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| In [[mathematics]], in the area of [[wavelet]] analysis, a '''refinable function''' is a function which fulfils some kind of self-similarity. A function <math>\varphi</math> is called refinable with respect to the mask <math>h</math> if
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| :<math>\varphi(x)=2\cdot\sum_{k=0}^{N-1} h_k\cdot\varphi(2\cdot x-k)</math>
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| This condition is called '''refinement equation''', '''dilation equation''' or '''two-scale equation'''.
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| Using the [[convolution]] (denoted by a star, *) of a function with a discrete mask and the dilation operator <math>D</math> you can write more concisely:
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| :<math>\varphi=2\cdot D_{1/2} (h * \varphi)</math>
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| It means that you obtain the function, again, if you convolve the function with a discrete mask and then scale it back.
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| There is an obvious similarity to [[iterated function systems]] and [[de Rham curve]]s.
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| The operator <math>\varphi\mapsto 2\cdot D_{1/2} (h * \varphi)</math> is linear.
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| A refinable function is an [[eigenfunction]] of that operator.
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| Its absolute value is not uniquely defined.
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| That is, if <math>\varphi</math> is a refinable function,
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| then for every <math>c</math> the function <math>c\cdot\varphi</math> is refinable, too.
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| These functions play a fundamental role in [[wavelet]] theory as [[Wavelet#Scaling_function|scaling functions]].
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| ==Properties==
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| ===Values at integral points===
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| A refinable function is defined only implicitly.
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| It may also be that there are several functions which are refinable with respect to the same mask.
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| If <math>\varphi</math> shall have finite support
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| and the function values at integer arguments are wanted,
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| then the two scale equation becomes a system of [[simultaneous linear equations]].
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| Let <math>a</math> be the minimum index and <math>b</math> be the maximum index
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| of non-zero elements of <math>h</math>, then one obtains
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| :<math>
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| \begin{pmatrix}
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| \varphi(a)\\
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| \varphi(a+1)\\
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| \vdots\\
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| \varphi(b)
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| \end{pmatrix}
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| =
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| \begin{pmatrix}
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| h_{a } & & & & & \\
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| h_{a+2} & h_{a+1} & h_{a } & & & \\
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| h_{a+4} & h_{a+3} & h_{a+2} & h_{a+1} & h_{a } & \\
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| \ddots & \ddots & \ddots & \ddots & \ddots & \ddots \\
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| & h_{b } & h_{b-1} & h_{b-2} & h_{b-3} & h_{b-4} \\
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| & & & h_{b } & h_{b-1} & h_{b-2} \\
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| & & & & & h_{b }
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| \end{pmatrix}
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| \cdot
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| \begin{pmatrix}
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| \varphi(a)\\
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| \varphi(a+1)\\
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| \vdots\\
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| \varphi(b)
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| \end{pmatrix}
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| </math>.
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| Using the [[Ideal sampler|discretization]]{{dn|date=June 2012}} operator, call it <math>Q</math> here, and the [[transfer matrix]] of <math>h</math>, named <math>T_h</math>, this can be written concisely as
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| :<math>Q\varphi = T_h \cdot Q\varphi</math>.
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| This is again a [[Fixed point (mathematics)|fixed-point equation]].
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| But this one can now be considered as an [[eigenvector]]-[[eigenvalue]] problem.
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| That is, a finitely supported refinable function exists only (but not necessarily),
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| if <math>T_h</math> has the eigenvalue 1.
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| ===Values at dyadic points===
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| From the values at integral points you can derive the values at dyadic points,
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| i.e. points of the form <math>k\cdot 2^{-j}</math>, with <math>k\in\mathbb{Z}</math> and <math>j\in\mathbb{N}</math>.
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| :<math>\varphi = D_{1/2} (2\cdot (h * \varphi))</math>
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| :<math>D_2 \varphi = 2\cdot (h * \varphi)</math>
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| :<math>Q(D_2 \varphi) = Q(2\cdot (h * \varphi)) = 2\cdot (h * Q\varphi)</math>
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| The star denotes the [[convolution]] of a discrete filter with a function.
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| With this step you can compute the values at points of the form <math>\frac{k}{2}</math>.
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| By replacing iteratedly <math>\varphi</math> by <math>D_2 \varphi</math> you get the values at all finer scales.
