
Frame of a vector space
Encyclopedia
In linear algebra
, a frame of a vector space V with an inner product can be seen as a generalization of the idea of a basis
to sets which may be linearly dependent. The key issue related to the construction of a frame appears when we have a sequence of vectors
, with each
and we want to express an arbitrary element
as a linear combination of the vectors
:

and want to determine the coefficients
. If the set
does not span
, then these coefficients cannot be determined for all such
. If
spans
and also is linearly independent, this set forms a basis
of
, and the coefficients
are uniquely determined by
: they are the coordinates of
relative to this basis. If, however,
spans
but is not linearly independent, the question of how to determine the coefficients becomes less apparent, in particular if
is of infinite dimension.
Given that
spans
and is linearly dependent, it may appear obvious that we should remove vectors from the set until it becomes linearly independent and forms a basis. There are some problems with this strategy:
In 1952, Duffin and Schaeffer gave a solution to this problem, by describing a condition on the set
that makes it possible to compute the coefficients
in a simple way. More precisely, a frame is a set
of elements of V which satisfy the so-called frame condition:
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
, a frame of a vector space V with an inner product can be seen as a generalization of the idea of a basis
Basis (linear algebra)
In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...
to sets which may be linearly dependent. The key issue related to the construction of a frame appears when we have a sequence of vectors





and want to determine the coefficients






Basis
Basis may refer to* Cost basis, in income tax law, the original cost of property adjusted for factors such as depreciation.* Basis of futures, the value differential between a future and the spot price...
of







Given that


- By removing vectors randomly from the set, it may lose its possibility to span
before it becomes linearly independent.
- Even if it is possible to devise a specific way to remove vectors from the set until it becomes a basis, this approach may become infeasible in practice if the set is large or infinite.
- In some applications, it may be an advantage to use more vectors than necessary to represent
. This means that we want to find the coefficients
without removing elements in
.
In 1952, Duffin and Schaeffer gave a solution to this problem, by describing a condition on the set



- There exist two real numbers, A and B such that
and
.
- This means that the constants A and B can be chosen independently of v: they only depend on the set
.
The numbers A and B are called lower and upper frame bounds.
It can be shown that the frame condition is both necessary and sufficient to form a frame a set of dual frame vectorswith the following property:
for any. This implies that a frame together with its dual frame has the same properties as a basis and its dual basis in terms of reconstructing a vector from scalar products.
Relation to bases
If the setis a frame of V, it spans V. Otherwise there would exist at least one non-zero
which would be orthogonal to all
. If we insert
into the frame condition, we obtain
therefore, which is a violation of the initial assumptions on the lower frame bound.
If a set of vectors spans V, this is not a sufficient condition for calling the set a frame. As an example, considerand the infinite set
given by
This set spans V but sincewe cannot choose
. Consequently, the set
is not a frame.
Tight frames
A frame is tight if the frame boundsand
are equal. This means that the frame obeys a generalized Parseval's identity
Parseval's identityIn mathematical analysis, Parseval's identity is a fundamental result on the summability of the Fourier series of a function. Geometrically, it is thePythagorean theorem for inner-product spaces....
. If, then a frame is either called normalized or Parseval. However, some of the literature refers to a frame for which
where
is a constant independent of
(see uniform below) as a normalized frame.
Uniform frames
A frame is uniform if each element has the same norm:where
is a constant independent of
.
A uniform normalized tight frame withis an orthonormal basis
Orthonormal basisIn mathematics, particularly linear algebra, an orthonormal basis for inner product space V with finite dimension is a basis for V whose vectors are orthonormal. For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of...
.
The dual frame
The frame condition is both sufficient and necessary for allowing the construction of a dual or conjugate frame,, relative the original frame,
. The duality of this frame implies that
is satisfied for all. In order to construct the dual frame, we first need the linear mapping:
defined as
From this definition ofand linearity in the first argument of the inner product, it now follows that
which can be inserted into the frame condition to get
The properties ofcan be summarised as follows:
-
is self-adjoint
Self-adjointIn mathematics, an element x of a star-algebra is self-adjoint if x^*=x.A collection C of elements of a star-algebra is self-adjoint if it is closed under the involution operation...
, positive definite, and has positive upper and lower bounds. This leads to - the inverse
of
exists and it, too, is self-adjoint, positive definite, and has positive upper and lower bounds.
The dual frame is defined by mapping each element of the frame with:
To see that this make sense, letbe arbitrary and set
It is then the case that
which proves that
Alternatively, we can set
By inserting the above definition ofand applying known properties of
and its inverse, we get
which shows that
This derivation of the dual frame is a summary of section 3 in the article by Duffin and Schaeffer. They use the term conjugate frame for what here is called dual frame.
History
Frames were introduced by Duffin and Schaeffer in their study on nonharmonic Fourier series. They remained obscure until MallatStéphane MallatStéphane G. Mallat made some fundamental contributions to the development of wavelet theory in the late 1980s and early 1990s...
, DaubechiesIngrid DaubechiesIngrid Daubechies is a Belgian physicist and mathematician. She was between 2004 and 2011 the William R. Kenan Jr. Professor in the mathematics and applied mathematics departments at Princeton University. In January 2011 she moved to Duke University as a Professor in mathematics. She is the first...
, and others used them to analyze wavelets in the 1980s. Some practical uses of frames today include robust codingError detection and correctionIn information theory and coding theory with applications in computer science and telecommunication, error detection and correction or error control are techniques that enable reliable delivery of digital data over unreliable communication channels...
and design and analysis of filter bankFilter bankIn signal processing, a filter bank is an array of band-pass filters that separates the input signal into multiple components, each one carrying a single frequency subband of the original signal. One application of a filter bank is a graphic equalizer, which can attenuate the components...
s.- This means that the constants A and B can be chosen independently of v: they only depend on the set