Orthogonal system

An orthogonal system of vectors is a set $ \{ x _ \alpha \} $ of non-zero vectors of a Euclidean (Hilbert) space with a scalar product $ ( \cdot , \cdot ) $ such that $ ( x _ \alpha , x _ \beta ) = 0 $ when $ \alpha \neq \beta $. If under these conditions the norm of each vector is equal to one, then $ \{ x _ \alpha \} $ is said to be an orthonormal system. A complete orthogonal (orthonormal) system of vectors $ \{ x _ \alpha \} $ is called an orthogonal (orthonormal) basis.

M.I. Voitsekhovskii

An orthogonal coordinate system is a coordinate system in which the coordinate lines (or surfaces) intersect at right angles. Orthogonal coordinate systems exist in any Euclidean space, but, generally speaking, do not exist in an arbitrary space. In a two-dimensional smooth affine space, orthogonal systems can always be introduced at least in a sufficiently small neighbourhood of every point. It is sometimes possible to introduce orthogonal coordinate systems in the large. In an orthogonal system, the metric tensor $ g _ {ij} $ is diagonal; the diagonal components $ g _ {ij} $ are called Lamé coefficients. The Lamé coefficients of an orthogonal system in space are expressed by the formulas

$$ L _ {u} = \sqrt {\left ( \frac{\partial x }{\partial u } \right ) ^ {2} + \left ( \frac{\partial y }{\partial u } \right ) ^ {2} + \left ( \frac{\partial z }{\partial u } \right ) ^ {2} } , $$

$$ L _ {v} = \sqrt {\left ( \frac{\partial x }{\partial v } \right ) ^ {2} + \left ( \frac{\partial y }{\partial v } \right ) ^ {2} + \left ( \frac{\partial z }{\partial v } \right ) ^ {2} } , $$

$$ L _ {w} = \sqrt {\left ( \frac{\partial x }{\partial w } \right ) ^ {2} + \left ( \frac{\partial y }{\partial w } \right ) ^ {2} + \left ( \frac{\partial z }{\partial w } \right ) ^ {2} } , $$

where $ x $, $ y $ and $ z $ are Cartesian coordinates. The Lamé coefficients are also used to express the line element:

$$ ds = \sqrt {L _ {u} ^ {2} du ^ {2} + L _ {v} ^ {2} dv ^ {2} + L _ {w} ^ {2} dw ^ {2} } , $$

the element of surface area:

$$ d \sigma = \sqrt {( L _ {u} L _ {v} du dv) ^ {2} + ( L _ {u} L _ {w} du dw) ^ {2} + ( L _ {v} L _ {w} dv dw) ^ {2} } , $$

the volume element:

$$ dV = L _ {u} L _ {v} L _ {w} du dv dw, $$

and the operations of vector analysis:

$$ \mathop{\rm grad} _ {u} \phi = \frac{1}{L _ {u} } \frac{\partial \phi }{\partial u } ,\ \ \mathop{\rm grad} _ {v} \phi = \frac{1}{L _ {v} } \frac{\partial \phi }{\partial v } , $$

$$ \mathop{\rm grad} _ {w} \phi = \frac{1}{L _ {w} } \frac{\partial \phi }{\partial w } , $$

$$ \mathop{\rm div} \mathbf a = \frac{1}{L _ {u} L _ {v} L _ {w} } \left [ \frac \partial {\partial u } ( a _ {u} L _ {v} L _ {w} ) + \frac \partial {\partial v } ( a _ {v} L _ {u} L _ {w} ) \right . + $$

$$ + \left . \frac \partial {\partial w } ( a _ {w} L _ {u} L _ {v} ) \right ] ; $$

$$ \mathop{\rm rot} _ {u} \mathbf a = \frac{1}{L _ {v} L _ {w} } \left [ \frac \partial {\partial v } ( a _ {w} L _ {w} ) - \frac \partial {\partial w } ( a _ {v} L _ {v} ) \right ] , $$

$$ \mathop{\rm rot} _ {v} \mathbf a = \frac{1}{L _ {u} L _ {w} } \left [ \frac \partial {\partial w } ( a _ {u} L _ {u} ) - \frac \partial {\partial u } ( a _ {w} L _ {w} ) \right ] , $$

