Kolmogorow-Riesz theorem

The set of Kolmogorov-Riesz (after Andrey Kolmogorov and Marcel Riesz ) is a theorem from the mathematical sub-region of the functional analysis , the a compactness criterion for subsets of L ^p represents -spaces. Depending on the degree of generalization, this theorem is also called M. Riesz's Theorem, Kolmogorow-Fréchet-Riesz's Theorem or Kolmogorow-Riesz-Weil's Theorem, which also honors contributions by the mathematicians Maurice René Fréchet and André Weil , also the names Jakob Davidowitsch Tamarkin and AN Tulajkov are mentioned by a few authors, the latter dealing with the special case . Such compactness criteria have many applications, particularly in the theory of partial differential equations . ${\displaystyle p=1}$

The sequence space l ^p

The situation for the sequence spaces is particularly simple, the following theorem was proven in 1908 for von Fréchet : ${\displaystyle \ell ^{p}}$ ${\displaystyle p=2}$

A subset ( ) is precompact if and only if the following two conditions are met: ${\displaystyle M\subset \ell ^{p}}$ ${\displaystyle 1\leq p<\infty }$

• ${\displaystyle \sup _{x\in M}|x_{k}|<\infty \,\,\forall k\in \mathbb {N} }$. Where the -th component of . ${\displaystyle x_{k}}$ ${\displaystyle k}$ ${\displaystyle x}$
• ${\displaystyle \sup _{x\in M}\sum _{k=n}^{\infty }|x_{k}|^{p}\,{\xrightarrow[{n\to \infty }]{}}\,0}$.

Function spaces L ^p

Elaborate are compactness criteria for L ^p -Spaces over non- discrete basic quantities. By tracing back to Arzelà-Ascoli's theorem , one can show:

Fréchet-Kolmogorov theorem : A subset , is if and precompact if ${\displaystyle M\subset L^{p}([0,1])}$ ${\displaystyle 1\leq p<\infty }$

• ${\displaystyle M}$is limited with regard to the norm , ${\displaystyle \|\cdot \|_{p}}$
• ${\displaystyle \sup _{f\in M}\int _{0}^{1}|f(t+h)-f(t)|^{p}{\mathrm {d} }t{\xrightarrow[{h\to 0}]{}}\,0}$.

It is for outside the unit interval to in the above formula to form. An analogous sentence naturally applies to for any . ${\displaystyle f(t):=0}$ ${\displaystyle t}$ ${\displaystyle f(t+h)}$ ${\displaystyle L^{p}([a,b])}$ ${\displaystyle a,b\in R,a<b}$

An extension of this theorem to include unrestricted areas requires an additional condition:

Theorem of M. Riesz : A subset ( ) is precompact if and only if ${\displaystyle M\subset L^{p}(\mathbb {R} ^{n})}$ ${\displaystyle 1\leq p<\infty }$

• ${\displaystyle M}$is limited with regard to the norm , ${\displaystyle \|\cdot \|_{p}}$
• ${\displaystyle \sup _{f\in M}\int _{\mathbb {R} ^{n}}|f(t+h)-f(t)|^{p}\mathrm {d} t\,{\xrightarrow[{h\to 0}]{}}\,0}$,
• ${\displaystyle \sup _{f\in M}\int _{\mathbb {R} ^{n}\setminus B_{r}(0)}|f(t)|^{p}\mathrm {d} t\,{\xrightarrow[{r\nearrow \infty }]{}}\,0}$.

Here stands for the sphere around 0 with radius . ${\displaystyle B_{r}(0)}$ ${\displaystyle r}$

Local compact Abelian groups

The set of M. Riesz not be applied to L ^p -Spaces on any measure spaces generalize, as in the second condition of the compactness criterion of addition and therefore of the group structure of the use is made. Now let a locally compact Abelian group and is a Haar measure on . Is a Banach space, so you can above the space of all measurable functions with form. The norm turns into a Banach space. This evidently generalizes the spaces considered above . Instead of the spheres around 0, we consider a network of compact sets in directed with respect to the union , so that every compact set from is contained in a set from . ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle G}$ ${\displaystyle \mu }$ ${\displaystyle G}$ ${\displaystyle Y}$ ${\displaystyle L^{p}(G,Y)}$ ${\displaystyle f\colon G\rightarrow Y}$ ${\displaystyle \int _{G}\|f(t)\|^{p}\,\mathrm {d} \mu (t)<\infty }$ ${\displaystyle \|f\|_{p}:=\left(\int _{G}\|f(t)\|^{p}\,\mathrm {d} \mu (t)\right)^{\frac {1}{p}}}$ ${\displaystyle L^{p}(G,Y)}$ ${\displaystyle L^{p}(\mathbb {R} ^{n},Y)}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle {\mathcal {C}}}$

