0 + \theta \, \theta_1.</math>
| |
| | |
| The density condition is always satisfied when {{nowrap| ''X''<sub>0</sub> ⊂ ''X''<sub>1</sub>}} or {{nowrap| ''X''<sub>1</sub> ⊂ ''X''<sub>0</sub>}}. | |
| | |
| == Duality ==
| |
| Let {{nowrap|(''X''<sub>0</sub>, ''X''<sub>1</sub>)}} be a compatible couple, and assume that {{nowrap|''X''<sub>0</sub> ∩ ''X''<sub>1</sub>}} is dense in ''X''<sub>0</sub> and in ''X''<sub>1</sub>. In this case, the restriction map from the (continuous) [[Dual space#Continuous dual space|dual]] {{nowrap|''X'' ′<sub>''j''</sub>}} of ''X''<sub>''j''</sub>, {{nowrap| ''j'' {{=}} 0, 1}}, to the dual of {{nowrap|''X''<sub>0</sub> ∩ ''X''<sub>1</sub>}} is one-to-one. It follows that the pair of duals {{nowrap|(''X'' ′<sub>0</sub>, ''X'' ′<sub>1</sub>)}} is a compatible couple continuously embedded in the dual {{nowrap|(''X''<sub>0</sub> ∩ ''X''<sub>1</sub>) ′}}.
| |
| | |
| For the complex interpolation method, the following duality result holds:
| |
| | |
| '''Theorem.'''<ref name="Cald">see 12.1 and 12.2, p. 121 in {{harvtxt|Calderón|1964}}.</ref>
| |
| Let {{nowrap|(''X''<sub>0</sub>, ''X''<sub>1</sub>)}} be a compatible couple of complex Banach spaces, and assume that {{nowrap|''X''<sub>0</sub> ∩ ''X''<sub>1</sub>}} is dense in ''X''<sub>0</sub> and in ''X''<sub>1</sub>. If ''X''<sub>0</sub> and ''X''<sub>1</sub> are [[Reflexive space|reflexive]], then the dual of the complex interpolation space is obtained by interpolating the duals,
| |
| | |
| :<math> ( (X_0, X_1)_\theta )' = (X'_0, X'_1)_\theta, \quad 0 < \theta < 1.</math>
| |
| | |
| In general, the dual of the space {{nowrap|(''X''<sub>0</sub>, ''X''<sub>1</sub>)<sub>θ</sub>}} is equal<ref name="Cald" />
| |
| to {{nowrap|(''X'' ′<sub>0</sub>, ''X'' ′<sub>1</sub>)<sup>θ</sup>}}, a space defined by a variant of the complex method.<ref>Theorem 4.1.4, p. 89 in {{harvtxt|Bergh|Löfström|1976}}.</ref>
| |
| The upper-θ and lower-θ methods do not coincide in general, but they do if at least one of ''X''<sub>0</sub>, ''X''<sub>1</sub> is a reflexive space.<ref>Theorem 4.3.1, p. 93 in {{harvtxt|Bergh|Löfström|1976}}.</ref>
| |
| | |
| For the real interpolation method, the duality holds provided that the parameter ''q'' is finite:
| |
| | |
| '''Theorem.'''<ref>see Théorème 3.1, p. 23 in {{harvtxt|Lions|Peetre|1964}}, or Theorem 3.7.1, p. 54 in {{harvtxt|Bergh|Löfström|1976}}.</ref>
| |
| Let {{nowrap| 0 < θ < 1}}, {{nowrap|1 ≤ ''q'' < ∞}} and {{nowrap|(''X''<sub>0</sub>, ''X''<sub>1</sub>)}} a compatible couple of real Banach spaces. Assume that {{nowrap|''X''<sub>0</sub> ∩ ''X''<sub>1</sub>}} is dense in ''X''<sub>0</sub> and in ''X''<sub>1</sub>. Then
| |
| | |
| :<math> ( (X_0, X_1)_{\theta, q} )' = (X'_0, X'_1)_{\theta, q'}, \ \ \text{where} \ \ 1/q' = 1 - 1/q.</math>
| |
| | |
| == Discrete definitions ==
| |
| Since the function {{nowrap|''t'' → ''K''(''x'', ''t'')}} varies regularly (it is increasing, but {{nowrap| ''K''(''x'', ''t'') / ''t'' }} is decreasing), the definition of the {{nowrap|''K''<sub>θ, ''q'' </sub>}}-norm of a vector ''x'', previously given by an integral, is equivalent to a definition given by a series.<ref>see chap. II in {{harvtxt|Lions|Peetre|1964}}.</ref>
| |
| This series is obtained by breaking (0, ∞) into pieces (2<sup>''n''</sup>, 2<sup>''n''+1</sup>) of equal mass for the measure {{nowrap|d ''t'' / ''t''}},
| |
| | |
| :<math> \|x\|_{\theta, q; K} \simeq \Bigl( \sum_{n \in \mathbf{Z}} \bigl( 2^{-\theta n} K(x, 2^n; X_0, X_1) \bigr)^q \Bigr)^{1/q}.</math>
| |
| | |
| In the special case where ''X''<sub>0</sub> is continuously embedded in ''X''<sub>1</sub>, one can omit the part of the series with negative indices ''n''. In this case, each of the functions {{nowrap| ''x'' → ''K''(''x'', 2<sup>''n''</sup>; ''X''<sub>0</sub>, ''X''<sub>1</sub>)}} defines an equivalent norm on ''X''<sub>1</sub>.
