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The Allahabad
Mathematical Society

10, C S P Singh Marg
Uttar Pradesh, India

Professor B. N. Prasad

The sudden death of Professor B.N. Prasad (his full name being BADRI NATH PRASAD) on the 18th of January, 1966, at the age of 67, has removed from the mathematical world as extremely forceful personality whose services to the cause of mathematical education and research in India will long be remembered. He was a distinguished mathematician whose creative researches won him international fame and brought credit to India. He was a great teacher and a pioneer of research, who inspired a large number of bright young men to devote themselves to mathematical research and created an active center of research at Allahabad, where he worked continuously for about four decades.

B.N. Prasad, was born on the 12th of January, 1899, in a respectable middle class family of Mohammadabad-Gohna in the Azamgarh District of Uttar Pradesh (India). He was the youngest son of his parents. His father, the late Sri Ram Lal, was a practical man of foresight who tried to give the best possible education to his sons. Young Badri Nath Prasad was full of promise in his boyhood. He had a remarkably strong memory. I remember his recalling to me once as to how in his school days he performed the amazing feats of committing to memory long verses by reading them only once or twice. His mother and his elder brother seem to have exercised stronger influence in shaping his personality than his father. In reminiscent moods B.N. Prasad used to talk with refreshing enthusiasm of the great qualities of head and heart possessed by his mother. He used to recall with a feeling of reverence that on every occasion he paid his obeisance to his mother. She gave him the blessing: " Be Victorious".

He received his early education through the medium of the Urdu language but later changed to Hindi, with the result that he had an almost equal command over both of these languages. He enjoyed reading both Urdu and Hindi poetry. His elder brother was a college student at Allahabad and B.N. Prasad also came to that town and joined the local C.A.V. High School, which has now grown into an Intermediate College. He passed the Matriculation Examination of the Calcutta University from the V.M.H.E. School of Swan in the present state of Bihar and the Intermediate (Science) and B.Sc. (Hons.) examination from the Patna College, Patna, which was then affiliated to the Calcutta University. He was a regular and hardworking student right from the beginning and had a uniformly brilliant academic career throughout. As a schoolboy Prasad used to take part in games as well and played hockey particularly well. He passed the M.Sc. examination in Mathematics of the Banaras Hindu University in the year 1921, obtaining a first class and securing the first position in order of merit among all the successful candidates at the Master's Degree Examination of that year in different subjects. As a student of the M.Sc. (Previous) class during the academic year 1919-20, Prasad came in contact with the late Professor Ganesh Prasad, who recognised his talents and inspired him to do creative research work in Mathematics. Those were the days when almost all the bright Indian students aspired to join the civil services. B.N. Prasad was actually offered a lucrative post by the then provincial government on the basis of his brilliant performance at the M.Sc. examination. This was not a small attraction, particularly when members of his family strongly desired that he should become an officer in the civil services, but his love for Mathematics and his devotion to his teacher Prof. Ganesh Prasad, who strongly advised him to take to the noble profession of teaching and research, were so great that he politely declined the invitation of the Government and decided to pursue his study of Mathematics. It must be said to the credit of the late Dr. Ganesh Prasad that he aroused a real interest in the minds of his students for doing creative work in Mathematics and succeeded in training a number of bright young men as researchers.

In July 1921, B.N. Prasad joined the Department of Mathematics of the Banaras Hindu University as a research scholar working under the supervision of Prof. Ganesh Prasad. He was appointed on the 8th of July 1922, as an Assistant Professor of Mathematics at the Banaras Hindu University. As is usually the lot of the young University lecturers, in India at least, B.N. Prasad, had to shoulder a heavy teaching load. But inspite of this, he continued his studies of advanced topics in Mathematics with unabated zeal and published during 1922-23 two original research papers on the properties of non-differentiable functions.

