Numerical Methods, Computational Mathematics

Numerical Mathematics is fascinating: optimization methods, linear and nonliear Equation Systems, least square methods, SVD, orthogonal polynomials, approximation theory and methods, etc. are my favorite areas of interest. Professors Gene Golub, Walter Murray, Carlo Tomasi, Paul Nevai, Peter Borwein, Vilmos Totik, and Tamás Erdélyi are my mentors and co-authors.

MR95j:41015, 41A17 (41A20)
Peter Borwein, Tamás Erdélyi and J. Zhang
Chebyshev polynomials and Markov-Bernstein type inequalities for rational spaces.
J. London Math. Soc. (2) 50 (1994), no. 3, 501--519.
Summary

MR94h:42041 , 42C05 (33C45)
J. Zhang
Relative growth of linear iterations and orthogonal polynomials on several intervals.
Linear Algebra Appl. 186 (1993), 97--115.
Summary

MR94f:42026 42C05 (39A10 41A17)
Peter Borwein, Tamás Erdélyi and J. Zhang
Müntz systems and orthogonal Müntz-Legendre polynomials.
Trans. Amer. Math. Soc. 342 (1994), no. 2, 523--542.
Summary

MR94d:42028 42C05 (33C45)
Paul Nevai and J. Zhang
Rate of relative growth of orthogonal polynomials.
J. Math. Anal. Appl. 175 (1993), no. 1, 10--24.
Summary

MR93a:42010 42C05
Tamás Erdélyi, Paul Nevai, J. Zhang and Jeff Geronimo
A simple proof of "Favard's theorem" on the unit circle.
Atti Sem. Mat. Fis. Univ. Modena 39 (1991), no. 2, 551--556.
Summary

MR92m:42025 42C05
Tamás Erdélyi, Paul Nevai, J. Zhang and Jeff Geronimo
A simple proof of "Favard's theorem" on the unit circle. Trends in functional analysis and approximation theory (Acquafredda di Maratea, 1989), 41--46,
Univ. Modena, Modena, 1991.
Summary

MR92k:42034 42C05
Paul Nevai, J. Zhang and Vilmos Totik
Orthogonal polynomials: their growth relative to their sums.
J. Approx. Theory 67 (1991), no. 2, 215--234.
Summary

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Summaries

95j:41015 41A17 (41A20)
Borwein, Peter(3-SFR); Erdélyi, Tamás(3-SFR); Zhang, J.(1-STF-C)
Chebyshev polynomials and Markov-Bernstein type inequalities for rational spaces. (English. English summary)
J. London Math. Soc. (2) 50 (1994), no. 3, 501--519.
Summary: "We consider the trigonometric rational system $$\Big \{1,\frac {1±\sin t}{\cos t-a\sb 1},\frac {1±\sin t}{\cos t-a\sb 2},\cdots \Big \}$$ on $\bold R \bmod 2\pi$ and the algebraic rational system $$\Big \{1,\frac {1}{x-a\sb 1},\frac {1}{x-a\sb 2},\cdots \Big \}$$ on the interval $[-1,1]$ associated with a sequence of distinct poles $(a\sb k)\sp \infty\sb {k=1}$ in $\bold R\sbs [-1,1]$. Chebyshev polynomials for the rational trigonometric system are explicitly found. Chebyshev polynomials of the first and second kinds for the algebraic rational system are also studied, as well as orthogonal polynomials with respect to the weight function $(1-x\sp 2)\sp {-1/2}$. Notice that in these situations, the `polynomials' are in fact rational functions. Several explicit expressions for these polynomials are obtained. For the span of these rational systems, an exact Bernstein-Szego type inequality is proved whose limiting case gives back the classical Bernstein-Szego inequality for trigonometric and algebraic polynomials. It gives, for example, the sharp Bernstein-type inequality $$\vert p'(x)\vert \leq \frac {1}{\sqrt {1-x\sp 2}}\sum\sp n\sb {k=1}\frac {\sqrt {a\sp 2\sb k-1}}{\vert a\sb k-x\vert } \max\sb {y\in [-1,1]}\vert p(y)\vert \quad \text {for} x\in [-1,1],$$ where $p$ is any real rational function of type $(n,n)$ with poles $a\sb k\in \bold R\sbs [-1,1]$. An asymptotically sharp Markov-type inequality is also established, which is at most a factor of $2n/(2n-1)$ away from the best possible result. With proper interpretation of $\sqrt {a\sp 2\sb k-1}$, most of the results are established for $(a\sb k)\sp \infty\sb {k=1}$ in $\bold C\sbs [-1,1]$ in a more general setting."

