J. Marshall Ash's Vita

Link to J. Marshall Ash's Homepage


S.B., University of Chicago, 1961, (Physics)

S.M., University of Chicago, 1963

Ph.D., University of Chicago, 1966

Academic Appointments:

DePaul University, Professor Emeritus, 2014-

DePaul University, Professor, 1975- 2014

Stanford University, Visiting Professor, 1977

DePaul University, Associate Professor, 1972-1975

DePaul University, Assistant Professor, 1969-1972

Columbia University, Joseph Fels Ritt Instructor, 1966-1969

Creative Activities:


Studies in Harmonic Analysis, Studies in Mathematics, Vol.13, Math. Assoc. of Amer., 1976. (Editor and Contributor)

Harmonic Analysis: Calderón-Zygmund and Beyond, a conference in honor of Stephen Vági's retirement, harmonic-analysis: Calderón-Zygmund and beyond, December 6--8, 2002, DePaul University, Chicago, Illinois/ J. M. Ash, R. L. Jones, eds., Contemporary Mathematics, v. 411, 2006.



 1. Generalizations of the Riemann derivative, Trans. Amer. Math. Soc., 126 (1967), 181-199.

 2. A characterization of the Peano derivative, Trans. Amer. Math. Soc., 149 (1970), 489-501.

 3. Convergence, summability, and uniqueness of multiple trigonometric series, Bull. Amer. Math. Soc., 77 (1971), 123-127. (Co-author: G.V. Welland).

 4. Convergence, uniqueness, and summability of multiple trigonometric series, Trans. Amer. Math. Soc., 163 (1972), 401- 436. (Co-author: G.V. Welland).

 5. A divergent multiple Fourier series of power series type, Studia Math., 44 (1972), 477- 491. (Co-author: L. Gluck).

 6. Very slowly varying functions, Aequationes Math., 10 (1974), 1-10. (Co-authors: P. Erdös and L. A. Rubel).

 7. Convergence and divergence of series conjugate to a convergent multiple Fourier series, Trans. Amer. Math. Soc., 207 (1975), 127-142. (Co-author: L. Gluck).

 8. Multiple trigonometric series, Studies in Harmonic Analysis, Studies in Mathematics, Vol.13, Math. Assoc. of Amer., J. M. Ash(Ed.), 1976, pp. 76-96.

 9. A characterization of isometries, J. Math. Anal. Appl., 60 (1977), 417-428. (Co-authors: Peter Ash and R. D. Ogden).

10. Singular integral operators with complex homogeneity,, Studia Math., 65 (1979), 31-50 (Co-authors: P.F. Ash, C. L. Fefferman, and R.L. Jones).

11. Termwise averages of two divergent series, L'Enseign. Math., 25 (1979), 189-192 (Co-author: Harlan Sexton).

12. Optimal numerical differentiation using three function evaluations, Math. Comp. 37 (1981), 159-167. (Co-author: Roger Jones).

13. Inegalites sur des sommes d'exponentielles, Comptes Rendus de l'Academie des Sciences, 296 (1983), 899-902. (Co-authors: Roger Jones and Bahman Saffari).

14. Decreasing rearrangements of Lp,q of the Bohr group, Studia Math. 78 (1984), 93-103. (Co-author: Kenneth A. Ross).

15. Weak restricted and very restricted operators on L2, Trans. Amer. Math. Soc., 281 (1984), 675-689.

16. Optimal numerical differentiation using n function evaluations , Estratto da Calcolo, 21 (1984), 151-169. (Co-authors: Svante Janson and Roger Jones)

17. A surface with one local minimum, Mathematics Magazine, 58 (1985), 147-149. (Co-author: Harlan Sexton).

18. Convergence of series conjugate to a convergent multiple trigonometric series, Bull. Sc. Math., 110 (1986), 177-224. (Co-author: Roger L. Jones)

19. Very generalized Riemann derivatives, generalized Riemann derivatives and associated summability methods, Real Analysis Exchange, 11 (1985-86), 10-29.