As he was preparing to get settled at Banaras, with regard to his research activities, his teacher Dr. Ganesh Prasad resigned his Professorship at Banaras in the year 1923 owing to some serious differences on questions of administrative policy with the university authorities and joined the Calcutta University as the Hardinge Professor of Higher Mathematics. Shortly after Dr. Gnash Prasad's departure from Banaras, some of the other teachers also left the department of Mathematics to join other universities and young B.N. Prasad had to shoulder almost the entire load of postgraduate teaching during that year. He had to teach topics as widely different as the Theory of Functions and Mechanics, but he did his job with creditable success. It appears that B.N. Prasad did not feel very happy to stay at Banaras after the departure of Dr. Ganesh Prasad and so he too left Banaras and joined the University of Allahabad as a Lecturer in Mathematics on the 17th of July, 1924, During the brief period of two years that he taught at Banaras, he became quite popular as a teacher and attracted a number of good students who were so devoted to him that some of them (M. L. Misra being one of them) migrated with him to the University of Allahabad.

He lectured at Allahabad to both under graduate and post graduate classes on different topics of pure and applied mathematics. His involvement in the teaching programme retarded the progress of his research to some extent. However, the few years spent at Allahabad before his departure to England in 1929, proved to be fruitful in that he made an intensive study of the work done on the theory of trigonometric series by W.H. Young, G.H. Hardy, J.E. Littlewood, E.C. Titchmarsh, H. Lebesgue, A. Denjoy, and other continental mathematicians and was fascinated by their profound researches. Obtaining study leave from the University of Allahabad, he went to the United Kingdom in the year 1929. He stayed at Edinburgh for a few months and did some work on the theory of Fourier series, but later proceeded to Liverpool to work with Prof. E.C. Titchmarsh. He completed his thesis in a short period of 1½ years and obtained the Ph.D. degree of the University of Liverpool in 1931. Professor Titchmarsh was highly impressed by the work of B.N. Prasad and writing about him he says, "I found him an extremely industrious and intelligent worker who had plenty of ideas of his own and merely asked for my advice and direction".

After the completion of his Ph.D. thesis at Liverpool, Prasad went to Paris, the great center for mathematical research and learning, and worked there with the celebrated French mathematician Arnaud Denjoy. His well-known thesis entitled, "Contribution á l'étude de la séries conjuguée d'une série de Fourier" prepared in less than a year under the guidance of Prof. Denjoy, earned him the degree of 'Docteur és Science' (State D.Sc.) of the University of Paris with mention 'très honorable' The 'Suotenance' (The French world Soutenace, which literally means defence, is uded to describe the ceremony of defending the thesis by a candidate in public before the Jury of Examiners.) of his thesis for the 'doctorat d'Etat'took place on the 4th of June 1932, before a Jury of which Prof. Emil Borel was the President and Prof. A. Denjoy and Prof. G. Valiron were the two examiners. His examiners were greatly impressed by the quality of his thesis and Prof. Borel while announcing the result of the "Soutenance" paid compliments to Prasad by saying: 'Nous ne pouvons que nous nous réjouir de voir I'un de plue distingués parmi les jeunes mathematiciens hindous se rattacher par ses travaux à I'école mathematique francaise' (English Translation: 'We can only rejoice to see one of the most distinguished of the young Indian mathematicians associating himself, through his work, with the French School of Mathematics'.) (quoted from LE TEMPS of Paris dated June 5, 1932 p.6). Bearing testimony to the merit of Prasad's thesis Prof. Borel wrote: "Le jury a été trés favourablement impressionné par ses qualitiés dont il fait preuve en traitant un subject difficile et en offerant des resultats importants et nouveaux dans une question déja travaillée par de nombreaux mathématiciens. II a fait preuve egalement, dans la soutenance, qualitiés de claritié et de precision qui montrent que il sera un excellent professur". (English Translation: "The jury have been very favourably impressed by his qualities of which he has given proof in treating a difficult subject and in obtaining new and important results in a field in which a number of mathematicians have already worked. During the Soutenance of his thesis he has equally well given proof of possessing qualities of clarity which show that he will be an excellent professor".).

