Hilbert basis

As a Hilbert space basis, a basis of a Hilbert space is called in the functional analysis. A Hilbert space is an (often infinite-dimensional ) vector space equipped with a scalar product and is fully equipped with the induced norm.

The natural base concept of a Hilbert space is a generalization of the orthonormal basis of the Euclidean geometry, the complete orthonormal system or the Hilbert basis. Sometimes, such as in the wavelet theory, one works with generating systems of a Hilbert space, of which the orthogonality is difficult or impossible to prove.

This article deals primarily with such a Hilbert space bases that are not orthonormal, so no Hilbert bases.

In the finite dimensional case, the alternative to an orthonormal basis is a general, non- orthogonal basis. For each base in the finite fall along the two characteristic properties: A basis is a maximal linearly independent system, while a minimal generating system.

In the infinite-dimensional case, the " stable " Diverting from the concept of Hilbert basis is not so simple. Of special cases apart, it requires a base, however, that any vector of the Hilbert space clearly has certain coordinates which are constantly changing with the vector, and that each vector is determined by its coordinates uniquely, even more, that there is any system of allowed coordinates an ever- dependent on those same coordinate vector is. In other words, there should be in a coordinate space is a bijective, continuous in both directions linear mapping of Hilbert space.

• 2.3.1 Definition ( Rieszsystem )

• 2.4.1 Definition (frame)
• 2.4.2 Definition ( Rieszbasis )

• 3.1 Rieszsysteme 3.1.1 Pseudo Inverse and best approximation

• 3.2.1 pseudoinverse
• 3.2.2 Smallest coefficient vector
• 3.2.3 Dual Frame
• 3.2.4 Parseval frame

Motivation in the Euclidean case

In an n-dimensional K- vector space V is a basis is characterized in particular in that to her a bijective mapping between the vector space V and the vector space model can be generated:

.

This figure, in turn, encodes the base, because the images of the canonical basis vectors of the column vector space are just the chosen basis vectors of V. The inverse mapping to this assigns to each vector of V its coordinate vector with respect to this basis.

In this sense, one can identify bijective maps from to V with bases of V. Is on V is a standard defined, it follows from the bijectivity that the coordinates of the unit vectors neither very small can still be very large.

Systems of vectors and their properties

Be a Hilbert space over the field or. Further, let a (finite, countable or even uncountable ) subset of the Hilbert space. To linguistically, to distinguish the subset of vector subspaces, X is called a system of vectors.

Coefficient space

At any finite number of vectors from X one can form linear combinations without any restriction. The coefficients of such a linear combination can be summed up in a function that is different only at a finite number of points of zero. The linear combination then takes the form

In the space of these functions with finite support, one can define a scalar product as

A finite number of terms different from zero, ie, the sum is defined as such.

Each inner product defines a norm and thus a metric. Be with the completion of the space with respect to this topology called. to later serve in the following as coefficients and as a coordinate space. If X is finite, then these coefficients space is isomorphic to a Euclidean space, X is countable for the coefficient space is isometrically isomorphic to the sequence space.

For simplicity, elements are referred to as coefficients vectors of component c to the " Index" is the value of x c (x). A coefficient vector c is called finite if the carrier of c is finite.

Linear combinations

The simplest requirement is now that there might be a linear combination of the system X to each coefficient vector. In general, however, the " sum "

Undefined. For each coefficient, there are vectors with finite support and a distance for which this linear combination is defined. The question now is, when these finite linear combinations have a common limit for.

Definition (Bessel system )

X is Bessel system, if the image is continuous with, that is, if there is a constant B

Note: This inequality must be satisfied for finite coefficient sequences or functions with finite support to already apply coefficient sequences or functions.

Under these circumstances, the image vectors of a sequence of finite approximations of a coefficient vector form a Cauchy sequence in the Hilbert space. Thus, this sequence has a limit, and this is independent of the chosen approximating sequence.