| |
| | |
| The interpolation space {{nowrap| (''X''<sub>0</sub>, ''X''<sub>1</sub>)<sub>θ, ''q''</sub>}} is a "diagonal subspace" of an ℓ<sup>''q''</sup>-sum of a sequence of Banach spaces (each one being isomorphic to ''X''<sub>0</sub> + ''X''<sub>1</sub>). Therefore, when ''q'' is finite, the dual of {{nowrap| (''X''<sub>0</sub>, ''X''<sub>1</sub>)<sub>θ, ''q''</sub> }} is a [[Banach space#General theory|quotient]] of the ℓ<sup>''p''</sup>-sum of the duals, {{nowrap| 1 / ''p'' + 1 / ''q'' {{=}} 1}}, which leads to the following formula for the discrete {{nowrap|''J''<sub>θ, ''p'' </sub>}}-norm of a functional ''x''' in the dual of {{nowrap| (''X''<sub>0</sub>, ''X''<sub>1</sub>)<sub>θ, ''q''</sub> }}: | |
| | |
| :<math> \|x'\|_{\theta, p; J} \simeq \inf \Bigl\{ \Bigl( \sum_{n \in \mathbf{Z}} \bigl( 2^{\theta n} \max(\|x'_n\|_{X'_0}, 2^{-n} \|x'_n\|_{X'_1}) \bigr)^p \Bigr)^{1/p} \!:\, x' = \sum_{n \in \mathbf{Z}} x'_n \Bigr\}.</math>
| |
| | |
| The usual formula for the discrete {{nowrap|''J''<sub>θ, ''p'' </sub>}}-norm is obtained by changing ''n'' to −''n''.
| |
| | |
| The discrete definition makes several questions easier to study, among which the already mentioned identification of the dual. Other such questions are compactness or weak-compactness of linear operators. Lions and Peetre have proved that:
| |
| | |
| '''Theorem.'''<ref>see chap. 5, Théorème 2.2, p. 37 in {{harvtxt|Lions|Peetre|1964}}.</ref>
| |
| If the linear operator ''T'' is [[Compact operator|compact]] from ''X''<sub>0</sub> to a Banach space ''Y'' and bounded from ''X''<sub>1</sub> to ''Y'', then ''T'' is compact from {{nowrap|(''X''<sub>0</sub>, ''X''<sub>1</sub>)<sub>θ, ''q'' </sub>}} to ''Y'' when {{nowrap| 0 < θ < 1}}, {{nowrap| 1 ≤ ''q'' ≤ ∞}}.
| |
| | |
| Davis, Figiel, Johnson and Pełczyński have used interpolation in their proof of the following result:
| |
| | |
| '''Theorem.'''<ref>{{citation
| |
| | last1 = Davis | first1 = William J.
| |
| | last2 = Figiel | first2 = Tadeusz
| |
| | last3 = Johnson | first3 = William B.
| |
| | author3-link = William B. Johnson (mathematician)
| |
| | last4 = Pełczyński | first4 = Aleksander
| |
| | year = 1974
| |
| | title = Factoring weakly compact operators
| |
| | journal = J. Functional Analysis
| |
| | volume = 17
| |
| | pages = 311–327
| |
| }}, see also Theorem 2.g.11, p. 224 in {{harvtxt|Lindenstrauss|Tzafriri|1979}}.</ref>
| |
| A bounded linear operator between two Banach spaces is [[Weak topology|weakly compact]] if and only if it factors through a [[reflexive space]].