B.N. Prasad was perhaps the first Indian to be awarded the State D.Sc. Degree in mathematics of the University of Paris and it was indeed a creditable achievement. During his stay in Europe, Prasad visited various centers of mathematical research in England, Scotland, Ireland, France, Germany and Italy and came in contact with mathematicians like Hadamard, Hilbert, Whittaker, Hardy, Lebesgue and others, in addition to those with whom he worked. He returned to India in July 1932, to resume his duties at the University of Allahabad as a Lecturer in Mathematics.

B.N. Prasad returned to India with great academic achievements to his credit, but the University of Allahabad failed to give due recognition to his merits and scholarship. He remained a Lecturer for fourteen long years after his return from France and was made a Reader only on April 8, 1946, when the then Vice-Chancellor, Dr. Amaranatha Jha, being conscious of the injustice done to Dr. Prasad, created a new post of Reader in the Mathematics Department. It was an irony of fate that a man who eminently deserved to hold a chair of Mathematics in any university in India remained only a Lecturer during the most creative period of his life.

On his return from Europe, Prasad had to face many difficulties. There were financial problems as also domestic ones. His hopes and claims for receiving a promotion in the university remained unfulfilled, but his spirit never got subdued. He had a keen desire to create an active centre of research in Analysis at Allahabad. He encouraged and inspired bright students to pursue higher studies and research. Those were the days of depression when it was difficult to get employment and money. Scholarships were too few. Most of the bright students were opting for the administrative services. It was very difficult to pursuade them to take to research as the prospects of good employment for a researcher in Mathematics were not very encouraging. B.N. Prasad's own example was there before them. Inspite of all these difficulties, he succeeded in training a few research workers during the years preceding the second world war. It was his indomitable courage and invincible determination that helped him in keeping the torch of research burning at Allahabad and it is gratifying to recall that his efforts bore rich fruits during the forties when a number of bright young researchers gathered round him and Allahabad thrived as a leading centre of research in Analysis in India. He succeeded in building at Allahabad a strong School of Research on Summability theory, which flourished and developed considerably during his life time and which is very appropriately associated with his name. It is perhaps true to say that B.N. prasad trained a larger number of researchers than any other individual mathematician in India. Some of his research students serving the cause of mathematical education and research in India are: M.L. Misra at Sagar ( retired recently as Professor of Mathematics), P.L. Bhatnagar at Bangalore, U.N. Singh at Baroda, J.A. Siddiqi and S.M. Mazhar at Aligarh, T. Pati at Jabalpur, S.R. Sinha, D.P. Gupta, S.N. Bhatt, T. Singh and N.D. Mehrotra at Allahabad, Mrs. Pramila Srivastava and L.M. Tripathi at Varanasi and Mrs. Sulaxana Kumari Gupta at Delhi.

Dr. Prasad was elected President of the Section of Mathematics and Statistics of the 32nd session of the Indian Science Congress held at Nagpur in January 1945. His Presidential Address delivered on this occasion, gave an excellent survey of the work done, upto the end of 1944, concerning the summability problems ( of various kinds) of a Fourier series and its Conjugate series. Written in an attractive style and marked for its clarity of expression and thoroughness of trcatment, this address gives an admirable account of the vast progress made upto that time in respect of researches in the theory of summability of a Fourier series. He gave several other addresses of this type later in his life including his last address which he delivered as the General President of the Indian Science Congress at Chandigarh on January 3, 1966, a forthnight before his death, and in which he gave a survey of the recent researches concerning the absolute summability of infinite series

B.N. Prasad had willingly chosen to stay at Allahabad all these years even though he received offers of higher posts from several other universities of India. However he yielded once to outside pressure, when he accepted the invitation of the Government of Bihar to work as Professor of Mathematics and Chairman of the Mathematics Department at the Science College of Patna. Soon after joining this post in the month of March 1949, he realized that the atmosphere there was not congenial to his academic pursuits. Consequently he resigned his professorship at Patna and came back to his old post at Allahabad in January 1951. He never entertained the idea of leaving Allahabad again thereafter.