$$ \mathop{\rm rot} _ {w} \mathbf a = \frac{1}{L _ {u} L _ {v} } \left [ \frac \partial {\partial u } ( a _ {v} L _ {v} ) - \frac \partial {\partial v } ( a _ {u} L _ {u} ) \right ] , $$

$$ \Delta \phi = \frac{1}{L _ {u} L _ {v} L _ {w} } \left [ \frac \partial { \partial u } \left ( \frac{L _ {v} L _ {w} }{L _ {u} } \frac{\partial \phi }{\partial u } \right ) + \frac \partial {\partial v } \left ( \frac{L _ {u} L _ {w} }{L _ {v} } \frac{\partial \phi }{\partial v } \right ) \right . + $$

$$ + \left . \frac \partial {\partial w } \left ( \frac{L _ {u} L _ {v} }{L _ {w} } \frac{\partial \phi }{\partial w } \right ) \right ] . $$

The most frequently used orthogonal coordinate systems are: on a plane — Cartesian coordinates; elliptic coordinates; parabolic coordinates; and polar coordinates; in space — cylinder coordinates; bicylindrical coordinates; bipolar coordinates; paraboloidal coordinates; and spherical coordinates.

D.D. Sokolov

An orthogonal system of functions is a finite or countable system of functions $ \{ \phi _ {i} \} $ belonging to a space $ L _ {2} ( X, S, \mu ) $ and satisfying the condition

$$ \int\limits _ { X } \phi _ {i} ( x) \overline \phi _ {j} ( x) d \mu ( x) = \ \left \{ \begin{array}{l} 0 \\ \lambda _ {i} > 0 \end{array} \ \begin{array}{l} \textrm{ if } i \neq j, \\ \textrm{ if } i = j . \end{array} \right .$$

If $ \lambda _ {i} = 1 $ for all $ i $, then the system is orthonormal. It is supposed that the measure $ \mu $ defined on the $ \sigma $-algebra $ S $ of subsets of the set $ X $ is countably additive, complete, and has a countable base. This definition encompasses all orthogonal systems studied in analysis. Such systems are obtained for various concrete realizations of the measure space $ ( X, S, \mu ) $.

The greatest interest is in complete orthonormal systems $ \{ \phi _ {n} \} $, which possess the property that for any function $ f \in L _ {2} ( X, S, \mu ) $ there is a unique series $ \sum c _ {n} \phi _ {n} $ which converges to $ f $ in the metric of the space $ L _ {2} ( X, S, \mu ) $. The coefficients $ c _ {n} $ are defined by the Fourier formula:

$$ c _ {n} = \int\limits _ { X } f \overline \phi _ {n} d \mu . $$

These systems exist by virtue of the separability of the space $ L _ {2} ( X, S, \mu ) $. A universal method of constructing complete orthonormal systems is given by the Gram–Schmidt orthogonalization method. This method can be applied to any complete linearly independent sequence $ \{ f _ {n} \} $ of functions in $ L _ {2} ( X, S, \mu ) $.

Important examples of orthogonal series are obtained by considering the space $ L _ {2} [ a, b] $ (in this case, $ X = [ a, b] $, $ S $ is the system of Lebesgue-measurable sets and $ \mu $ is the Lebesgue measure). Many theorems on the convergence or summability of a series $ \sum a _ {n} \phi _ {n} $ with respect to a general orthogonal system $ \{ \phi _ {n} \} $ in the space $ L _ {2} [ a, b] $ are also valid for series with respect to orthonormal systems in the space $ L _ {2} ( X, S, \mu ) $. Moreover, in this particular case, interesting concrete orthogonal systems have been constructed which possess some nice properties. These systems include the Haar, Rademacher, Walsh–Paley, and Franklin systems.

