Page 203 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 203
ORTHOGONAL POLYNOMIALS AND SPECIAL FUNCTIONS (MS-10)
of situations covering not only OPUC, but also orthogonal polynomials on the real line (OPRL),
matrix valued measures built out of scalar OPUC and OPRL, as well as matrix valued OPUC
and OPRL. This talk will give an overview of these different versions of Khrushchev formula,
pointing also to their implications in other areas such as harmonic analysis or quantum renewal
theory.
Dual bases and orthogonal polynomials
Paweł Woz´ny, pwo@cs.uni.wroc.pl
University of Wrocław, Poland
Let b0, b1, . . . , bn be linearly independent functions. Let us consider the linear space Bn gener-
ated by these functions with an inner product ·, · : Bn × Bn → C. We say that the functions
Dn := {d(0n), d1(n), . . . , dn(n)} form a dual basis of the space Bn with respect to the inner product
·, · , if the following conditions hold:
d0(n), d(1n), . . . , d(nn) = Bn,
span
bi, d(jn) = δij (0 ≤ i, j ≤ n),
where δii = 1, and δij = 0 for i = j.
In general, the dual basis Dn can be found with O(n2) computational complexity. However,
if the dual basis Dn is known, it is possible to construct the dual basis Dn+1 faster, i.e., with
O(n) computational complexity. Dual bases have many applications in numerical analysis,
approximation theory or in computer aided geometric design. For example, skillful use of these
bases often results in less costly algorithms which solve some computational problems.
It is also important that dual bases are very closely related to orthogonal bases. In the
first part of the talk, we present general results on dual bases. Next, we focus on some im-
portant families of polynomial dual bases and their connections with classical, discrete and
q-orthogonal polynomials. For example, the so-called dual Bernstein polynomials are related
to orthogonal Hahn, dual Hahn and Jacobi polynomials. Using some of these connections, one
can find differential-recurrence formulas, differential equation or recurrence relation satisfied
by dual Bernstein polynomials. There also exists a first-order non-homogeneous recurrence
relation linking dual Bernstein and orthogonal Jacobi polynomials. When used properly, it al-
lows to propose fast and numerically efficient algorithms for evaluating all n + 1 dual Bernstein
polynomials of degree n with O(n) computational complexity.
201
of situations covering not only OPUC, but also orthogonal polynomials on the real line (OPRL),
matrix valued measures built out of scalar OPUC and OPRL, as well as matrix valued OPUC
and OPRL. This talk will give an overview of these different versions of Khrushchev formula,
pointing also to their implications in other areas such as harmonic analysis or quantum renewal
theory.
Dual bases and orthogonal polynomials
Paweł Woz´ny, pwo@cs.uni.wroc.pl
University of Wrocław, Poland
Let b0, b1, . . . , bn be linearly independent functions. Let us consider the linear space Bn gener-
ated by these functions with an inner product ·, · : Bn × Bn → C. We say that the functions
Dn := {d(0n), d1(n), . . . , dn(n)} form a dual basis of the space Bn with respect to the inner product
·, · , if the following conditions hold:
d0(n), d(1n), . . . , d(nn) = Bn,
span
bi, d(jn) = δij (0 ≤ i, j ≤ n),
where δii = 1, and δij = 0 for i = j.
In general, the dual basis Dn can be found with O(n2) computational complexity. However,
if the dual basis Dn is known, it is possible to construct the dual basis Dn+1 faster, i.e., with
O(n) computational complexity. Dual bases have many applications in numerical analysis,
approximation theory or in computer aided geometric design. For example, skillful use of these
bases often results in less costly algorithms which solve some computational problems.
It is also important that dual bases are very closely related to orthogonal bases. In the
first part of the talk, we present general results on dual bases. Next, we focus on some im-
portant families of polynomial dual bases and their connections with classical, discrete and
q-orthogonal polynomials. For example, the so-called dual Bernstein polynomials are related
to orthogonal Hahn, dual Hahn and Jacobi polynomials. Using some of these connections, one
can find differential-recurrence formulas, differential equation or recurrence relation satisfied
by dual Bernstein polynomials. There also exists a first-order non-homogeneous recurrence
relation linking dual Bernstein and orthogonal Jacobi polynomials. When used properly, it al-
lows to propose fast and numerically efficient algorithms for evaluating all n + 1 dual Bernstein
polynomials of degree n with O(n) computational complexity.
201