Reviewed by Dinh Dung

94h:42041 42C05 (33C45)
Zhang, J.(1-OHS)
Relative growth of linear iterations and orthogonal polynomials on several intervals. (English. English summary)
Linear Algebra Appl. 186 (1993), 97--115.
Summary: "Assume $A\in {C}\sp {2\times2}$ and $y\sb 0\in {C}\sp 2$. Let $\{y\sb n=y\sb n(A,y\sb 0)\}$ be defined by the linear iteration $y\sb n=Ay\sb {n-1}$, $n=1,2\cdots$. Then for every $p>0$ and for every norm $\Vert ·\Vert $ in ${C}\sp 2$, there is a constant $C$, depending only on $p$ and $\Vert ·\Vert $, such that $$\sup\sb {\rho(A)\leq 1,y\sb 0\not=0}{\Vert y\sb L(A,y\sb 0)\Vert \sp p\over \sum\sb {k=0}\sp L\Vert y\sb k(A,y\sb 0)\Vert \sp p}\leq\frac CL,\quad L=1,2,\cdots,$$ where $\rho(A)$ denotes the spectral radius of $A$. Applications to orthogonal polynomials $\{p\sb n\}$ with respect to a positive Borel measure $\alpha$ on several intervals are studied. In particular, the growth of orthogonal polynomials relative to their sums satisfies $$\lim\sb {n\to\infty}\sup\sb {x\in{\rm supp}\,(\alpha)}{\vert p\sb n(x)\vert \sp p\over \sum\sp {n-1}\sb {k=0}\vert p\sb k(x)\vert \sp p}=0,$$ provided that the three-term recurrence coefficients of $\{p\sb n\}$ are asymptotically periodic."

Reviewed by G. Gasper

Cited in reviews: 96a:42032

94f:42026 42C05 (39A10 41A17)
Borwein, Peter(3-SFR); Erdélyi, Tamás(3-SFR); Zhang, J.(1-STF-C)
Müntz systems and orthogonal Müntz-Legendre polynomials. (English. English summary)
Trans. Amer. Math. Soc. 342 (1994), no. 2, 523--542.
For distinct, complex $\lambda\sb i$ for which Re $\lambda\sb i >1/2$, let $M\sb n (\Lambda)$ denote the span of $\{x\sp {\lambda\sb 0}, x\sp {\lambda\sb 1},\cdots, x\sp {\lambda\sb n}\}$. The authors define certain elements of $M\sb n(\Lambda)$, Muntz-Legendre polynomials, on $0<x\leq 1$, via a simple contour integral, show that they are, in fact, Laguerre polynomials evaluated at $(-\log x)$, and develop both differential and integral difference formulae. A reproducing kernel, similar to that appearing for "ordinary" polynomials, is given for elements in $M\sb n(\Lambda)$. One principal result, $L\sp 2$ Markov inequalities, is given by Theorem 3.4: an upper bound (in terms of $\vert \lambda\sb i\vert $ and Re$(\lambda\sb i)$) is found for the supremum, over all $p$ in $M\sb n (\Lambda)$, of $\Vert xp'(x)\Vert /\Vert p\Vert $. Sections 4 and 5 focus on real $\lambda\sb i$ and provide some results about bounding or interlacing of zeros of Muntz polynomials in $M\sb n(\Lambda)$ vis-a-vis zeros of those in $M\sb n(\Delta)$ when, e.g., $\lambda\sb i \leq \delta\sb i$ for all $i$. Christoffel functions permit development of four interesting equivalent statements, for the $\lambda\sb i >0$ situation, summarized in Theorem 5.1---one pair expresses the fact that $M\sb n(\Lambda)$ is not dense in $C[0,1]$ in the uniform norm iff $\sum\sp \infty\sb {k=1} 1/\lambda\sb k <\infty$.

