Riesz-Thorin theorem
Encyclopedia
In mathematics
, the Riesz–Thorin theorem, often referred to as the Riesz–Thorin interpolation theorem or the Riesz–Thorin convexity theorem is a result about interpolation of operators. It is named after Marcel Riesz
and his student G. Olof Thorin.
This theorem bounds the norms of linear maps acting between
Lp
spaces. Its usefulness stems from the fact that some of these spaces have rather simpler structure than others. Usually that refers to L2 which is a Hilbert space
, or to L1 and L∞. Therefore one may prove theorems about the more complicated cases by proving them in two simple cases and then using the Riesz–Thorin theorem to pass from the simple cases to the complicated cases. The Marcinkiewicz theorem
is similar but applies to non-linear maps.
This is informal because an operator cannot formally be defined on two different spaces at the same time. To formalize it we need to say: let T be a linear operator defined on a family F of functions that is dense in both
and 
(for example, the family of all simple functions). And assume that Tƒ is in both 
and 
for any ƒ in F, and that T is bounded in both norms. Then for any r between p and q we have that F is dense in 
, that Tƒ is in 
for any ƒ in F and that T is bounded in the 
norm. These three ensure that T can be extended to an operator from 
to 
.
In addition an inequality for the norms holds, namely
A version of this theorem exists also when the domain
and range
of T are not identical. In this case, if T is bounded from
to 
then one should draw the point 
in the unit square
. The two q-s give a second point. Connect them with a straight line segment and you get the r-s for which T is bounded. Here is again the almost formal version
Theorem: Assume T is a bounded linear operator from
to 
and at the same time from 
to 
. Then it is also a bounded operator from 
to 
with
and t is any number between 0 and 1.
The perfect formalization is done as in the simpler case.
One last generalization is that the theorem holds for
for any measure space Ω. In particular it holds for the 
spaces.
to be the operator norm of a continuous extension of T to
if such an extension exists, and ∞ otherwise. Then the theorem asserts that the function
is convex in the rectangle (a,b) ∈ [0,1]×[0,1].
, namely let T be the operator that takes a function
on the unit circle
and outputs
the sequence of its Fourier coefficients
Parseval's theorem
shows that T is bounded from
to 
with norm 1. On the other hand, clearly,
so T is bounded from
to 
with norm 1. Therefore we may invoke the Riesz–Thorin theorem to get, for any 1 < p < 2 that T, as an operator from 
to 
, is bounded with norm 1, where
In a short formula, this says that
This is the Hausdorff–Young inequality
.
It is well known that T is bounded from
to 
and it is trivial that it is bounded from L∞ to L∞ (both bounds are by 
). Therefore the Riesz–Thorin theorem gives
We take this inequality and switch the role of the operator and the operand, or in other words, we think of S as the operator of convolution with g, and get that S is bounded from
to 
. Further, since g is in 
we get,
in view of Hölder's inequality, that S is bounded from
to L∞, where again 
. So interpolating we get
where the connection between p, r and s is
Here is a brief sketch of that proof:
One of its main ingredients is the following rather well known result about analytic functions. Suppose that
is a bounded analytic function on the two lines
and 
and on the strip between these two lines.
Suppose also that
at every point 
on those two lines.
Then, by applying the Phragmén–Lindelöf principle (a kind of maximum principle for infinite domains) one gets that
at every point between these two lines, and in particular at the point 
.
Thorin ingeniously defined a special analytic function
connected with the operator 
. He used the fact that 
is bounded on
to deduce that 
on the line
, and, analogously, he used the boundedness of 
on
to deduce that 
on the line
. Then, after using the result mentioned above to give that
, he was able to show that this implies that
is bounded on
.
Thorin obtained this function
with the help of a generalized notion of an analytic function whose values are elements of 
spaces instead of being complex numbers.
In the 1960s Alberto Calderón
adapted and generalized Thorin's ideas to develop
the method of complex interpolation
.
