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Gokulananda Das

Research & Publications

Analysis, summability and the limits of sequences

Dr. Gokulananda Das made substantial research contributions in Analysis and Functional Analysis, publishing 128 papers over more than four decades.

Dr. Das seated with faculty colleagues of the Department of Mathematics

Major thrust areas

Nine threads of a research life

01

Tauberian Theory & Absolute Summability

Pioneered a new technique for absolute summability problems, published in the Proceedings of the London Mathematical Society (1969).

02

Fourier Series & Nörlund Summability

Determined necessary and sufficient conditions for the absolute Nörlund summability of Fourier series — the first result of its kind (Proc. London Math. Soc., 1980).

03

Summability Factor Problems

Landmark results on summability factors for Nörlund means, in collaboration with Prof. L. S. Bosanquet (Proc. London Math. Soc., 1979).

04

Sublinear Functionals & Banach Limits

Identified new sublinear functionals generating Banach limits (J. London Math. Soc., 1973), opening a long programme of work with his students.

05

Sequence Spaces & Absolute Almost Convergence

Introduced the sequence space of absolute almost convergence — the absolute analogue of unique Banach limits — first presented at the British Mathematical Colloquium (1968) and published in Transactions of the AMS (1984).

06

Abelian Theorems & New Summability Methods

Twin papers in the Quarterly Journal of Mathematics (Oxford) and the introduction of several new methods of summation.

07

Ergodic Theorems & Invariant Measures

Determination of an invariant measure space through a sublinear functional (Int. J. Math. & Math. Sci., 1989).

08

Fixed Point Theory & Probability

Iteration methods for contraction mappings; a new concept of uniform convergence in probability and convergence of random Fourier–Stieltjes series.

09

Degree of Approximation

Approximation through orthogonal systems — Fourier series, Walsh–Fourier series and wavelets — in Hölder and generalised Hölder metrics; the focus of his final decade of supervision.

Selected high-impact publications

Ten papers that defined the field

  1. 1969

    Tauberian theorems for absolute Nörlund summability

    Proceedings of the London Mathematical Society (3) 19, 357–384

  2. 1979

    Absolute summability factors for Nörlund means

    Proceedings of the London Mathematical Society 38, 1–52 · with L. S. Bosanquet

  3. 1980

    A necessary and sufficient condition for absolute summability of Fourier series

    Proceedings of the London Mathematical Society 41, 217–253 · with P. C. Mohapatra

  4. 1973

    Banach and other limits

    Journal of the London Mathematical Society (2) 7, 501–507

  5. 1984

    Some sequence spaces and absolute almost convergence

    Transactions of the American Mathematical Society 283, 729–739 · with B. Kuttner and S. Nanda

  6. 1966

    On some methods of summability

    Quarterly Journal of Mathematics (Oxford) (2) 17

  7. 1974

    Non-absolute Nörlund summability of Fourier series

    Pacific Journal of Mathematics 51(1), 49–55 · with R. N. Mohapatra

  8. 1986

    Uniform law of large numbers

    Indian Journal of Pure and Applied Mathematics 17, 26–37 · with S. K. Mohanty

  9. 1989

    Sublinear functional, ergodicity and finite invariant measures

    International Journal of Mathematics and Mathematical Sciences 12, 809–820 · with B. K. Patel

  10. 1998

    Degree of approximation of functions with Hardy–Littlewood series in the Hölder metric by Borel means

    Journal of Mathematical Analysis and Applications 219, 279–293 · with A. K. Ojha and B. K. Ray

International collaborators

Working alongside eminent mathematicians

Prof. T. Pati
Prof. L. S. Bosanquet
Prof. B. Kuttner
  • Prof. T. PatiAllahabad University · Doctoral guide
  • Prof. L. S. BosanquetUniversity College London, UK
  • Prof. B. KuttnerUniversity of Birmingham, UK
  • Prof. S. SimonsUniversity of California, Santa Barbara, USA
  • Prof. Ram Narayan MohapatraUniversity of Central Florida, USA
  • Prof. Sudarsan NandaIIT Kharagpur
  • Dr. B. K. RayRavenshaw College
  • Prof. Swadheen PattanayakDirector, Institute of Mathematics and Applications, Bhubaneswar

Complete bibliography

128 research publications, 1965 – 2008

Transcribed from Dr. Das's own bio-data. Journal names follow his abbreviations.