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| :<math>Q(D_{2^{j+1}}\varphi) = 2\cdot (h * Q(D_{2^j}\varphi))</math>
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| ===Convolution===
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| If <math>\varphi</math> is refinable with respect to <math>h</math>,
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| and <math>\psi</math> is refinable with respect to <math>g</math>,
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| then <math>\varphi*\psi</math> is refinable with respect to <math>h*g</math>.
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| ===Differentiation===
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| If <math>\varphi</math> is refinable with respect to <math>h</math>,
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| and the derivative <math>\varphi'</math> exists,
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| then <math>\varphi'</math> is refinable with respect to <math>2\cdot h</math>.
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| This can be interpreted as a special case of the convolution property,
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| where one of the convolution operands is a derivative of the [[Dirac delta function|Dirac impulse]].
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| ===Integration===
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| If <math>\varphi</math> is refinable with respect to <math>h</math>,
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| and there is an antiderivative <math>\Phi</math> with
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| <math>\Phi(t) = \int_0^{t}\varphi(\tau)\mathrm{d}\tau</math>,
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| then the antiderivative <math>t \mapsto \Phi(t) + c</math>
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| is refinable with respect to mask <math>\frac{1}{2}\cdot h</math>
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| where the constant <math>c</math> must fulfill
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| <math>c\cdot(1 - \sum_j h_j) = \sum_j h_j \cdot \Phi(-j)</math>.
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| If <math>\varphi</math> has [[compact support|bounded support]],
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| then we can interpret integration as convolution with the [[Heaviside function]] and apply the convolution law.
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| ===Scalar products===
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| Computing the scalar products of two refinable functions and their translates can be broken down to the two above properties.
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| Let <math>T</math> be the translation operator. It holds
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| :<math>\langle \varphi, T_k \psi\rangle = \langle \varphi * \psi^*, T_k\delta\rangle = (\varphi*\psi^*)(k)</math>
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| where <math>\psi^*</math> is the [[adjoint filter|adjoint]] of <math>\psi</math> with respect to [[convolution]],
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| i.e. <math>\psi^*</math> is the flipped and [[complex conjugate]]d version of <math>\psi</math>,
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| i.e. <math>\psi^*(t) = \overline{\psi(-t)}</math>.
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| Because of the above property, <math>\varphi*\psi^*</math> is refinable with respect to <math>h*g^*</math>,
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| and its values at integral arguments can be computed as eigenvectors of the transfer matrix.
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| This idea can be easily generalized to integrals of products of more than two refinable functions.<ref>{{Cite journal
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| | last1 = Dahmen | first1 = Wolfgang
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| | last2 = Micchelli | first2 = Charles A.
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| | title = Using the refinement equation for evaluating integrals of wavelets
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| | journal = Journal Numerical Analysis
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| | volume = 30
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| | pages = 507–537
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| | publisher = SIAM
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| | year = 1993
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| }}
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| </ref>
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| ===Smoothness===
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| A refinable function usually has a fractal shape.
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| The design of continuous or smooth refinable functions is not obvious.
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| Before dealing with forcing smoothness it is necessary to measure smoothness of refinable functions.
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| Using the Villemoes machine<ref>
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| {{Cite web
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| | last = Villemoes
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| | first = Lars
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| | title = Sobolev regularity of wavelets and stability of iterated filter banks
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| | url = http://www.math.kth.se/~larsv/paper3.ps.Z
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| | format = PostScript
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| | accessdate = 2006}}
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| </ref>
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| one can compute the smoothness of refinable functions in terms of [[Sobolev space|Sobolev exponents]].
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| In a first step the refinement mask <math>h</math> is divided into a filter <math>b</math>, which is a power of the smoothness factor <math>(1,1)</math> (this is a binomial mask) and a rest <math>q</math>.
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| Roughly spoken, the binomial mask <math>b</math> makes smoothness and
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| <math>q</math> represents a fractal component, which reduces smoothness again.
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| Now the Sobolev exponent is roughly
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| the order of <math>b</math> minus [[logarithm]] of the [[spectral radius]] of <math>T_{q*q^*}</math>.
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| ==Generalization==
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| The concept of refinable functions can be generalized to functions of more than one variable,
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| that is functions from <math>\R^d \to \R</math>.
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| The most simple generalization is about [[tensor product]]s.
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| If <math>\varphi</math> and <math>\psi</math> are refinable with respect to <math>h</math> and <math>g</math>, respectively, then <math>\varphi\otimes\psi</math>
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| is refinable with respect to <math>h\otimes g</math>.