Since a linear operator between two Hilbert spaces, there is an adjoint operator. After defining a adjoint operator is determined to this. If X is a Bessel system, then the adjoint operator satisfies a Bessel's inequality: The constant B> 0 holds for arbitrary vectors, the inequality

Linear Independence

In many cases, the definition is not sufficient to ensure that no non-trivial linear combination of X is the zero vector. So it may be the case in spite of this property, that there are arbitrarily small linear combinations in which the coefficient vector has length 1. It is so aggravating to demand that X is a Bessel system and it gives a lower bound A > 0 such that

Applies to all coefficient vectors.

Definition ( Rieszsystem )

A system of X vectors of a Hilbert space is Rieszsystem, if there is a finite constant, allowing for finite coefficient vectors and thus for all the inequalities coefficient vectors

Are fulfilled.

Generating system

A generating set X in the Hilbert space can be characterized in that the orthogonal complement of X consists only of the zero vector. X is additionally a Bessel system, the scalar products are the components of the vector. In other words, be any vector with has the zero vector.

Again, this characterization is not sufficient in many cases, since it is possible that takes on the unit sphere arbitrarily small values ​​. To prevent this, the proposal calls for the existence of a lower bound A> 0 for the values ​​on the unit sphere, except for those with the inequality

Met.

Definition (frame)

A system of vectors in X is a Hilbert space frame ( for frame En ) if there is a finite constant, the frame constants, such that, for each vector, the inequalities

Are fulfilled. Applies even so is called a tight frame X ( engl. "tight frame" ).

In particular this property implies the existence of a continuous pseudo-inverse operator ( see below).

Definition ( Rieszbasis )

A system of X vectors in a Hilbert space is Rieszbasis, if it is a Rieszsystem and a frame at the same time.

Conclusions

For Rieszsysteme

Pseudoinverse and best approximation

A Rieszsystem X spans a closed subspace in Hilbert space. There is a best approximation to any vector in this subspace, ie a coefficient vector for which the distance is minimal. This coefficient vector is determined as

The occurring in this expression inverse operator exists because the composite is limited, self-adjoint and positive definite. The inverse operator may be constructed as the Neumann series, because it is

Because the term

Has an operator norm less than 1

The operator must be the pseudo-inverse operator, there are the two identities

• Is the identity in the space of coefficient vectors and
• Is the orthogonal projector on the image.

For frames

Pseudoinverse

As a result of Frameungleichung the operator is surjective. For the orthogonal complement of the image is exactly the center of, and towards the left inequation, each vector in the core, the length of zero.

Analogous to the consideration for Rieszsystem is now the self-adjoint operator, bounded and positive definite. There is the inverse operator, with which, in turn, the pseudo- inverse operator can be formed. In this case, the identities hold

• Is the identity of the Hilbert space, and
• Is the projection onto the image of the adjoint operator, which is the orthogonal complement of the core at the same time.

Lowest coefficient vector

With a frame X, any vector X are represented as a linear combination of the system. There are often multiple coefficient vectors that satisfy this task. Among all these coefficient vectors is the smallest.

Dual frame

There is a frame X is a dual frame, where R is to the above-defined inverse operator. This system is actually a frame with constants, it is dual in the sense that the identity can be designed to

Ie the scalar products with the vectors of the dual frames to yield v. the components of the smallest coefficient vector

Parseval frame

A tight frame X, the frame constants are both equal to 1 is called Parsevalframe, because in him the Parseval equation

Applies. This is equivalent to saying that X is its own dual frame, ie, any vector can be developed as

It is the sentence: If the vectors of X Parsevalframes all unit vectors, then X is already a Hilbert basis.

For Rieszbasen

In a Rieszbasis the constants of inequality are consistent from the definition of Rieszsystems with the frame constants and the pseudo- inverse operator is actually already the inverse operator to.

If also A = B = 1, then X is already a complete orthonormal system, ie a Hilbert basis. In this case, both the Parseval equation

Leading to equivalent

Is; as well as

Equivalent to