a) The Haar system $ \{ \chi _ {n} \} _ {n=1} ^ \infty $: $ \chi _ {1} ( x) = 1 $, $ x \in [ 0, 1] $,

$$ \chi _ {m} ( x) = \left \{ \begin{array}{ll} \sqrt {2 ^ {n} } & \textrm{ if } x \in \left ( 2k- \frac{2}{2 ^ {n+1} } , 2k- \frac{1}{2 ^ {n+1} } \right ) , \\ - \sqrt {2 ^ {n} } & \textrm{ if } x \in \left ( 2k- \frac{1}{2 ^ {n+1} } ,\ \frac{2k}{2 ^ {n+1} } \right ) , \\ 0 & \textrm{ at the remaining points of } \ [ 0, 1], \\ \end{array} \right. $$

where $ m = 2 ^ {n} + k $, $ 1 \leq k \leq 2 ^ {n} $, $ m = 2, 3 ,\dots $. Series with respect to the Haar system are typical examples of martingales (cf. Martingale) and thus the general theorems of martingale theory are also correct for them. Moreover, the system $ \{ \chi _ {n} \} _ {n=1} ^ \infty $ is a basis in $ L _ {p} [ 0, 1] $, $ p \geq 1 $, and the Fourier series with respect to the Haar system of any integrable function converges almost-everywhere.

b) The Rademacher system $ \{ r _ {n} \} _ {n=0} ^ \infty $:

$$ r _ {n} ( x) = \mathop{\rm sign} \sin 2 ^ {n+1} \pi x,\ x \in [ 0, 1], $$

is an important example of a stochastically-independent orthogonal system of functions and is used both in probability theory and in the theory of orthogonal and general series of functions.

c) The Walsh–Paley system $ \{ W _ {n} \} _ {n=0} ^ \infty $ is defined using the Rademacher functions:

$$ W _ {0} ( x) = 1,\ \ W _ {n} ( x) = \prod _ { k= 0} ^ { m } [ r _ {k} ( x)] ^ {q _ {k} } ,\ \ x \in [ 0, 1], $$

where the numbers $ m $ and $ q _ {k} $ are defined using the binary expansion of the number $ n $:

$$ n = \sum _ { k= 0} ^ { m } q _ {k} 2 ^ {k} . $$

d) The Franklin system $ \{ \Phi _ {n} ( x) \} _ {n=1} ^ \infty $ is obtained by Gram–Schmidt orthogonalization of the sequence of functions

$$ u _ {1} ( x) = x,\ \ u _ {2} ( x) = 1- x, $$

$$ u _ {n} ( x) = \int\limits _ { 0 } ^ { x } \chi _ {n-1} ( t) dt,\ n \geq 3 ,\ x \in [ 0, 1]. $$

It is an example of an orthogonal basis of the space $ C[ 0, 1] $ of continuous functions.

In the theory of multiple orthogonal series, function systems of the form

$$ \phi _ {n _ {1} } ( x _ {1} ), \dots, \phi _ {n _ {m} } ( x _ {m} ) , $$

$$ x _ {i} \in [ a, b],\; n _ {i} = 1, 2, \dots \; 1 \leq i \leq m, $$

are examined, where $ \{ \phi _ {n} \} _ {n=1} ^ \infty $ is an orthonormal system in $ L _ {2} [ a, b] $. These systems are orthonormal on the $ m $-dimensional cube $ J _ {m} = [ a, b] \times \dots \times [ a, b] $, and are complete if the system $ \{ \phi _ {n} \} $ is complete.

References

[1]	S. Kaczmarz, H. Steinhaus, "Theorie der Orthogonalreihen" , Chelsea, reprint (1951)
[2]	B.I. Golubov, "Series with respect to the Haar system" J. Soviet Math. , 1 : 6 (1973) pp. 704–726 Itogi Nauk. Mat. Anal. 1970 (1971) pp. 109–146
[3]	L.A. Balashov, A.I. Rubenshtein, "Series with respect to the Walsh system and their generalizations" J. Soviet Math. , 1 : 6 (1973) pp. 727–763 Itogi Nauk. Mat. Anal. 1970 (1971) pp. 147–202
[4]	J.L. Doob, "Stochastic processes" , Chapman & Hall (1953)
[5]	M. Loève, "Probability theory" , Springer (1977)
[6]	A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988)

A.A. Talalyan

Comments

A complete system of elements $ \{ \phi _ \alpha \} $ in a Hilbert space, or, more generally, an inner product space $ V $, is a set of elements such that for any $ \phi \in V $, if $ \langle \phi , \phi _ \alpha \rangle = 0 $ for all $ \alpha $, then $ \phi = 0 $.

Cf. also Complete system of functions. The Walsh–Paley system is a complete orthonormal system in $ L _ {2} ( 0, 1) $.