| |
| | |
| === A general interpolation method ===
| |
| The space ℓ<sup>''q''</sup> used for the discrete definition can be replaced by an arbitrary [[sequence space]] ''Y'' with [[Schauder basis#Unconditionality|unconditional basis]], and the weights {{nowrap| ''a''<sub>''n''</sub> {{=}} 2<sup>−θ''n''</sup>}}, {{nowrap| ''b''<sub>''n''</sub> {{=}} 2<sup>(1−θ)''n''</sup>}}, that are used for the {{nowrap|''K''<sub>θ, ''q'' </sub>}}-norm,
| |
| can be replaced by general weights
| |
| | |
| :<math>a_n, b_n > 0, \ \ \sum_{n=1}^\infty \min(a_n, b_n) < \infty.</math>
| |
| | |
| The interpolation space {{nowrap| ''K''(''X''<sub>0</sub>, ''X''<sub>1</sub>, ''Y'', {''a''<sub>''n''</sub>}, {''b''<sub>''n''</sub>}) }} consists of the vectors ''x'' in {{nowrap| ''X''<sub>0</sub> + ''X''<sub>1</sub>}} such that<ref>
| |
| {{citation
| |
| | last1 = Johnson
| |
| | first1 = William B.
| |
| | last2 = Lindenstrauss
| |
| | first2 = Joram
| |
| | contribution = Basic concepts in the geometry of Banach spaces
| |
| | title = Handbook of the geometry of Banach spaces, Vol. I
| |
| | pages = 1–84
| |
| | publisher = North-Holland
| |
| | location = Amsterdam
| |
| | year = 2001
| |
| }}, and section 2.g in {{harvtxt|Lindenstrauss|Tzafriri|1979}}.
| |
| </ref>
| |
| | |
| :<math> \|x\|_{K(X_0, X_1)} = \sup_{m \ge 1} \Bigl\| \sum_{n=1}^m a_n K(x, b_n / a_n; X_0, X_1) \, y_n\Bigr\|_Y < \infty,</math>
| |
| | |
| where {''y''<sub>''n''</sub>} is the unconditional basis of ''Y''. This abstract method can be used, for example, for the proof of the following result:
| |
| | |
| '''Theorem.'''<ref>see Theorem 3.b.1, p. 123 in {{citation
| |
| | last1 = Lindenstrauss
| |
| | first1 = Joram | author1-link = Joram Lindenstrauss
| |
| | last2 = Tzafriri | first2 = Lior
| |
| | location = Berlin
| |
| | publisher = Springer-Verlag
| |
| | series = Ergebnisse der Mathematik und ihrer Grenzgebiete
| |
| | title = Classical Banach Spaces I, Sequence Spaces
| |
| | volume = 92
| |
| | pages = xiii+188
| |
| | isbn = 3-540-08072-4
| |
| | year = 1977}}.</ref>
| |
| A Banach space with unconditional basis is isomorphic to a complemented subspace of a space with [[Schauder basis#Unconditionality|symmetric basis]].
| |
| | |
| == Interpolation of Sobolev and Besov spaces ==
| |
| Several interpolation results are available for [[Sobolev space]]s and [[Besov space]]s on '''R'''<sup>''n''</sup>,<ref>Theorem 6.4.5, p. 152 in {{harvtxt|Bergh|Löfström|1976}}.</ref>
| |
| | |
| :<math> H^s_p, \ \ s \in \mathbf{R}, \ 1 \le p \le \infty \, ; \quad B^s_{p, q}, \ \ s \in \mathbf{R}, \ 1 \le p, q \le \infty.</math>
| |
| | |
| These spaces are spaces of [[measurable function]]s on '''R'''<sup>''n''</sup> when {{nowrap|''s'' ≥ 0}}, and of [[Distribution (mathematics)|tempered distributions]] on '''R'''<sup>''n''</sup> when {{nowrap|''s'' < 0}}. For the rest of the section, the following setting and notation will be used:
| |
| | |
| :<math> 0 < \theta < 1, \ \ 1 \le p, p_0, p_1, q, q_0, q_1 \le \infty, \ \ s, s_0, s_1 \in \mathbf{R},</math>
| |
| and
| |
| | |
| :<math> \frac 1 {p_\theta} = \frac{1 - \theta}{p_0} + \frac{\theta}{p_1}, \ \frac 1 {q_\theta} = \frac{1 - \theta}{q_0} + \frac{\theta}{q_1}, \ \ s_\theta = (1 - \theta) s_0 + \theta s_1.</math>
| |
| | |
| Complex interpolation works well on the class of Sobolev spaces <math>H^{s}_{p}</math> (the [[Sobolev space#Bessel potential spaces|Bessel potential spaces]]),
| |
| | |
| :<math>(H^{s_0}_{p_0}, H^{s_1}_{p_1})_\theta = H^{s_\theta}_{p_\theta}, \quad s_0 \ne s_1, \ 1 < p_0, p_1 < \infty,</math>
| |
| | |
| and it also works well with the class of Besov spaces,
| |
| | |
