Namespaces
Variants
Actions

Difference between revisions of "Fourier series in orthogonal polynomials"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
Line 1: Line 1:
 +
{{TEX|done}}
 
A series of the form
 
A series of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f0411001.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$\sum_{n=0}^\infty a_nP_n\tag{1}$$
  
where the polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f0411002.png" /> are orthonormal on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f0411003.png" /> with weight function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f0411004.png" /> (see [[Orthogonal polynomials|Orthogonal polynomials]]) and the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f0411005.png" /> are calculated from the formula
+
where the polynomials $\{P_n\}$ are orthonormal on an interval $(a,b)$ with weight function $h$ (see [[Orthogonal polynomials|Orthogonal polynomials]]) and the coefficients $\{a_n\}$ are calculated from the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f0411006.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$a_n=\int\limits_a^bh(x)f(x)P_n(x)dx.\tag{2}$$
  
Here, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f0411007.png" /> belongs to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f0411008.png" /> of functions that are square summable (Lebesgue integrable) with weight function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f0411009.png" /> over the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110010.png" /> of orthogonality.
+
Here, the function $f$ belongs to the class $L_2=L_2[(a,b),h]$ of functions that are square summable (Lebesgue integrable) with weight function $h$ over the interval $(a,b)$ of orthogonality.
  
As for any orthogonal series, the partial sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110011.png" /> of (1) are the best-possible approximations to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110012.png" /> in the metric of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110013.png" /> and satisfy the condition
+
As for any orthogonal series, the partial sums $\{s_n(x,f)\}$ of \ref{1} are the best-possible approximations to $f$ in the metric of $L_2$ and satisfy the condition
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$\lim_{n\to\infty}a_n=0.\tag{3}$$
  
For a proof of the convergence of the series (1) at a single point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110015.png" /> or on a certain set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110016.png" /> one usually applies the equality
+
For a proof of the convergence of the series \ref{1} at a single point $x$ or on a certain set in $(a,b)$ one usually applies the equality
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110017.png" /></td> </tr></table>
+
$$f(x)-s_n(x,f)=\mu_n[a_n(\phi_x)P_{n+1}-a_{n+1}(\phi_x)P_n(x)],$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110018.png" /> are the Fourier coefficients of an auxiliary function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110019.png" />, given by
+
where $\{a_n(\phi_x)\}$ are the Fourier coefficients of an auxiliary function $\phi_x$, given by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110020.png" /></td> </tr></table>
+
$$phi_x(t)=\frac{f(x)-f(t)}{x-t},\quad t\in(a,b),$$
  
for fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110021.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110022.png" /> is the coefficient given by the [[Christoffel–Darboux formula|Christoffel–Darboux formula]]. If the interval of orthogonality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110023.png" /> is bounded, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110024.png" /> and if the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110025.png" /> is bounded at the given point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110026.png" />, then the series (1) converges to the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110027.png" />.
+
for fixed $x$, and $\mu_n$ is the coefficient given by the [[Christoffel–Darboux formula|Christoffel–Darboux formula]]. If the interval of orthogonality $[a,b]$ is bounded, if $\phi_x\in L_2$ and if the sequence $\{P_n\}$ is bounded at the given point $x$, then the series \ref{1} converges to the value $f(x)$.
  
The coefficients (2) can also be defined for a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110028.png" /> in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110029.png" />, that is, for functions that are summable with weight function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110030.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110031.png" />. For a bounded interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110032.png" />, condition (3) holds if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110033.png" /> and if the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110034.png" /> is uniformly bounded on the whole interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110035.png" />. Under these conditions the series (1) converges at a certain point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110036.png" /> to the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110037.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110038.png" />.
+
The coefficients \ref{2} can also be defined for a function $f$ in the class $L_1=L_1[(a,b),h]$, that is, for functions that are summable with weight function $h$ over $(a,b)$. For a bounded interval $[a,b]$, condition \ref{3} holds if $f\in L_1[(a,b),h]$ and if the sequence $\{P_n\}$ is uniformly bounded on the whole interval $[a,b]$. Under these conditions the series \ref{1} converges at a certain point $x\in[a,b]$ to the value $f(x)$ if $\phi_x\in L_1[(a,b),h]$.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110039.png" /> be a part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110040.png" /> on which the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110041.png" /> is uniformly bounded, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110042.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110043.png" /> be the class of functions that are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110044.png" />-summable over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110045.png" /> with weight function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110046.png" />. If, for a fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110047.png" />, one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110049.png" />, then the series (1) converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110050.png" />.
+
Let $A$ be a part of $(a,b)$ on which the sequence $\{P_n\}$ is uniformly bounded, let $B=[a,b]\setminus A$ and let $L_p(A)=L_p[A,h]$ be the class of functions that are $p$-summable over $A$ with weight function $h$. If, for a fixed $x\in A$, one has $\phi_x\in L_1(A)$ and $\phi_x\in L_2(B)$, then the series \ref{1} converges to $f(x)$.
  