20. Mean value theorems for generalized Riemann derivatives, Proc. Amer. Math. Soc., 101 (1987), 263-271. (Co-author: Roger L. Jones)

21. Generalized differentiation and summability, Real Analysis Exchange, 12 (1986-87), 366-371.

22. Fourier series with positive coefficients, Proc. Amer. Math. Soc., 101 (1987), 392-393. (Co-authors: Michael Rains and Stephen Vági)

23. Uniqueness of representation by trigonometric series, Amer. Math. Monthly, 96 (1989), 873-885.

24. A multidimensional Taylor's Theorem with minimal hypothesis, Colloq. Math., 60-61 (1990), 245-252. (Co-authors: A. E. Gatto and S. Vági)

25. A new proof of uniqueness for (multiple) trigonometric series, Proc. Amer. Math. Soc., 107 (1989), 409-410.

25a. Erratum: "A new proof of uniqueness for multiple trigonometric series" [Proc. Amer. Math. Soc. 107 (1989), 409--410], Proc. Amer. Math. Soc. 108 (1990), 571.

26. Generalizations of the wave equation, Trans. Amer. Math. Soc., 338 (1993), 57-75. (Co-authors: J. Cohen, C. Freiling, and D. Rinne)

27. Generalized Derivatives, IMA Volumes in Mathematics and its Applications, Partial Differential Equations with Minimal Smoothness and Applications, B. Dahlberg, E. Fabes, R. Fefferman, D. Jerison, C. Kenig and J. Pipher(Eds.), Vol. 42, Springer-Verlag, New York, 1992. pp. 25-32. (Co-authors: J. Cohen, C. Freiling, A. E. Gatto, and D. Rinne)

28. Uniqueness and nonuniqueness for harmonic functions with zero nontangential limits, ICM-90 Satellite Conference Proceedings, Harmonic Analysis, S. Igari(Ed.), Springer-Verlag, Tokyo, 1991, pp.30-40. (Co-author: Russell Brown)

29. Uniqueness of rectangularly convergent trigonometric series, Annals of Math., 37 (1993), 145-166. (Co-authors: C. Freiling and D. Rinne)

30. The Cantor-Lebesgue Property, Israel Journal of Math., 84 (1993), 179-191. (Co-authors: R. P. Kaufman and E. Rieders)

31. Characterizations and generalizations of continuity, Proceedings of the Amer. Math. Soc., 121 (1994), 833-842. (Co-authors: J. Cohen, C. Freiling, L. Gluck, E. Rieders, D. Rinne, and G. Wang)

32. A new, harder proof that continuous functions with Schwarz derivative 0 are lines,  Fourier analysis (Orono, ME, 1992)/ eds. W.O. Bray, P.S. Milojević, Č.V. Stanojević. Lecture Notes in Pure and Appl. Math. Vol.157. Marcel Dekker, New York, 1994. pp. 35-46.

33. Remarks on  W. H. Young's "On multiple Fourier series," Journal Expositiones Mathematicae, 18 (2000), 317-322.

34. The limit of x^(x^...^x) as x tends to zero, Mathematics Magazine, 69 (1996), 207-209.

35. A Cantor-Lebesgue theorem with variable coefficients, Proceedings of the Amer. Math. Soc., 125 (1997), 219-228. (Co-authors: G. Wang and D. Weinberg)

36. On strongly interacting internal solitary waves, Journal of Fourier Analysis and Applications, 2 (1996), 507-517. (Co-authors: J. Cohen and G. Wang)

37. One and two dimensional Cantor-Lebesgue type theorems, Trans. Amer. Math. Soc., 349 (1997), 1663-1674. (Co-author: G. Wang)

38. A survey of uniqueness questions in multiple trigonometric series, A Conference in Harmonic Analysis and Non-linear Differential Equations in Honor of Victor L. Shapiro, Contemporary Math., 208 (1997), 35-71. (Co-author: G. Wang)

39. An extremal problem for trigonometric polynomials, Proc. of  the Amer. Math. Soc.,127 (1999), pp. 211-216. (Coauthor:  Michael Ganzburg)

40. The probability of a tie in an n-game match, Amer. Math. Monthly, 105 (1998), 844-846.

41. Neither a worst convergent nor a best divergent series exists, College Math. J., 28 (1997), 296-297.

42. New uniqueness theorems for trigonometric series, Proc. Amer.  Math. Soc., 128 (2000), 2627-2636. (Coauthor: Shakro Tetunashvili)

