Obituary: Frank Stenger

Frank Stenger, professor emeritus at the University of Utah’s Kahlert School of Computing and a giant in the field of numerical analysis, passed away peacefully on October 23, 2024. He was 86 years old.

Frank was born in Magyarpolány, Hungary, on July 6, 1938, to György Stenger and Katalin Görgy. The Stenger family was forced to flee Hungary in 1947 and ultimately moved to Canada.

Frank graduated from the University of Alberta with a bachelor’s degree in engineering physics and a master’s degree in numerical analysis and engineering for control theory. He then earned his Ph.D. in mathematics at Alberta, where he studied asymptotics under Ian Whitney. During that time, he also worked with Frank Olver of the U.S. National Bureau of Standards to research error boundaries for second-order differential equations. After receiving his Ph.D., Frank taught as an assistant professor at Alberta, the University of Michigan, and the University of Utah before eventually becoming a professor at Utah, where he subsequently earned tenure.

Frank’s significant contributions to Sinc-related research began during his Ph.D. studies, when he encountered Edmund Whittaker’s work on cardinal functions \(C(f,h)(x)\). When Whitney and John McNamee submitted a joint article on the function \(C(f,h)\) to a SIAM journal in 1964, it was rejected. McNamee then handed Frank a copy of the paper as well as the reviewer’s objections. After reading the paper three times, Frank’s suggestions for improvement included the Paley-Wiener theorem. McNamee and Whitney agreed to Frank’s request to be a co-author on the article; with its publication in Mathematics of Computation in 1971 [2], the following cardinal function became an important component of Frank’s computational method:

\[C(f,h)(x)=\sum _{k=-\infty }^{\infty } f(k \, h)S(k,h)(x), \tag1\]

\[S(k,h)(x)=\text{Sinc}\left(\frac{x}{h}-k\right), \tag2\]

\[\text{Sinc}(x)=\frac{\sin (\pi \, x)}{\pi  x}. \tag3\]

Here, \(\text{Sinc}(x)\) denotes the scaled sinc function, whereas Frank defined \(S(k,h)\) as Sinc functions. The 1971 paper described the coinage of Sinc functions as “a function of royal blood whose distinguished properties set it apart from its bourgeois brethren.” Sinc techniques always remained at the heart of Frank’s research, and he continued to work in this field until his death.

While conducting research at Michigan, Frank discovered a novel elliptic function to verify the error formula for a rational approximation [3]. As a visiting professor at the University of Montreal in 1970, he reproduced this function and later established the Sinc computation discipline with his 1975 article in the SIAM Journal on Numerical Analysis [4]. 

This same publication also invalidated a previous hypothesis of numerous mathematicians on the calculation of the best error \(\sigma (n)\) of an integral approximation, which Herbert Wilf defined in 1964 [11]. Wilf estimated this error as \(\sigma (n)=\mathcal{O}\left((\log (n)/n)^{1/2}\right)\), after which three independent research articles each produced an estimate of \(\sigma (n)=\mathcal{O}\left(n^{-1/2}\right)\) and conjectured it to be the best possible bound. However, Frank was able to compute \(\sigma (n)=\mathcal{O}\left(\exp(-\pi (n/2)^{1/2})\right)\) by using an explicit “q-series” type quadrature formula [4].

Additionally, Frank obtained several explicit q-series type quadrature formulae via transformations of simple conformal maps in his characteristic function: (i) an arc of the unit circle, (ii) the interval \((-1,1)\), (iii) the interval \((0,\infty)\), and (iv) the real line \(\mathbb{R}=(-\infty ,\infty)\) [4]. The latter was simply the trapezoidal rule with approximately the same error bound as in [2]. This relationship to the aforementioned cardinal series prompted McNamee to recommend that Frank utilize the series (rather than elliptic functions) to derive similar formulae, since the cardinal series was accessible to a larger audience.