Suppose that
and 
are two Banach space
s which are continuously contained in some suitable larger space.
Calderon's method enables one to construct a family of new Banach spaces
,
for each
with 
which are ``between"
and 
and have the ``interpolation" property that
every linear operator
which is bounded on 
and on 
is also bounded on each of the complex interpolation spaces 
.
Calderon's spaces have many applications. See for example Sobolev space.
in the special case of spaces of sequences with unconditional bases (cf. below).
Assume
, 
. Then 
for any unconditional Banach space of sequences 
(that is, for any 
and any 
, 
).
The proof is based on the Krein–Milman theorem.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Riesz–Thorin theorem, often referred to as the Riesz–Thorin interpolation theorem or the Riesz–Thorin convexity theorem is a result about interpolation of operators. It is named after Marcel Riesz
Marcel Riesz
Marcel Riesz was a Hungarian mathematician who was born in Győr, Hungary . He moved to Sweden in 1908 and spent the rest of his life there, dying in Lund, where he was a professor from 1926 at Lund University...
and his student G. Olof Thorin.
This theorem bounds the norms of linear maps acting between
Lp
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
spaces. Its usefulness stems from the fact that some of these spaces have rather simpler structure than others. Usually that refers to L2 which is a Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
, or to L1 and L∞. Therefore one may prove theorems about the more complicated cases by proving them in two simple cases and then using the Riesz–Thorin theorem to pass from the simple cases to the complicated cases. The Marcinkiewicz theorem
Marcinkiewicz theorem
In mathematics, the Marcinkiewicz interpolation theorem, discovered by , is a result bounding the norms of non-linear operators acting on Lp spaces....
is similar but applies to non-linear maps.
Definition
A slightly informal version of the theorem can be stated as follows:- Theorem: Assume T is a bounded linear operator from Lp to Lp and at the same time from Lq to Lq. Then it is also a bounded operator from Lr to Lr for any r between p and q.
This is informal because an operator cannot formally be defined on two different spaces at the same time. To formalize it we need to say: let T be a linear operator defined on a family F of functions that is dense in both
and (for example, the family of all simple functions). And assume that Tƒ is in both and for any ƒ in F, and that T is bounded in both norms. Then for any r between p and q we have that F is dense in , that Tƒ is in for any ƒ in F and that T is bounded in the norm. These three ensure that T can be extended to an operator from to .In addition an inequality for the norms holds, namely
A version of this theorem exists also when the domain
Domain (mathematics)
In mathematics, the domain of definition or simply the domain of a function is the set of "input" or argument values for which the function is defined...
and range
Range (mathematics)
In mathematics, the range of a function refers to either the codomain or the image of the function, depending upon usage. This ambiguity is illustrated by the function f that maps real numbers to real numbers with f = x^2. Some books say that range of this function is its codomain, the set of all...
of T are not identical. In this case, if T is bounded from
to then one should draw the point in the unit squareUnit square
In mathematics, a unit square is a square whose sides have length 1. Often, "the" unit square refers specifically to the square in the Cartesian plane with corners at , , , and .-In the real plane:...
. The two q-s give a second point. Connect them with a straight line segment and you get the r-s for which T is bounded. Here is again the almost formal version
Theorem: Assume T is a bounded linear operator from
to and at the same time from to . Then it is also a bounded operator from to withand t is any number between 0 and 1.
The perfect formalization is done as in the simpler case.
One last generalization is that the theorem holds for
for any measure space Ω. In particular it holds for the spaces.Convexity
Another more general form of the theorem is as follows . Suppose that μ1 and μ2 are two measures on possibly different measure spaces. Let T be a linear mapping from the space of μ1-integrable functions into the space of μ2-measurable functions, and for 1 ≤ p,q ≤ ∞, define to be the operator norm of a continuous extension of T toif such an extension exists, and ∞ otherwise. Then the theorem asserts that the function
is convex in the rectangle (a,b) ∈ [0,1]×[0,1].