Grouped by year

1965

  1. 1.On |R, log n, 1| summability factors of power series on the circle of convergence. Math. Zeit. 90, 319–324 (with V. P. Srivastava and R. N. Mohapatra)

1966

  1. 2.On absolute Nörlund summability factors of infinite series. J. London Math. Soc. 41, 685–692
  2. 3.On some methods of summability. Quart. J. Math. (Oxford) (2) 17
  3. 4.Functional Nörlund methods for infinite integrals. Rendiconti del Circolo Matematico di Palermo (2) 15, 310–318
  4. 5.On Nörlund method of summation. Journal of Mathematics 2, 23–30
  5. 6.On discontinuous Riesz means. Mathematics Student 34, 152–171

1967

  1. 7.On a theorem of Hardy and Littlewood. Proc. Cambridge Phil. Soc. 62, 707–713
  2. 8.On absolute summability factors and its application to Fourier series. Proc. Cambridge Phil. Soc. 63, 107–118 (with V. P. Srivastava and R. N. Mohapatra)
  3. 9.A note on Nörlund means. Rivista di Matematica Università di Parma (2) 8, 65–75
  4. 10.Some theorems in Nörlund summability. J. Indian Math. Soc. 31, 1–9
  5. 11.On functional methods. J. Indian Math. Soc. 31, 81–93
  6. 12.On absolute summability factors of infinite series. J. Indian Math. Soc. 31, 189–200 (with V. P. Srivastava and R. N. Mohapatra)
  7. 13.A new method of summability. J. Indian Math. Soc. 31, 149–160

1968

  1. 14.On some methods of summability (II). Quart. J. Math. (Oxford) (2) 19, 417–431
  2. 15.A note on Riesz means. Proc. Cambridge Phil. Soc. 64, 389–392
  3. 16.Product of Nörlund methods — I. Indian J. Math. 31, 25–43
  4. 17.Product of Nörlund methods — II. J. Indian Math. Soc. 31, 155–171

1969

  1. 18.Tauberian theorems for absolute Nörlund summability. Proc. London Math. Soc. (3) 19, 357–384

1970

  1. 19.A Tauberian theorem for absolute summability. Proc. Cambridge Phil. Soc. 67, 321–348
  2. 20.The summability (L) of Fourier series and first differential Fourier series. Indian J. Math. 12, 125–135 (with M. M. Nanda)
  3. 21.Summability (L) of Fourier series. Proc. Cambridge Phil. Soc. 67, 327–331 (with M. M. Nanda)

1971

  1. 22.On absolute Nörlund summability factors of infinite series. J. London Math. Soc. (2) 4, 193–214
  2. 23.Some inclusion theorems for Nörlund summability. J. Indian Math. Soc. 35, 241–248

1973

  1. 24.Banach and other limits. J. London Math. Soc. (2) 7, 501–507
  2. 25.Convergence, absolute convergence and strong convergence with periodicity. J. London Math. Soc. (2) 7
  3. 26.Inclusion theorems for an absolute method of summability. J. London Math. Soc. (2) 6, 467–472
  4. 27.Regularity and inverse regularity of Nörlund means. J. Indian Math. Soc. 37
  5. 28.Correction: Some inclusion theorems for Nörlund summability. J. Indian Math. Soc. (N.S.) 37

1974

  1. 29.Non-absolute Nörlund summability of Fourier series. Pacific J. Math. 51(1), 49–55 (with R. N. Mohapatra)
  2. 30.On a theorem of Sunouchi. J. Indian Math. Soc. 38, 155–174

