Fourier series in orthogonal polynomials

A series of the form

(1)

where the polynomials are orthonormal on an interval with weight function (see Orthogonal polynomials) and the coefficients are calculated from the formula

(2)

Here, the function belongs to the class of functions that are square summable (Lebesgue integrable) with weight function over the interval of orthogonality.

As for any orthogonal series, the partial sums of (1) are the best-possible approximations to in the metric of and satisfy the condition

(3)

For a proof of the convergence of the series (1) at a single point or on a certain set in one usually applies the equality

where are the Fourier coefficients of an auxiliary function , given by

for fixed , and is the coefficient given by the Christoffel–Darboux formula. If the interval of orthogonality is bounded, if and if the sequence is bounded at the given point , then the series (1) converges to the value .

The coefficients (2) can also be defined for a function in the class , that is, for functions that are summable with weight function over . For a bounded interval , condition (3) holds if and if the sequence is uniformly bounded on the whole interval . Under these conditions the series (1) converges at a certain point to the value if .

Let be a part of on which the sequence is uniformly bounded, let and let be the class of functions that are -summable over with weight function . If, for a fixed , one has and , then the series (1) converges to .

For the series (1) the localization principle for conditions of convergence holds: If two functions and in coincide in an interval , where , then the Fourier series of these two functions in the orthogonal polynomials converge or diverge simultaneously at . An analogous assertion is valid if and belong to and and .

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).

Uniform convergence of the series (1) over the whole bounded interval of orthogonality , or over part of it, is usually investigated using the Lebesgue inequality

where the Lebesgue function

does not depend on and is the best uniform approximation (cf. Best approximation) to the continuous function on by polynomials of degree not exceeding . The sequence of Lebesgue functions can grow at various rates at the various points of , depending on the properties of . However, for the whole interval one introduces the Lebesgue constants

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

is satisfied, then the series (1) converges uniformly to on the whole interval . On the other hand, the rate at which the sequence tends to zero depends on the differentiability properties of . 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 (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 on a finite interval belonging to the Szegö class, i.e. , 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