| :<math>(B^{s_0}_{p_0, q_0}, B^{s_1}_{p_1, q_1})_\theta = B^{s_\theta}_{p_\theta, q_\theta}, \quad s_0 \ne s_1, \ 1 \le p_0, p_1, q_0, q_1 \le \infty.</math>
| |
| | |
| Real interpolation between Sobolev spaces may give Besov spaces, except when {{nowrap|''s''<sub>0</sub> {{=}} ''s''<sub>1</sub>}},
| |
| | |
| :<math>(H^{s}_{p_0}, H^{s}_{p_1})_{\theta, p_\theta} = H^{s}_{p_\theta}, \quad 1 \le p_0, p_1 \le \infty.</math>
| |
| | |
| When {{nowrap|''s''<sub>0</sub> ≠ ''s''<sub>1</sub>}} but {{nowrap|''p''<sub>0</sub> {{=}} ''p''<sub>1</sub>}}, real interpolation between Sobolev spaces gives a Besov space:
| |
| | |
| :<math>(H^{s_0}_p, H^{s_1}_p)_{\theta, q} = B^{s_\theta}_{p, q}, \quad s_0 \ne s_1, \ 1 \le p, q \le \infty.</math>
| |
| | |
| Also,
| |
| :<math>(B^{s_0}_{p, q_0}, B^{s_1}_{p, q_1})_{\theta, q} = B^{s_\theta}_{p, q}, \quad s_0 \ne s_1, \ 1 \le p, q, q_0, q_1 \le \infty,</math>
| |
| and
| |
| | |
| :<math>(B^{s}_{p, q_0}, B^{s}_{p, q_1})_{\theta, q} = B^{s}_{p, q_\theta}, \quad 1 \le p, q_0, q_1 \le \infty,</math>
| |
| | |
| :<math>(B^{s_0}_{p_0, q_0}, B^{s_1}_{p_1, q_1})_{\theta, q_\theta} = B^{s_\theta}_{p_\theta, q_\theta}, \quad s_0 \ne s_1, \ p_\theta =q_\theta, \ 1 \le p_0, p_1, q_0, q_1 \le \infty.</math>
| |
| | |
| == See also ==
| |
| *[[Riesz–Thorin theorem]]
| |
| * [[Marcinkiewicz interpolation theorem]]
| |
| | |
| == Notes ==
| |
| {{Reflist}}
| |
| | |
| == References ==
| |
| *{{citation
| |
| | last = Calderón | first = Alberto P. | author-link = Alberto Calderón
| |
| | title = Intermediate spaces and interpolation, the complex method
| |
| | journal = Studia Math.
| |
| | volume = 24
| |
| | year = 1964
| |
| | pages = 113–190
| |
| }}.
| |
| *{{citation
| |
| | last1 = Lions |first1 = Jacques-Louis.
| |
| | author-link1 = Jacques-Louis Lions
| |
| | last2 = Peetre |first2 = Jaak
| |
| | title = Sur une classe d'espaces d'interpolation
| |
| | language = French
| |
| | journal = Inst. Hautes Études Sci. Publ. Math.
| |
| | volume = 19
| |
| | year = 1964
| |
| | pages = 5–68
| |
| }}.
| |
| *{{citation
| |
| | last1 = Bennett | first1 = Colin
| |
| | last2 = Sharpley |first2 = Robert
| |
| | title = Interpolation of operators
| |
| | series = Pure and Applied Mathematics
| |
| | volume = 129
| |
| | publisher = Academic Press, Inc., Boston, MA,
| |
| | year = 1988
| |
| | pages = xiv+469
| |
| | ISBN = 0-12-088730-4
| |
| }}.
| |
| *{{citation
| |
| | last1 = Bergh | first1 = Jöran
| |
| | last2 = Löfström | first2 = Jörgen
| |
| | title = Interpolation spaces. An introduction
| |
| | series = Grundlehren der Mathematischen Wissenschaften
| |
| | volume = 223
| |
| | publisher = Springer-Verlag
| |
| | location = Berlin-New York
| |
| | year = 1976
| |
| | pages = x+207
| |
| | ISBN = 3-540-07875-4
| |
| }}.
| |
| *{{citation
| |
| | last1 = Lindenstrauss
| |
| | first1 = Joram | author1-link = Joram Lindenstrauss
| |
| | last2 = Tzafriri | first2 = Lior
| |
| | title = Classical Banach spaces. II. Function spaces
| |
| | series = Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas]
| |
| | volume = 97
| |
| | publisher = Springer-Verlag
| |
| | location = Berlin-New York
| |
| | year = 1979
| |
| | pages = x+243
| |
| | isbn = 3-540-08888-1 | |
| }}.
| |
| *{{citation|last=Tartar|first=Luc|title=An Introduction to Sobolev Spaces and Interpolation |publisher=Springer|year=2007| ISBN=978-3-540-71482-8 }}.
| |
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| [[Category:Banach spaces]]
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| [[Category:Sobolev spaces]]
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| [[Category:Fourier analysis]]
| |