1975

  1. 31.Summability factors for lower semimatrix transformations. Monatshefte für Mathematik 79, 307–315 (with R. N. Mohapatra)

1977

  1. 32.Absolute summability factors involving Nörlund means. Nanta Mathematica 12, 94–101 (with R. N. Mohapatra)

1978

  1. 33.Absolute summability factors — II. Tamkang J. Math. 9, 157–169 (with R. N. Mohapatra)

1979

  1. 34.Absolute summability factors for Nörlund means. Proc. London Math. Soc. 38, 1–52 (with L. S. Bosanquet)

1980

  1. 35.A necessary and sufficient condition for absolute summability of Fourier series. Proc. London Math. Soc. 41, 217–253 (with P. C. Mohapatra)
  2. 36.On a generalized harmonic Cesàro method of summation and its application to Fourier series. Indian J. Math. 22, 33–47 (with P. C. Mohapatra)
  3. 37.On some sequence-to-function transforms. Analyse Numérique et Théorie de l'Approximation 9(2), 233–243

1981

  1. 38.Inclusion theorems for summability factors. Math. Japonica 26, 607–612 (with R. N. Mohapatra)
  2. 39.A note on a theorem of Maddox on strong almost convergence. Math. Proc. Cambridge Phil. Soc. 89, 393–396 (with S. K. Mishra)

1982

  1. 40.Generalised harmonic Cesàro method of summation and its modified mean. Journal of Mathematics 24, 145–163 (with P. C. Mohapatra)

1983

  1. 41.Banach limits and lacunary strong almost convergence. J. Orissa Math. Soc. 2, 61–70 (with S. K. Mishra)

1984

  1. 42.Some sequence spaces and absolute almost convergence. Trans. Amer. Math. Soc. 283, 729–739 (with B. Kuttner and S. Nanda)
  2. 43.On common fixed points of hemicontractive mappings. Indian J. Pure Appl. Math. 15, 713–718 (with J. P. Debata)
  3. 44.On two new methods of summability. Indian J. Pure Appl. Math. 15, 1340–1351 (with K. C. Panda and S. Sahoo)

1985

  1. 45.Invariant linear functionals and infinite matrices. Indian J. Pure Appl. Math. 16, 1307–1316 (with S. L. Mishra)
  2. 46.Convergence of Ishikawa iteration of quasi-contractive mappings. Bull. Inst. Math. Academia Sinica 13, 297–302 (with J. P. Debata)
  3. 47.A note on fixed points of commuting mappings of contractive type. Indian J. Math. 27, 49–51 (with J. P. Debata)
  4. 48.Infinite matrices and summable series. J. Orissa Math. Soc. 4(1), 5–14 (with E. Savas)

1986

  1. 49.Uniform law of large numbers. Indian J. Pure Appl. Math. 17, 26–37 (with S. K. Mohanty)
  2. 50.Convergence in probability of weighted sums involving conservative matrices. Bull. Inst. Math. Academia Sinica 14, 75–82 (with S. K. Mohanty)
  3. 51.Space of absolute almost convergence. Indian J. Math. 28, 241–257 (with B. Kuttner)
  4. 52.Inverse and regularity of quasi-matrix transformations. Indian J. Pure Appl. Math. 16(9), 1123–1130
  5. 53.Fixed points of quasi-nonexpansive mappings. Indian J. Pure Appl. Math. 17(11) (with J. P. Debata)
  6. 54.Extension of the concept of weak law of large numbers. J. Orissa Math. Soc. 5, 1–10 (with M. N. Mishra)
  7. 55.On summability of Fourier and allied series by (D, γ, δ) method. J. Indian Math. Soc. 50, 167–179 (with K. C. Panda)

