# Fourier series in orthogonal polynomials

A series of the form

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

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

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

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 \eqref{1} are the best-possible approximations to $f$ in the metric of $L_2$ and satisfy the condition

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

For a proof of the convergence of the series \eqref{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 \eqref{1} converges to the value $f(x)$.

The coefficients \eqref{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 \eqref{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 \eqref{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 \eqref{1} converges to $f(x)$.

For the series \eqref{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 \eqref{1} (see Equiconvergent series).

Uniform convergence of the series \eqref{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 \eqref{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.

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=44702
This article was adapted from an original article by P.K. Suetin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article