Nicolae Dinculeanu has proven the following generalization of the above compactness criterion:

Theorem : A subset ( , locally compact Abelian group, Banach space) is precompact if and only if ${\displaystyle M\subset L^{p}(G,Y)}$ ${\displaystyle 1\leq p<\infty }$ ${\displaystyle G}$ ${\displaystyle Y}$

• For all measurable subsets with is precompact, ${\displaystyle A\subset G}$ ${\displaystyle \mu (A)<\infty }$ ${\displaystyle \{\int _{A}f(t)\mathrm {d} \mu (t);f\in M\}\subset Y}$
• ${\displaystyle \sup _{f\in M}\int _{G}\|f(t+h)-f(t)\|^{p}\,\mathrm {d} \mu (t)\,{\xrightarrow[{h\to 0}]{}}\,0}$,
• ${\displaystyle \sup _{f\in M}\int _{G\setminus C}\|f(t)\|^{p}\,\mathrm {d} \mu (t)\,{\xrightarrow[{C\in {\mathcal {C}}}]{}}\,0}$.

This version was proven for the case , i.e. for scalar-valued functions, by M. Riesz. A version that goes back to Kolmogorov and JD Tamarkin and uses an approximation of the one was also generalized to the Banach space-valued case by N. Dinculeanu. For the following presentation of this result, let us assume a zero neighborhood basis from relatively compact, open sets in . For each choose a function that is bounded, measurable and symmetric (i.e., ) with support in and . One can choose, for example , where is the characteristic function of . For and let the convolution be defined. Then is , and ; that is, in this sense the network is an approximation of one. The following applies ${\displaystyle Y=\mathbb {R} }$ ${\displaystyle {\mathcal {V}}}$ ${\displaystyle G}$ ${\displaystyle V\in {\mathcal {V}}}$ ${\displaystyle u_{V}\colon G\rightarrow \mathbb {R} _{0}^{+}}$ ${\displaystyle u_{V}(t)=u_{V}(-t)\,\forall t\in G}$ ${\displaystyle {\overline {V}}}$ ${\displaystyle \int _{G}^{}u_{V}(t)\mathrm {d} \mu (t)=1}$ ${\displaystyle u_{V}={\frac {1}{2\mu (V)}}(\chi _{V}+\chi _{-V})}$ ${\displaystyle \chi _{V}}$ ${\displaystyle V}$ ${\displaystyle f\in L^{p}(G,Y)}$ ${\displaystyle t\in G}$ ${\displaystyle u_{v}\star f(t):=\int _{G}u_{V}(s)f(t-s)\mathrm {d} \mu (s)}$ ${\displaystyle u_{V}\in L^{p}(G,Y)}$ ${\displaystyle u_{V}\star f\in L^{p}(G,Y)}$ ${\displaystyle \|u_{V}\star f-f\|_{p}\,{\xrightarrow[{V\in {\mathcal {V}}}]{}}\,0}$ ${\displaystyle (u_{V})_{V\in {\mathcal {V}}}}$

Theorem : A subset ( , locally compact Abelian group, Banach space) is precompact if and only if ${\displaystyle M\subset L^{p}(G,Y)}$ ${\displaystyle 1\leq p<\infty }$ ${\displaystyle G}$ ${\displaystyle Y}$

• For all measurable subsets with is precompact, ${\displaystyle A\subset G}$ ${\displaystyle \mu (A)<\infty }$ ${\displaystyle K(A):=\{\int _{A}f(t)\mathrm {d} \mu (t);f\in M\}}$
• ${\displaystyle \sup _{f\in M}\int _{G}\|(u_{V}\star f)(t)-f(t)\|^{p}\,\mathrm {d} \mu (t)\,{\xrightarrow[{V\in {\mathcal {V}}}]{}}\,0}$,
• ${\displaystyle \sup _{f\in M}\int _{G\setminus C}\|f(t)\|^{p}\,\mathrm {d} \mu (t)\,{\xrightarrow[{C\in {\mathcal {C}}}]{}}\,0}$.

