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.

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