While serving as a visiting professor at the University of British Columbia (UBC) during the 1975-1976 academic year, Frank used different conformal maps of the series in \((1)\) to generate explicit, precise Sinc-type methods for approximation across arbitrary intervals and contours [5]. When the series converges, the resultant function is an entire function of order \(1\) and type \(\pi /h\). If \(f\) is uniformly bounded on \(\mathbb{R}\) and an entire function of order \(1\) and type \(\pi /h\), then the function \(C(f,h)\) in \((1)\) fulfills the identity \(C(f,h)=f\). Furthermore, \(C(f,h)\) has several identities in this space that are derived from operations on \(C(f,h)\), such as differentiation, orthogonality, the delta-function-like behavior of Sinc functions, Fourier transforms, and Hilbert transforms. 

These identities become very precise approximations if \(f\) is not analytic in the whole complex plane, but rather analytic and uniformly limited to a strip: a region that arose naturally in the derivation of quadrature rules [4]. A conformal map \(\phi\) of another region automatically yields methods of interpolation over a contour \(\Gamma=\phi ^{-1}(\mathbb{R})\) of the form

\[F\approx C(F,h)\circ \phi =\sum _{k=-\infty }^{\infty } F\left(z_k\right)S(k,h)\circ \phi, \tag 4\]

\[\int _{\Gamma }F(x)dx\approx \int _{\Gamma }C(F,h)\circ \phi (x)dx\approx h\sum _{k=-\infty }^{\infty } \frac{F(x_k)}{\phi '(x_k)}, \tag5\]

where \(x_k=\phi ^{-1}(k h)\) denotes the Sinc points. The identities for the function \(C(f,h)\) remain valid. We find the same exact bounds on the errors of approximation and consequently obtain an explicit family of formulae for interpolation, quadrature, differentiation, Hilbert transforms, and so on for arbitrary bounded, semi-infinite, and infinite intervals, as well as analytic arcs \(\Gamma\). This result is even valid for a finite number of Sinc points \(m=2N+1\).

Established in 1984, Sinc spaces anticipate consistent accuracy over a set of points. These spaces exist in the interior of a set and are calculated via Sinc techniques, which achieve approximations within a relative error range even when they approximate an operation with unbounded results at a set’s endpoint. Examples of this behavior include differentiation, Laplace transform inversion, Hilbert transform approximation, and the approximation of Abel-type integrals.

In the 1970s, the truncation of infinite Sinc series to finite ones—as well as a slight alteration of the bases from \(S(k,h)\circ \phi\) to \(\omega_k\) [6]—resulted in uniformly accurate Sinc approximation over \(\Gamma\) with respect to functions that are bounded but non-zero at the endpoints of \(\Gamma\).

The Sinc spaces also house solutions to differential equations [6]. The approximation of such functions leads to exponential convergence, even though we do not know the exact nature of the singularities at the end of intervals or on the boundary of a region in more than one dimension when solving partial differential equations (PDEs). For instance, if \(d\) and \(\alpha\) are positive constants, then the choice \(h=c/N^{1/2}\) produces an error of the kind

\[\underset{x\in \mathbb{R}}{\sup }\left| F(x)-\sum _{k=-N}^N F(x_k)\omega _k(x)\right| =\mathcal{O}\left(\exp \left(-c N^{1/2}\right)\right), \enspace N\to \infty , \tag6\]

where \(c\) is a positive constant. The best estimation is \(c=\sqrt{\pi\, d\,\alpha}\) with \(h=\sqrt{\pi\, d/(\alpha\, N)}\).

During his stint at UBC, Stenger discovered the important Sinc indefinite integration matrices \(A^{\pm}\) for the approximation of indefinite integrals \(\mathcal{J}^{+}(f)(x)=\int_a^x f(\xi) d\xi\) and \(\mathcal{J}^{-}(f)(x)=\int_x^b f(\xi) d\xi\). By this time, the research community understood that matrix techniques can easily handle Sinc methods; basis functions are only necessary to generate matrices that approximate calculus operations. Although the discrete formulae were found in the late 1970s, they were initially published without proof in 1981. Several proofs have since followed.

Sinc indefinite integration can uniformly approximate indefinite integrals over arbitrary intervals and contours, even if the intervals are unbounded at the endpoints of \(\Gamma\). It also provides a unique approach for tackling ordinary differential equation (ODE) initial value problems over arbitrary intervals. The Sinc ODE technique eliminates stability and stiffness concerns due to Sinc points that “bunch up” near the interval’s endpoints. 