Hausdorff−Young inequality
We consider the Fourier operatorFourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
, namely let T be the operator that takes a function
on the unit circleUnit circle
In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...
and outputs
the sequence of its Fourier coefficients
Parseval's theorem
Parseval's theorem
In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum of the square of a function is equal to the sum of the square of its transform. It originates from a 1799 theorem about series by Marc-Antoine Parseval, which was later...
shows that T is bounded from
to with norm 1. On the other hand, clearly,so T is bounded from
to with norm 1. Therefore we may invoke the Riesz–Thorin theorem to get, for any 1 < p < 2 that T, as an operator from to , is bounded with norm 1, whereIn a short formula, this says that
This is the Hausdorff–Young inequality
Hausdorff–Young inequality
In mathematics, the Hausdorff−Young inequality bounds the Lq-norm of the Fourier coefficients of a periodic function for q ≥ 2. proved the inequality for some special values of q, and proved it in general...
.
Convolution operators
Let f be a fixed integrable function and let T be the operator of convolution with f, i.e., for each function g we haveIt is well known that T is bounded from
to and it is trivial that it is bounded from L∞ to L∞ (both bounds are by ). Therefore the Riesz–Thorin theorem givesWe take this inequality and switch the role of the operator and the operand, or in other words, we think of S as the operator of convolution with g, and get that S is bounded from
to . Further, since g is in we get,in view of Hölder's inequality, that S is bounded from
to L∞, where again . So interpolating we getwhere the connection between p, r and s is
Thorin's contribution
The original proof of this theorem, published in 1926 by Marcel Riesz, was a long and difficult calculation. Riesz' student G. Olof Thorin subsequently discovered a far more elegant proof and published it in 1939. The English mathematician J. E. Littlewood once enthusiastically referred to Thorin's proof as "the most impudent idea in mathematics".Here is a brief sketch of that proof:
One of its main ingredients is the following rather well known result about analytic functions. Suppose that
is a bounded analytic function on the two lines and and on the strip between these two lines.Suppose also that
at every point on those two lines.Then, by applying the Phragmén–Lindelöf principle (a kind of maximum principle for infinite domains) one gets that
at every point between these two lines, and in particular at the point .Thorin ingeniously defined a special analytic function
connected with the operator . He used the fact that is bounded on to deduce that on the line, and, analogously, he used the boundedness of on to deduce that on the line. Then, after using the result mentioned above to give that, he was able to show that this implies that is bounded on.Thorin obtained this function
with the help of a generalized notion of an analytic function whose values are elements of spaces instead of being complex numbers.In the 1960s Alberto Calderón
Alberto Calderón
Alberto Pedro Calderón was an Argentine mathematician best known for his work on the theory of partial differential equations and singular integral operators, and widely considered as one of the 20th century's most important mathematicians...
adapted and generalized Thorin's ideas to develop
the method of complex interpolation
Complex interpolation
In the field of mathematical analysis, an interpolation space is a space which lies "in between" two other spaces. The main applications are in Sobolev spaces, where spaces of functions that have a noninteger number of derivatives are interpolated from the spaces of functions with integer number of...
.
Suppose that
and are two Banach spaceBanach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
s which are continuously contained in some suitable larger space.
Calderon's method enables one to construct a family of new Banach spaces
,for each
with which are ``between" and and have the ``interpolation" property thatevery linear operator
which is bounded on and on is also bounded on each of the complex interpolation spaces .Calderon's spaces have many applications. See for example Sobolev space.
Mityagin's theorem
B.Mityagin extended the Riesz–Thorin theorem; we formulate the extensionin the special case of spaces of sequences with unconditional bases (cf. below).
Assume
, . Then for any unconditional Banach space of sequences (that is, for any and any , ).The proof is based on the Krein–Milman theorem.