Reviewed by A. G. Law

94d:42028 42C05 (33C45)
Nevai, Paul(1-OHS); Zhang, J.(1-OHS)
Rate of relative growth of orthogonal polynomials. (English. English summary)
J. Math. Anal. Appl. 175 (1993), no. 1, 10--24.
Let a sequence of polynomials $\{p\sb n\}$ be defined by the recurrence formula (1) $xp\sb n(x)=a\sb {n+1}p\sb {n+1}(x)+b\sb np\sb n(x)+a\sb np\sb {n-1}(x)$, where $p\sb {-1}=0$, $p\sb 0=1$, $a\sb n\not=0$ and $\{a\sb n\}$ and $\{b\sb n\}$ are otherwise arbitrary complex sequences. The authors investigate the rate of growth (with respect to $n$) of the expression (2) $\vert p\sb n(x)\vert \sp p/\sum\sp n\sb {k=0}\vert p\sb k(x)\vert \sp p$. For complex numbers $a$ and $b$, the interval $[b-a,b+a]$ is defined to be the segment $\{b+ta\colon\;-1\leq t\leq 1\}$ and the class ${\rm CM}(b,a)$ is the set of all pairs of complex sequences $\{a\sb n\}\sp \infty\sb {n=0}$, $\{b\sb n\}\sp \infty\sb {n=0}$ which converge to $a/2$, $b/2$, respectively (with $a\sb n\not=0$). For $\{a\sb n,b\sb n\}\sp \infty\sb {n=0}\in {\rm CM}(b,a)$, define $\epsilon\sb n=\vert 2a\sb n-a\vert +2\vert b\sb n-b\vert +\vert 2a\sb {n+1}-a\vert $. The main result of the authors is a theorem which, for every positive integer $n$ and every $L\leq \sqrt n$, places an upper bound on the maximum of the expression in (2), the maximum being computed for $x\in [b-a,b+a]$. This maximum is a somewhat complicated expression involving $p$, $n$, $L$, and $\epsilon\sb k$. It was shown previously that the expression in (2) converges to 0 uniformly for $x\in [b-a,b+a]$.

When the $b\sb n$ are real and $a\sb n>0$, the polynomials become real orthogonal polynomials and the authors specialize to the case $\epsilon\sb n=O(1/n\sp \delta)$ to obtain bounds which apply to the Jacobi, sieved ultraspherical, and Pollaczek polynomials.

Reviewed by T. S. Chihara

93a:42010 42C05
Erdélyi, Tamás(1-OHS); Nevai, Paul(1-OHS); Zhang, J.(1-OHS); Geronimo, Jeffrey S.(1-GAIT)
A simple proof of "Favard's theorem" on the unit circle.
Atti Sem. Mat. Fis. Univ. Modena 39 (1991), no. 2, 551--556.
Favard's theorem on the unit circle states that a sequence of monic polynomials satisfying $\Phi\sb n(z)=z\Phi\sb {n-1}(z)+a\sb n\Phi\sb {n-1}\sp *(z)$ is an orthogonal system on the unit circle with respect to some positive Borel measure if and only if $\vert a\sb n\vert <1$. The result is well known but not in the standard books on orthogonal polynomials. Some proofs exist in the literature (see the references), but the present proof is short and constructive and in fact a nice exercise for graduate students interested in orthogonal polynomials.