1987

  1. 56.Sublinear functionals and a class of conservative matrices. Bull. Inst. Math. Academia Sinica 15, 89–106
  2. 57.Some Tauberian theorems for (D, r, s) methods of summation. Indian J. Math. 29, 23–25 (with K. C. Panda and S. Sahoo)
  3. 58.On a generalised absolute harmonic summability of Fourier series. Indian J. Pure Appl. Math. 18(8), 705–722 (with K. C. Panda)
  4. 59.Almost sure convergence of factored random Fourier series. Bull. Inst. Math. Academia Sinica 15(2), 221–229
  5. 60.Functional Nörlund methods. J. Nat. Acad. Math. 5, 19–28 (with N. Guru)

1988

  1. 61.Fixed points on unit interval using infinite matrices. J. Indian Math. Soc. 53, 167–176 (with J. P. Debata)
  2. 62.Addendum: A note on fixed points of commuting mappings of contractive type. Indian J. Math. 30(3), 249–250
  3. 63.Quasi Cesàro summability of Fourier series. Indian J. Math. 30(2), 161–173 (with S. N. Pujari)
  4. 64.On (D, γ) summability of Fourier series. Bull. Inst. Math. Academia Sinica 16(2), 167–175 (with K. C. Panda)
  5. 65.Absolute almost convergence with an index. Rendiconti di Matematica VII, 8, 501–510 (with S. K. Mishra)

1989

  1. 66.Modified means. Indian J. Pure Appl. Math. 20(3), 347–385
  2. 67.Lacunary distribution of sequences. Indian J. Pure Appl. Math. 20(1), 64–74
  3. 68.Sublinear functional, ergodicity and finite invariant measures. Int. J. Math. Math. Sci. 12, 809–820 (with B. K. Patel)
  4. 69.Functional limits. Indian J. Pure Appl. Math. 20(8), 812–819 (with S. Nanda)
  5. 70.Quasi-convex sequence spaces and matrix transformations. Bull. Inst. Math. Academia Sinica 17, 269–284 (with K. Rao)
  6. 71.Distribution of sequences and infinite matrices. J. Orissa Math. Soc. 8 (with B. K. Patel)
  7. 72.On lacunary strong almost convergence. J. Orissa Math. Soc. 8 (with S. K. Mishra)
  8. 73.Spaces of strongly almost summable sequences. J. Orissa Math. Soc. 8, 83–91

1990

  1. 74.Summability of Fourier and allied series by (D, r, s) method. Indian J. Pure Appl. Math. 21(5), 535–556 (with K. C. Panda)

1991

  1. 75.Inclusion theorems for absolute harmonic–Cesàro methods of summation. Indian J. Math. 2, 155–164 (with K. C. Panda)
  2. 76.Convergence in probability of matrix transformation of random Fourier–Stieltjes series. J. Indian Math. Soc. 56, 133–143 (with S. K. Mohanty)
  3. 77.Degree of approximation of functions by the F(a, q) transform of its Fourier series in the Hölder metric. Proc. Nat. Acad. Sci. India 61(A), 345–356 (with M. Mohapatra)
  4. 78.On absolute almost convergence. J. Math. Anal. Appl. 161, 50–56 (with B. Kuttner and S. Nanda)

1992

  1. 79.On some sequence spaces. J. Math. Anal. Appl. 163 (with S. K. Sahoo)
  2. 80.A generalisation of strong and absolute almost convergence. J. Indian Math. Soc. 58(2), 65–74 (with S. K. Sahoo)
  3. 81.A generalisation of quasi-convex sequence spaces and its application to trigonometric series. J. Indian Math. Soc. 58(3), 159–186 (with K. Rao)
  4. 82.Degree of approximation of functions in the Hölder metric. J. Indian Math. Soc. 58(3), 39–50 (with M. Mohapatra)
  5. 83.Absolute almost convergence. Indian J. Math. 34, 99–110 (with S. Nanda)
  6. 84.Degree of approximation of functions by the (Z, λ) transform of its Fourier series. Indian J. Math. 34, 217–223 (with M. Mohapatra)
  7. 85.A paranormed sequence space. Proc. Nat. Acad. Sci. India 62(A), 177–190 (with K. Rao)