43. Nonuniqueness for a bounded, differentiable complete orthonormal system in L^2[0,1],  J. d'Analyse Math., 81 (2000), 185-207. (Coauthor: Gang Wang)

44. Exponential sums with coefficients 0 or 1 and concentrated  L^p normsAnnal. Inst. Fourier, 57 (2007), 1377-1404. (Coauthors: B. Anderson, R. Jones, D. G. Rider, B.  Saffari)

45. Some spherical uniqueness theorems for multiple trigonometric series, Ann. Math., 151 (2000), 1-33. (Coauthor: Gang Wang)

46. Uniqueness for Spherically Convergent Multiple Trigonometric Series, Handbook of Analytic-Computational Methods in Applied Mathematics, G. Anastassiou ed., Chapman & Hall/CRC, Boca Raton, 2000, pp. 309-355.

47. L^p norm local estimates for exponential sums, Comptes Rendus de l'Academie des Sciences, 330 (2000), 765-769. (Coauthors: B. Anderson, R. Jones, D. G. Rider, B. Saffari)

48. Sets of uniqueness of the power of the continuum, Proceedings of A. Razmadze Mathematical Institute, 122 (2000), 15-19. (Coauthor: Gang Wang)

49. Sets of uniqueness for spherically convergent multiple trigonometric series, Trans. Amer. Math. Soc., 354 (2002), 4769-4788. (Coauthor: Gang Wang)

50. Once in a while, differentiation is multiplicative, Math. Mag., 73 (2000), 302.                                                                                                                                                                                                                                                                                                                                                                                                                                                            

51. Review of Steven Krantz's A Panorama of Harmonic Analysis, Math. Intelligencer, 22 (2000), no. 4, 75-77.

52. Uniqueness for higher dimensional trigonometric series, Cubo, 4 (2002), 97-125.

53. On the nth quantum derivative, J. London Math. Soc., 66 (2002), 114-130. (Coauthors: Stefan Catoiu, Ricardo Rios)


54. Tiling deficient rectangles with trominoes, Math. Mag., 77 (2004), 46-55.  (Coauthor: Solomon W. Golomb)


55. Counterexamples without cases, Proceedings of A. Razmadze Mathematical Institute, 131 (2003), 17-19.


56. Symmetric and quantum symmetric derivatives of Lipschitz functions, J. Math. Anal. Appl., 288 (2003), 717-721.


57. Topics in generalized differentiation, In: D. G. Alvarez, G. L. Acedo, R. V. Caro, eds. Seminar of Mathematical Analysis: Proceedings, Universities of Malaga and Seville (Spain). Seville: University of Seville; 2003. pp. 57-79.


58. Telescoping, rational-valued series, and zeta functions, Trans. Amer. Math. Soc., 357 (2005), 3339-3358. (Coauthor: Stefan Catoiu)


59. Uniqueness for multiple trigonometric and Walsh series with convergent rearranged square partial sums, Proc. Amer. Math. Soc. 134 (2006), 1681-1686.  (Coauthor: Sh. T. Tetunashvili)


60. Linearizing mile run times, College Math. J., 35 (2004), 370-374. (Coauthors: Garrett Ash and Stefan Catoiu)


61. Approximate and Lp Peano derivatives of non-integral order, Studia Math., 170 (2005), 241-258. (Coauthor: Hajrudin Fejzić)


62. Mean value theorems for differences, Elemente der Mathematik, 63 (2008), 122-125.


63. An Lp differentiable non-differentiable function, Real Analysis Exchange, 30 (2004-2005), 747-754.


64. Triangular Dirichlet kernels and growth of Lp Lebesgue constants, J. Fourier Anal. Appl., 16 (2010), 1053-1069.


65. Quantum symmetric Lp derivatives, Trans. Amer. Math. Soc. 360 (2008), 959-987.  (Coauthor: Stefan Catoiu)


66. Growth of Lp Lebesgue constants for convex polyhedra and other regions, Trans. Amer. Math. Soc., 361 (2009), 4215-4232. (Coauthor: Laura De Carli)