For the series (1) the localization principle for conditions of convergence holds: If two functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110052.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110053.png" /> coincide in an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110054.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110055.png" />, then the Fourier series of these two functions in the orthogonal polynomials converge or diverge simultaneously at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110056.png" />. An analogous assertion is valid if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110058.png" /> belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110059.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110061.png" />.
+
For the series \ref{1} the localization principle for conditions of convergence holds: If two functions $f$ and $g$ in $L_2$ coincide in an interval $(x-\delta,x+\delta)$, where $x\in A$, then the Fourier series of these two functions in the orthogonal polynomials converge or diverge simultaneously at $x$. An analogous assertion is valid if $f$ and $g$ belong to $L_1(A)$ and $L_2(B)$ and $x\in A$.
  
For the classical orthogonal polynomials the theorems on the equiconvergence with a certain associated trigonometric Fourier series hold for the series (1) (see [[Equiconvergent series|Equiconvergent series]]).
+
For the classical orthogonal polynomials the theorems on the equiconvergence with a certain associated trigonometric Fourier series hold for the series \ref{1} (see [[Equiconvergent series|Equiconvergent series]]).
  
Uniform convergence of the series (1) over the whole bounded interval of orthogonality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110062.png" />, or over part of it, is usually investigated using the Lebesgue inequality
+
Uniform convergence of the series \ref{1} over the whole bounded interval of orthogonality $[a,b]$, or over part of it, is usually investigated using the Lebesgue inequality
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110063.png" /></td> </tr></table>
+
$$\left|f(x)-\sum_{k=0}^na_kP_k(x)\right|\leq[1+L_n(x)]E_n(f),\quad x\in[a,b],$$
  
 
where the Lebesgue function
 
where the Lebesgue function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110064.png" /></td> </tr></table>
+
$$L_n(x)=\int\limits_a^bh(t)\left|\sum_{k=0}^nP_k(x)P_k(t)\right|dt$$
  
does not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110065.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110066.png" /> is the best uniform approximation (cf. [[Best approximation|Best approximation]]) to the continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110067.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110068.png" /> by polynomials of degree not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110069.png" />. The sequence of Lebesgue functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110070.png" /> can grow at various rates at the various points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110071.png" />, depending on the properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110072.png" />. However, for the whole interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110073.png" /> one introduces the Lebesgue constants
+
does not depend on $f$ and $E_n(f)$ is the best uniform approximation (cf. [[Best approximation|Best approximation]]) to the continuous function $f$ on $[a,b]$ by polynomials of degree not exceeding $n$. The sequence of Lebesgue functions $\{L_n\}$ can grow at various rates at the various points of $[a,b]$, depending on the properties of $h$. However, for the whole interval $[a,b]$ one introduces the Lebesgue constants
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110074.png" /></td> </tr></table>
+
$$L_n=\max_{x\in[a,b]}L_n(x),$$
  
which increase unboundedly as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110075.png" /> (for different systems of orthogonal polynomials the Lebesgue constants can increase at different rates). The Lebesgue inequality implies that if the condition
+
which increase unboundedly as $n\to\infty$ (for different systems of orthogonal polynomials the Lebesgue constants can increase at different rates). The Lebesgue inequality implies that if the condition
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110076.png" /></td> </tr></table>
+
$$\lim_{n\to\infty}L_nE_n(f)=0$$
  