1993

  1. 89.On absolute quasi-Nörlund means. J. Analysis 1, 149–155 (with P. Mohapatra)
  2. 90.Functional Banach limits. J. Indian Math. Soc. 59, 61–78 (with S. Nanda)

1994

  1. 91.Some generalisation of strong and absolute almost convergence. J. Indian Math. Soc. 60, 225–246 (with K. Rao)
  2. 92.On the A-continuity of real functions. Istanbul Univ. Fen Fak. Mat. Der. 53, 61–66 (with E. Savas)

1995

  1. 93.A convergence theorem for non-expansive mappings in Hilbert space. Proc. Nat. Acad. Sci. India 65(A) IV, 445–453 (with Manoja Manjari Swain)
  2. 94.Degree of approximation of functions in the Hölder metric by (e, c) means. Proc. Indian Acad. Sci. (Math. Sci.) 105(3), 315–327 (with Tulika Ghosh and B. K. Ray)
  3. 95.A Tauberian theorem for absolute quasi-Nörlund means. Selecta Analytica (dedicated to the memory of Prof. Brian Kuttner)

1996

  1. 96.Degree of approximation of functions by their series in the generalised Hölder metric. Proc. Indian Acad. Sci. (Math. Sci.) 106(2), 139–153 (with Tulika Ghosh and B. K. Ray)
  2. 97.A Tauberian theorem for quasi-Nörlund summability. Analysis 16, 213–221
  3. 98.Degree of approximation of functions associated with Hardy–Littlewood series in the Hölder metric by Euler means. Proc. Indian Acad. Sci. (Math. Sci.) 106(3), 227–243 (with A. K. Ojha and B. K. Ray)
  4. 102.Saturation theorems by matrix transformation. J. Indian Math. Soc. 1–4, 9–28 (with Tulika Ghosh)
  5. 103.Fixed point as a best approximation. Indian J. Math. 38(3), 275–286 (with Manoja Manjari Swain)

1997

  1. 104.Degree of approximation of functions in the Hölder metric by Borel's means. Proc. Indian Acad. Sci. (Math. Sci.) 107(2), 169–182 (with A. K. Ojha and B. K. Ray)
  2. 105.Abelian and Tauberian theorems for a new trigonometric method of summation. Proc. Indian Acad. Sci. (Math. Sci.) 107, 391–403 (with B. K. Ray)
  3. 106.Degree of approximation of functions of BV class using Cesàro mean of Fourier series. Proc. National Seminar, Berhampur University (Narosa) (with B. K. Ray and B. N. B. Ray)
  4. 107.Degree of approximation in the Hölder metric by a new trigonometric method of summation. J. Analysis 5, 127–137 (with B. K. Ray)
  5. 108.Degree of approximation in the sense of convergence in probability of weighted random Fourier–Stieltjes series associated with stable process. Fourier Analysis, Approximation Theory and Applications (New Age International), eds. Z. U. Ahmad, N. K. Govil, P. K. Jain

1998

  1. 109.Degree of approximation of functions with Hardy–Littlewood series in the Hölder metric by Borel means. J. Math. Anal. Appl. 219, 279–293 (with A. K. Ojha and B. K. Ray)
  2. 110.Degree of approximation of functions associated with Hardy–Littlewood series in the generalised Hölder metric. Proc. Indian Acad. Sci. (Math. Sci.) 108(2), 109–120 (with A. K. Ojha and B. K. Ray)
  3. 111.Degree of approximation of functions in H*p class by (D) mean of Fourier series. Indian J. Math. 40(3), 283–297 (with B. K. Ray)

1999

  1. 116.Degree of approximation of functions in Lp by power series method of summation. Sequence Spaces and Applications (Narosa), eds. P. K. Jain and E. Malkowsky (with Ellipse Das and B. K. Ray)