Reviewed by Walter Van Assche

92m:42025 42C05
Erdélyi, Tamás(1-OHS); Nevai, Paul(1-OHS); Zhang, J.(1-OHS); Geronimo, Jeffrey S.(1-GAIT)
A simple proof of "Favard's theorem" on the unit circle. Trends in functional analysis and approximation theory (Acquafredda di Maratea, 1989), 41--46,
Univ. Modena, Modena, 1991.
A short constructive proof of the following "Favard theorem" on the unit circle is given: Let $\{\epsilon\sb n\colon\;n=1,2,\cdots\}$ be a sequence of complex numbers with $\vert \epsilon\sb n\vert <1$ for all $n$, and let the sequence of polynomials $\{\Phi\sb n\colon\;n=0,1,2,\cdots\}$ satisfy the Szego recursion $\Phi\sb n(z)=z\Phi\sb {n-1}(z)+\epsilon\sb n\Phi\sp *\sb {n-1}(z)$, $\Phi\sb 0(z)=1$, where $\Phi\sp *\sb {n-1}(z)=z\sp {n-1}\overline{\Phi\sb {n-1}(1/\overline z)}$. Then there exists a unique finite positive Borel measure $µ$ on the unit circle $T$ with infinite support such that $\{\Phi\sb n\colon\; n=0,1,2,\cdots\}$ is orthogonal with respect to $µ$. The orthonormal polynomials $\phi\sb n$ are given by $$\phi\sb n(z)=\Phi\sb n(z)\Big[\prod\sp n\sb {k=1}(1-\vert \epsilon\sb k\vert \sp 2)\sp {-1/2}\Big].$$

The construction of $µ$ is as follows: The distribution function $µ\sb n$ is defined by $µ\sb n(\tau)=\frac1{2\pi}\int\sp \tau\sb 0\vert \phi\sb n(e\sp {i\theta})\vert \sp {-2}\,d\theta$. It follows by properties obtained from the recurrence relation that the system $\{\phi\sb n\colon\;0,1,\cdots,n\}$ is orthonormal with respect to $µ\sb n$. By Helly's selection and convergence theorems there eixsts a subsequence $\{n\sb k\}$ such that $µ\sb {n\sb k}(\theta)$ tends to a distribution function $µ(\theta)$ for which $\{\phi\sb n\colon\; n=0,1,2,\cdots\}$ is an orthonormal system. The uniqueness follows by the unique representation of bounded linear functionals on $C(T)$.

The authors point out that, unknown to them and to many other workers in the field, similar methods were used by P. Delsarte, Y. V. Genin and Y. G. Kamp [IEEE Trans. Circuits and Systems CAS-25 (1978), no. 3, 149--160; MR 58 #1981].

\{For the entire collection see MR 92f:00028\}.

Reviewed by Olav Njastad

92k:42034 42C05
Nevai, Paul(1-OHS); Zhang, J.(1-OHS); Totik, Vilmos(H-SZEG-B)
Orthogonal polynomials: their growth relative to their sums.
J. Approx. Theory 67 (1991), no. 2, 215--234.
The uniform convergence $ \lim\sb {n \to \infty} { \vert p\sb n(x)\vert \sp p /\sum\sb {k=0}\sp {n-1} \vert p\sb k(x)\vert \sp p } = 0$, for $x \in [b-a,b+a]$ and $0 < p< \infty$, is considered for solutions of a three-term recurrence relation $xp\sb n(x) = a\sb {n+1}p\sb {n+1}(x) + b\sb np\sb n(x) + a\sb np\sb {n-1}(x)$, with the property that $a\sb n$ $(\neq 0)$, $b\sb n \in {\bold C}$ with $a\sb n \to a/2$ and $b\sb n \to b$. Almost all initial conditions $p\sb 0$ and $p\sb {-1}$ are allowed. The special case $p\sb {-1}=0, p\sb 0=1$ gives orthogonal polynomials whenever $a\sb n>0$ and $b \in {\bold R}$. This paper thus treats three generalizations of an important result in the theory of orthogonal polynomials: the relative growth is considered for every $p \in (0,\infty)$ and not for $p=2$ alone; complex coefficients $a\sb n \neq 0$ and $b\sb n$ are allowed and a great variety of initial conditions $p\sb {-1},p\sb 0$ lead to the same uniform convergence.

Reviewed by Walter Van Assche

Cited in reviews: 96a:42032 94b:39005