In the earlier versions for and the nets and were used. If one applies this theorem to the locally compact Abelian group , then the first condition is equivalent to , for every set of finite measure is finite; the second condition is empty if you choose the net , and the last condition becomes too if you set. With a suitable isomorphism between and one obtains exactly the sentence about spaces cited at the beginning . ${\displaystyle G=\mathbb {R} }$ ${\displaystyle Y=\mathbb {R} }$ ${\displaystyle u_{n}={\frac {1}{2n}}\chi _{[-n,n]}}$ ${\displaystyle C_{n}=[-n,n]}$ ${\displaystyle G=\mathbb {Z} }$ ${\displaystyle \sup _{x\in M}|x_{k}|<\infty \,\,\forall k\in \mathbb {N} }$ ${\displaystyle u_{V}=u_{\{0\}}}$ ${\displaystyle \sup _{x\in M}\sum _{|k|>n}|x_{k}|^{p}\,{\xrightarrow[{n\to \infty }]{}}\,0}$ ${\displaystyle C_{n}=\{-n,\ldots ,n\}}$ ${\displaystyle \ell ^{p}}$ ${\displaystyle L^{p}(\mathbb {Z} )}$ ${\displaystyle \ell ^{p}}$

Further generalizations

Further generalizations to non-commutative locally compact groups were found by Josh Isralowitz . An expansion of compactness criteria of this type to other function spaces defined by locally compact groups can be found in Hans G. Feichtinger .

Individual evidence

1. ↑ H. Hanche-Olsen, Helge Holden: The Kolmogorov-Riesz Compactness Theorem , 4. A Bit of History
2. ↑ M. Fréchet: Essai de geometrie analytique , Nouv. ann. Math. 4 (1908) 97-116, 289-317.
3. ↑ Joseph Wloka : Functional Analysis and Applications , §22, Sentence 1
4. ↑ Jürgen Appell, Martin Väth : Elements of functional analysis. Vector spaces, operators and fixed point theorem , Theorem 3.2
5. ↑ Joseph Wloka : Functional Analysis and Applications , §22, sentence 3
6. ↑ N. Dinculeanu: On Kolmogorov-Tamarkin and M. Riesz Compactness Criteria in Function Spaces Over a Locally Compact Group ., J. Math Anal. Appl. 87, pp. 67-85 (1982)
7. ↑ Josh Isralowitz: A characterization of norm compactness in the Bochner space L ^p (G; B) for an arbitrary locally compact group , J. Math. Anal. Appl. 323,2 (2005), pp. 1007-1017
8. ↑ Hans G. Feichtinger: Compactness in Translation Invariant Banach Spaces of Distributions and Compact Multipliers , Journal of Mathematical Analysis and Applications (1984), Volume 102, pages 289-327, Theorem 2.2

swell

• Hans Wilhelm Alt: Lineare functional analysis , Springer-Verlag (2006) ISBN 3-540-34186-2
• Jürgen Appell, Martin Väth : Elements of functional analysis. Vector spaces, operators and fixed point sentences , Vieweg + Teubner (2005), ISBN 3-528-03222-7
• N. Dinculeanu: On Kolmogorov-Tamarkin and M. Riesz Compactness Criteria in Function Spaces Over a Locally Compact Group , J. Math. Anal. Appl. 87, pp. 67-85 (1982)
• Hans G. Feichtinger: Compactness in Translation Invariant Banach Spaces of Distributions and Compact Multipliers , Journal of Mathematical Analysis and Applications (1984), Volume 102, pages 289-327. (also available online) (PDF; 2.6 MB)
• H. Hanche-Olsen, Helge Holden: The Kolmogorov – Riesz Compactness Theorem (PDF; 398 kB)
• AN Kolmogorow: On the compactness of the sets of functions in the mean convergence , Nachr. Akad. Wiss. Göttingen Math. Phys. Kl. II (1931), pages 60-63
• JD Tamarkin: On the compactness of the space L , Bull. Amer. Math. Soc. Volume 38 (1932) pages 79-84
• AN Tulajkow: On compactness in space L ^p for p = 1 , Göttinger. Nachrichten (1933), pages 167-170
• Joseph Wloka : Functional Analysis and Applications , ISBN 3-110-01989-2