67. Constructing a quadrilateral inside another one, Mathematical Gazette, November, 2009, 93 (2009), 522-532. (Coauthors: Michael A. Ash and Peter F. Ash)


68. Uniqueness questions for Multiple Trigonometric Series, Topics in Harmonic Analysis and Ergodic Theory, 129-165, Contemp. Math., 444, Amer. Math. Soc., Providence, RI, 2007. (Coauthor: Gang Wang)


69. Series involving iterated logarithms, College Math. J., 40 (2009), 40-42.


70. On the boundary behavior of special classes of C -functions and analytic functions, International Mathematical Forum, 7 (2012), 153 - 166.(Coauthor: M. T. Karaev)


71. On concentrating idempotents, a survey, Topics in Classical Analysis and Applications in Honor of Daniel Waterman, edited by L. De Carli, K. Kazarian, & M. Milman, World Scientific, 2008. pp. 31-44.


72. Wiener's positive Fourier coefficients theorem in variants of Lp spaces, Michigan Math. J., 59 (2010), 143-151. (Coauthors: S. Tikhonov and J. Tung)


73. Plausible and genuine extensions of L'Hospital's Rule, Math Mag., 85 (2012), 52-60. (Coauthors: Allan Berele and Stefan Catoiu)


74. How to concentrate idempotents, Real Analysis Exchange, 35 (2009/2010), 1-20.


75. Many proofs that the primes are infinite, J. of Recreational Mathematics, 36 (2007), 294-298. (Coauthor: T. Kyle Petersen)


76. A rational number of the form aa with a irrational, Mathematical Gazette, 96 (2012), 106-108. (Coauthor: Yiren Tan)


77. A survey of multidimensional generalizations of Cantor's uniqueness theorem for trigonometric series, Recent Advances in Harmonic Analysis and Application: In Honor of Konstantin Oskolkov, Bilyk, D.; De Carli, L.; Petukhov, A.; Stokolos, A.M.; Wick, B.D. (Eds.), Springer, 2013. pp. 47-59.


78. A balanced Diplomacy tournament, J. of Combinatorial Math. and Combinatorial Computing, 96 (2016), 245-264. (Coauthors: Andrew Ash, Timothy L. McMurry, Allen Schwenk, Bridget E. Tenner)


79. The limit comparison test needs positivity, Math. Mag., 85 (2012), 374-375.


80.  Remarks on various generalized derivatives, Special Functions, Partial Differential Equations, and Harmonic Analysis:  in honor of Calixto P. Calderón. Georgakis, C.; Stokolos, A. M.; and Urbina, W.(Eds.), Springer proceedings in Math. & Stat., Springer, Cham, 2014. pp. 25-39.


81.  Victor Shapiro and the theory of uniqueness for multiple trigonometric series, Harmonic Analysis and Operator Theory: Cora Sadosky Memorial Seminar in Analysis, Vol. I, Stokolos, A. M.; Marcantognini, S.; Urbina, W.; Pereyra, M. C. (Eds.), Springer, 2016, pp. 123­­­-132.


82.  Discontinuous functions as limits of compactly supported formulas, College Math. J., to appear.


83. Generalized vs. ordinary differentiation, Proc. Amer. Math. Soc. 145 (2017), 1553-1565. (Coauthors: Stefan Catoiu and Marianna Csörnyei)


84. The classification of generalized Riemann derivatives, Proc. Amer. Math. Soc., to appear. (Coauthors: Stefan Catoiu and William Chin)


85. New definitions of continuity, Real Analysis Exchange, 40(2) (2014/2015), 403--420. (Coauthors:  Arlene Ash and Stefan Catoiu)


86. Multidimensional Riemann derivatives, Studia Math., 235 (2016), 87--100. (Coauthor: Stefan Catoiu)


87. Directional differentiability in the Euclidean plane, Real Analysis Exchange, 42(1) (2017), 185--192. (Coauthor: Stefan Catoiu)


88. The classification of generalized Riemann derivatives, preprint. (Coauthors: Stefan Catoiu and William Chin)



Items 49, 50, 52-61 are based upon work supported by the National Science Foundation under Grant No. DMS 9707011. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the Discontinuities as limits of compactly supported formulas