is satisfied, then the series (1) converges uniformly to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110077.png" /> on the whole interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110078.png" />. On the other hand, the rate at which the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110079.png" /> tends to zero depends on the differentiability properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110080.png" />. Thus, in many cases it is not difficult to formulate sufficient conditions for the right-hand side of the Lebesgue inequality to tend to zero as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110081.png" /> (see, for example, [[Legendre polynomials|Legendre polynomials]]; [[Chebyshev polynomials|Chebyshev polynomials]]; [[Jacobi polynomials|Jacobi polynomials]]). In the general case of an arbitrary weight function one can obtain specific results if one knows asymptotic formulas or bounds for the orthogonal polynomials under consideration.
+
is satisfied, then the series \ref{1} converges uniformly to $f$ on the whole interval $[a,b]$. On the other hand, the rate at which the sequence $\{E_n(f)\}$ tends to zero depends on the differentiability properties of $f$. Thus, in many cases it is not difficult to formulate sufficient conditions for the right-hand side of the Lebesgue inequality to tend to zero as $n\to\infty$ (see, for example, [[Legendre polynomials|Legendre polynomials]]; [[Chebyshev polynomials|Chebyshev polynomials]]; [[Jacobi polynomials|Jacobi polynomials]]). In the general case of an arbitrary weight function one can obtain specific results if one knows asymptotic formulas or bounds for the orthogonal polynomials under consideration.
  
 
====References====
 
====References====
Line 57: Line 58:
  
 
====Comments====
 
====Comments====
See also [[#References|[a1]]], Chapt. 4 and [[#References|[a2]]], part one. Equiconvergence theorems have been proved more generally for the case of orthogonal polynomials with respect to a weight function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110082.png" /> on a finite interval belonging to the Szegö class, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041100/f04110083.png" />, cf. [[#References|[a2]]], Sect. 4.12. For Fourier series in orthogonal polynomials with respect to a weight function on an unbounded interval see [[#References|[a2]]], part two.
+
See also [[#References|[a1]]], Chapt. 4 and [[#References|[a2]]], part one. Equiconvergence theorems have been proved more generally for the case of orthogonal polynomials with respect to a weight function $h$ on a finite interval belonging to the Szegö class, i.e. $\log h\in L$, cf. [[#References|[a2]]], Sect. 4.12. For Fourier series in orthogonal polynomials with respect to a weight function on an unbounded interval see [[#References|[a2]]], part two.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Freud,  "Orthogonal polynomials" , Pergamon  (1971)  (Translated from German)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Nevai,  G. Freud,  "Orthogonal polynomials and Christoffel functions (A case study)"  ''J. Approx. Theory'' , '''48'''  (1986)  pp. 3–167</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Freud,  "Orthogonal polynomials" , Pergamon  (1971)  (Translated from German)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Nevai,  G. Freud,  "Orthogonal polynomials and Christoffel functions (A case study)"  ''J. Approx. Theory'' , '''48'''  (1986)  pp. 3–167</TD></TR></table>

Revision as of 11:33, 21 November 2018

A series of the form

$$\sum_{n=0}^\infty a_nP_n\tag{1}$$

where the polynomials $\{P_n\}$ are orthonormal on an interval $(a,b)$ with weight function $h$ (see Orthogonal polynomials) and the coefficients $\{a_n\}$ are calculated from the formula

$$a_n=\int\limits_a^bh(x)f(x)P_n(x)dx.\tag{2}$$

Here, the function $f$ belongs to the class $L_2=L_2[(a,b),h]$ of functions that are square summable (Lebesgue integrable) with weight function $h$ over the interval $(a,b)$ of orthogonality.

As for any orthogonal series, the partial sums $\{s_n(x,f)\}$ of \ref{1} are the best-possible approximations to $f$ in the metric of $L_2$ and satisfy the condition

$$\lim_{n\to\infty}a_n=0.\tag{3}$$

For a proof of the convergence of the series \ref{1} at a single point $x$ or on a certain set in $(a,b)$ one usually applies the equality

$$f(x)-s_n(x,f)=\mu_n[a_n(\phi_x)P_{n+1}-a_{n+1}(\phi_x)P_n(x)],$$

where $\{a_n(\phi_x)\}$ are the Fourier coefficients of an auxiliary function $\phi_x$, given by

$$phi_x(t)=\frac{f(x)-f(t)}{x-t},\quad t\in(a,b),$$

for fixed $x$, and $\mu_n$ is the coefficient given by the Christoffel–Darboux formula. If the interval of orthogonality $[a,b]$ is bounded, if $\phi_x\in L_2$ and if the sequence $\{P_n\}$ is bounded at the given point $x$, then the series \ref{1} converges to the value $f(x)$.