Frank’s 1995 discovery of the formula for indefinite convolutions [7] has inspired many important novel formulae in applications that were previously considered difficult. For example, the solution of PDEs in the form of integrals of Green’s function is a straightforward application of Sinc convolution. However, this approach does require the multidimensional Laplace transform of the function. Frank successfully determined explicit formulae for all of the free space multidimensional Green’s functions for Poisson, biharmonic, wave, and heat problems. He also provided explicit procedures for the evaluation of Green’s function convolution integrals across rectangular and curved areas. As such, it is now possible to obtain a highly efficient and accurate approximation of multidimensional Green’s function convolution integrals by conducting a minimal number of multiplications of one-dimensional matrices. 

Frank received numerous honors throughout his life, including the 1996 First Degree Prize from the Polish Secretary of Education and a 1998 Distinguished Research Award from the University of Utah. In 1987, he was named a distinguished visiting professor by the University of Tsukuba in Japan. Later, in 2004, SIAM acknowledged his significant efforts to advance numerical analysis with an in-depth interview as part of its collection of oral histories.

After his retirement in 2008, Frank focused on advanced Sinc techniques. He published the Handbook of Sinc Numerical Methods in 2010 [8]; this text served as the foundation for the use of Sinc methodology to solve some of the Clay Mathematics Institute’s Millennium Prize Problems. In 2016, Frank conducted numerical and analytical analyses of Charles Fefferman’s formulation of the Navier-Stokes problem [10].

During retirement, Frank also pursued application-oriented research for singular integral equations in the fractal description of the Fokker-Planck and Riemann-Liouville equations, as well as in quantum mechanics applications that solve the Schrödinger equation in three dimensions. The development of Poly-Sinc techniques that utilize polynomials and orthogonal polynomials began in 2013 and resulted in the invention of novel approximation methods, which were published in a 2021 anthology in honor of Frank’s 80th birthday [1]. 

Frank’s ultimate goal was to find the answer to the Riemann hypothesis. He was sure that Sinc methods could provide a solution, and this conjecture lingered with him throughout his life. At the age of 85, Frank published a paper on the subject [9]. These findings complete a circle that lasted from the beginning of Frank’s work until the very end.

Fond memories of Frank will live on in the works of his students, friends, and followers.

References
[1] Baumann, G. (2021). New Sinc methods of numerical analysis: Festschrift in honor of Frank Stenger’s 80th birthday. Cham, Switzerland: Birkhäuser.
[2] McNamee, J., Stenger, F., & Whitney, E.L. (1971). Whittaker’s cardinal function in retrospect. Math. Comput., 25(113), 141-154.
[3] Stenger, F. (1971). Constructive proofs for approximation by inner functions. J. Approx. Theory, 4(4), 372-386.
[4] Stenger, F. (1975). An analytic function which is an approximate characteristic function. SIAM J. Numer. Anal., 12(2), 239-254.
[5] Stenger, F. (1976). Approximations via Whittaker’s cardinal function. J. Approx. Theory, 17(3), 222-240.
[6] Stenger, F. (1993). Numerical methods based on Sinc and analytic functions. In Springer series in computational mathematics (Vol. 20). New York, NY: Springer.
[7] Stenger, F. (1995). Collocating convolutions. Math. Comput., 64(209), 211-235.
[8] Stenger, F. (2010). Handbook of Sinc numerical methods. Boca Raton, FL: CRC Press.
[9] Stenger, F. (2023). All zeros of the Riemann zeta function in the critical strip are located on the critical line and are simple. Adv. Pure Appl., 13(6), 402-411.
[10] Stenger, F., Tucker, D., & Baumann, G. (2016). Navier-Stokes equations on \(\mathbb{R}^3 \times [0, T]\). Cham, Switzerland: Springer Nature.
[11] Wilf, H.S. (1964). Exactness conditions in numerical quadrature. Numer. Math., 6, 315-319.