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| The scheme can be generalized even more to different scaling factors with respect to different dimensions or even to mixing data between dimensions.<ref>
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| {{Citation
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| | last1 = Berger | first = Marc A.
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| | last2 = Wang | first2 = Yang
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| | chapter = Multidimensional two-scale dilation equations (chapter IV)
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| | publisher = Academic Press, Inc.
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| | year = 1992
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| | volume = 2
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| | series = Wavelet Analysis and its Applications
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| | booktitle = Wavelets: A Tutorial in Theory and Applications
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| | editor-last = Chui | editor-first = Charles K.
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| | pages = 295–323 }}
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| </ref>
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| Instead of scaling by scalar factor like 2 the signal the coordinates are transformed by a matrix <math>M</math> of integers.
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| In order to let the scheme work, the absolute values of all eigenvalues of <math>M</math> must be larger than one.
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| (Maybe it also suffices that <math>|\det M|>1</math>.)
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| Formally the two-scale equation does not change very much:
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| :<math>\varphi(x)=|\det M|\cdot\sum_{k\in\Z^d} h_k\cdot\varphi(M\cdot x-k)</math>
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| :<math>\varphi=|\det M|\cdot D_{M^{-1}} (h * \varphi)</math> | |
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| ==Examples==
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| * If the definition is extended to [[Distribution_(mathematics)|distributions]], then the [[Dirac delta function|Dirac impulse]] is refinable with respect to the unit vector <math>\delta</math>, that is known as [[Kronecker delta]]. The <math>n</math>-th derivative of the Dirac distribution is refinable with respect to <math>2^{n}\cdot\delta</math>.
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| * The [[Heaviside function]] is refinable with respect to <math>\frac{1}{2}\cdot\delta</math>.
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| * The [[truncated power function]]s with exponent <math>n</math> are refinable with respect to <math>\frac{1}{2^{n+1}}\cdot\delta</math>.
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| * The [[triangular function]] is a refinable function.<ref>
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| {{Cite web
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| | last = Nathanael
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| | first = Berglund
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| | title = Reconstructing Refinable Functions
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| | url = http://www.math.gatech.edu/~berglund/Refinable/index.html
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| | accessdate = 2010-12-24}}
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| </ref> [[B-Spline]] functions with successive integral nodes are refinable, because of the convolution theorem and the refinability of the [[indicator function|characteristic function]] for the interval <math>[0,1)</math> (a [[boxcar function]]).
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| * All [[polynomial function]]s are refinable. For every refinement mask there is a polynomial that is uniquely defined up to a constant factor. For every polynomial of degree <math>n</math> there are many refinement masks that all differ by a mask of type <math>v * (1,-1)^{n+1}</math> for any mask <math>v</math> and the convolutional power <math>(1,-1)^{n+1}</math>.<ref>
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| {{cite arxiv
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| | last = Thielemann | first = Henning
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| | title = How to refine polynomial functions
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| | date = 2012-01-29
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| | eprint = 1012.2453
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| }}</ref>
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| * A [[rational function]] <math>\varphi</math> is refinable if and only if it can be represented using [[partial fraction]]s as <math>\varphi(x) = \sum_{i\in\mathbb{Z}} \frac{s_i}{(x-i)^k}</math>, where <math>k</math> is a [[positive number|positive]] [[natural number]] and <math>s</math> is a real sequence with finitely many non-zero elements (a [[Laurent polynomial]]) such that <math>s | (s \uparrow 2)</math> (read: <math>\exists h(z)\in\mathbb{R}[z,z^{-1}]\ h(z)\cdot s(z) = s(z^2)</math>). The Laurent polynomial <math>2^{k-1}\cdot h</math> is the associated refinement mask.<ref>
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| {{Citation
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| | last1 = Gustafson | first1 = Paul
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| | last2 = Savir | first2 = Nathan
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| | last3 = Spears | first3 = Ely
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| | title = A Characterization of Refinable Rational Functions
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| | journal = American Journal of Undergraduate Research
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| | volume = 5
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| | issue = 3
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| | pages = 11–20
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| | date = 2006-11-14
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| | month = November
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| | url = http://www.uni.edu/ajur/v5n3/Gufstafson%20et%20al%20new%20pp%2011-20.pdf}}
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| </ref>
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| == References ==
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| {{reflist}}
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| ==See also==
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| * [[Subdivision surface|Subdivision scheme]]
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| [[Category:Wavelets]]
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