The coefficients \ref{2} can also be defined for a function $f$ in the class $L_1=L_1[(a,b),h]$, that is, for functions that are summable with weight function $h$ over $(a,b)$. For a bounded interval $[a,b]$, condition \ref{3} holds if $f\in L_1[(a,b),h]$ and if the sequence $\{P_n\}$ is uniformly bounded on the whole interval $[a,b]$. Under these conditions the series \ref{1} converges at a certain point $x\in[a,b]$ to the value $f(x)$ if $\phi_x\in L_1[(a,b),h]$.

Let $A$ be a part of $(a,b)$ on which the sequence $\{P_n\}$ is uniformly bounded, let $B=[a,b]\setminus A$ and let $L_p(A)=L_p[A,h]$ be the class of functions that are $p$-summable over $A$ with weight function $h$. If, for a fixed $x\in A$, one has $\phi_x\in L_1(A)$ and $\phi_x\in L_2(B)$, then the series \ref{1} converges to $f(x)$.

For the series \ref{1} the localization principle for conditions of convergence holds: If two functions $f$ and $g$ in $L_2$ coincide in an interval $(x-\delta,x+\delta)$, where $x\in A$, then the Fourier series of these two functions in the orthogonal polynomials converge or diverge simultaneously at $x$. An analogous assertion is valid if $f$ and $g$ belong to $L_1(A)$ and $L_2(B)$ and $x\in A$.

For the classical orthogonal polynomials the theorems on the equiconvergence with a certain associated trigonometric Fourier series hold for the series \ref{1} (see Equiconvergent series).

Uniform convergence of the series \ref{1} over the whole bounded interval of orthogonality $[a,b]$, or over part of it, is usually investigated using the Lebesgue inequality

$$\left|f(x)-\sum_{k=0}^na_kP_k(x)\right|\leq[1+L_n(x)]E_n(f),\quad x\in[a,b],$$

where the Lebesgue function

$$L_n(x)=\int\limits_a^bh(t)\left|\sum_{k=0}^nP_k(x)P_k(t)\right|dt$$

does not depend on $f$ and $E_n(f)$ is the best uniform approximation (cf. Best approximation) to the continuous function $f$ on $[a,b]$ by polynomials of degree not exceeding $n$. The sequence of Lebesgue functions $\{L_n\}$ can grow at various rates at the various points of $[a,b]$, depending on the properties of $h$. However, for the whole interval $[a,b]$ one introduces the Lebesgue constants

$$L_n=\max_{x\in[a,b]}L_n(x),$$

which increase unboundedly as $n\to\infty$ (for different systems of orthogonal polynomials the Lebesgue constants can increase at different rates). The Lebesgue inequality implies that if the condition

$$\lim_{n\to\infty}L_nE_n(f)=0$$

is satisfied, then the series \ref{1} converges uniformly to $f$ on the whole interval $[a,b]$. On the other hand, the rate at which the sequence $\{E_n(f)\}$ tends to zero depends on the differentiability properties of $f$. Thus, in many cases it is not difficult to formulate sufficient conditions for the right-hand side of the Lebesgue inequality to tend to zero as $n\to\infty$ (see, for example, Legendre polynomials; Chebyshev polynomials; Jacobi polynomials). In the general case of an arbitrary weight function one can obtain specific results if one knows asymptotic formulas or bounds for the orthogonal polynomials under consideration.

References

[1] G. Szegö, "Orthogonal polynomials" , Amer. Math. Soc. (1975)
[2] Ya.L. Geronimus, "Polynomials orthogonal on a circle and interval" , Pergamon (1960) (Translated from Russian)
[3] P.K. Suetin, "Classical orthogonal polynomials" , Moscow (1979) (In Russian)

See also the references to Orthogonal polynomials.


Comments

See also [a1], Chapt. 4 and [a2], part one. Equiconvergence theorems have been proved more generally for the case of orthogonal polynomials with respect to a weight function $h$ on a finite interval belonging to the Szegö class, i.e. $\log h\in L$, cf. [a2], Sect. 4.12. For Fourier series in orthogonal polynomials with respect to a weight function on an unbounded interval see [a2], part two.

References

[a1] G. Freud, "Orthogonal polynomials" , Pergamon (1971) (Translated from German)
[a2] P. Nevai, G. Freud, "Orthogonal polynomials and Christoffel functions (A case study)" J. Approx. Theory , 48 (1986) pp. 3–167
How to Cite This Entry:
Fourier series in orthogonal polynomials. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fourier_series_in_orthogonal_polynomials&oldid=13878
This article was adapted from an original article by P.K. Suetin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article