2000

  1. 117.Approximating fixed points of non-expansive mappings. Proc. Nat. Acad. Sci. India 70(A) III, 295–308 (with Manoja Manjari Swain)

2001

  1. 118.Degree of approximation of functions of bounded variation by a new trigonometric method of summation. J. Analysis 9, 1–20 (with Anusaya Nath and B. K. Ray)

2002

  1. 119.An estimate of the rate of convergence of Fourier series in the generalised Hölder metric. Narosa Publishing House, eds. H. P. Dikshit and P. K. Jain (with Anusaya Nath and B. K. Ray)
  2. 120.A new trigonometric method of summation and its application to the degree of approximation. Proc. Indian Acad. Sci. (Math. Sci.) 112(2), 299–319 (with Anusaya Nath and B. K. Ray)

2003

  1. 126.Invariant limits. J. Orissa Math. Soc., 171–179 (with N. Tripathy)

2005

  1. 124.A Tauberian theorem for absolute logarithmic summability. Analysis and Application — Proc. of the Conference on the Birthday of Professor G. Das (IMA / Narosa) (with B. K. Ray)
  2. 125.Degree of approximation of Fourier series of functions of bounded variation by Abel mean. Analysis and Application — Proc. of the Conference on the Birthday of Professor G. Das (IMA / Narosa) (with B. K. Ray and B. N. B. Ray)

2007

  1. 127.A family of new trigonometric means. Analytica Mathematica (with A. Nath and B. K. Ray)

2008

  1. 128.A Tauberian theorem for quasi-Nörlund means. Analysis

1991–92

  1. 86.A new convergence criterion for Fourier series. J. Orissa Math. Soc. 11–12, 1–6 (with B. K. Ray)
  2. 87.Summability (L) of r-th derived Fourier series. J. Orissa Math. Soc. 11–12, 29–38 (with Sudhakar Sahoo)
  3. 88.Double scale difference as a generalisation of difference of fractional order (I). J. Orissa Math. Soc. 11–12 (with K. Rao)

1993–96

  1. 99.Stochastic integral and random Fourier–Stieltjes series. J. Orissa Math. Soc. 12–15 (Prof. R. Mohanty 85th birthday volume) (with Ellipse Das and S. Pattanaik)
  2. 100.Degree of approximation of functions associated with Hardy–Littlewood series with Hölder metric by (e, c) means. J. Orissa Math. Soc. 12–15, 177–194 (with A. K. Ojha and B. K. Ray)

1992–96

  1. 101.Degree of approximation by Euler means of r-th derived Fourier series in Hölder metric. J. Nat. Acad. Math. 10, 82–111 (with Sanjita Mishra and B. K. Ray)

1998–2001

  1. 112.Degree of approximation of functions by their conjugate Fourier series in the generalized Hölder metric. J. Orissa Math. Soc. 17–20 (with R. C. Dash and B. K. Ray)
  2. 113.Degree of approximation of conjugate Fourier series of functions of bounded variation by Cesàro mean. J. Orissa Math. Soc. 17–20 (with B. N. B. Ray and B. K. Ray)
  3. 114.Degree of approximation by Cesàro means of r-th derived Fourier series in the Hölder metric. J. Orissa Math. Soc. 17–20 (with Sangita Mishra and B. K. Ray)
  4. 115.Rate of convergence of a series associated with Hardy–Littlewood series. J. Orissa Math. Soc. 17–20 (with B. K. Ray and Pratima Sarangi)

2002–03

  1. 121.Composition operators on harmonic Bergman space. J. Orissa Math. Soc. 21–22, 149–158 (with S. Pattanayak and Sumitra Patel)
  2. 122.Invariant limits. J. Orissa Math. Soc. 21–22, 171–179 (with Nandita Tripathy)
  3. 123.Riemann–Liouville integral mean and approximation. J. Orissa Math. Soc. 21–22, 43–51 (with B. K. Ray)