B.N. Prasad was married on the 29th May, 1923, to Smt. Lakshmi Devi, a very accomplished lady, highly cultured and kind-hearted. The Prasads had three children: one son Prakash Chandra and two daughters, Indu Prabha and Arun Prabha. They lived happily till the time of Mrs. Prasad's death. She passed away on the 15th of September 1954, at Patna, after a prolonged illness. This tragedy gave a stunning blow to Dr. Prasad. It created a void in his life which tormented his mind constantly, but he concealed his grief well within himself, remaining outwardly calm and composed. Only those who were in very close contact with him, could perceive how lonely he felt within himself after the death of his wife. This was, perhaps the main reason for his keeping himself extremely busy and accepting many outstation engagements.

In October, 1954, Prasad went to Montevideo (Uruguay, South America) to attend the Eighth General Conference of the UNESCO as a member of the Government of India's delegation which was headed by the then Vice-President of India, Dr. S. Radhakrishnan. This trip to South America gave him an opportunity to visit various universities in the U.S.A., where he was invited to give research lectures. He re-visited Paris after an interval of about 22 years and renewed his contacts with mathematician friends there.

After gaining independence, India embarked on a big programme of planned economy. One of the first steps taken in planning scientific education on modern lines was to build a chain of national laboratories. However, no national institutes for higher studies and research in Mathematics were created besides the Tata Institute of Fundamental Research, which was founded by a private trust at Bombay and has subsequently been financed almost entirely by the Central Government. People felt that it was desirable to have a few more institutes for promoting quality research in Mathematics. B.N. Prasad had slightly different ideas in this regard. He was convinced that the universities were the fountainheads of higher learning and research and that they should never be allowed to starve in respect of facilities. By offering better conditions of service, higher scales of pay and better facilites to work, the national laboratories and institutes would attract talents from the universities causing a serious drain on their resources and, as an inevitable consequence, damaging the quality of research and teaching in the universities. He, therefore, desired that good University Departments should be given facilities to function as centres of Advanced Research and learning and advocated a healthy co-ordination between research and teaching. He wanted India to have institutions like the "École Normale Superieur of Paris. He expressed his ideas on these questions very clearly and emphatically in his General Presidential Address, delivered at the 53rd session of the Indian Science Congress, at Chandigarh on January 3, 1966.

B.N. Prasad was also not quite happy with the working and the affairs of the Indian Mathematical Society. During his tenure of office as the President of the Society in the year 1961, he remedied many of its organizational defects. He believed that the duties of a mathematical society should not only be limited to organizing annual conferences and publishing journals, but they should also devise and successfully execute constructive programmes for promoting quality research in Mathematics. With this end in view, he founded the Allahabad Mathematical Society at Allahabad in December 1958. He ardently desired this Society to function as an effective forum for promoting the cause of advanced studies and research in Mathematics and, as the President of this Society, he constantly endeavoured to achieve this objective. The Allahabad Mathematical Society started the publication of a research journal called the Indian Journal of Mathematics. This Journal attained an international status very soon. Prasad completely identified himself with the welfare of the Society (even the office of the Society was accommodated in his own house at Allahabad), and had a very bright vision regarding its future growth. It is, indeed, unfortunate for the society that he is no more to guide its destiny. He has left behind many ideas and plans unfulfilled; however, he had the satisfaction of putting the Society on a firm footing and of giving it the initial inspiration and impetus.

B.N. Prasad acted as officiating Professor and Head of the Department of Mathematics in the University of Allahabad for about two years beginning from November 1, 1958. He was appointed Professor of Mathematics and Head of the Mathematics Department on a permanent basis with effect from August 16, 1960, just about five months before his retirement from the service of the University. During the brief period that he was the Head of the Department, he succeeded in infusing fresh vigour into the academic life of the Department. He retired from the active service of the University of Allahabad on the 11th of January 1961, after rendering to the university, for over 36 years, a service of rare distinction of which any mar can justly be proud. On the day of his retirement he was given a memorable and touching farewell by his pupils and friends. That was, in a sense, a measure of the high esteem in which he was held by them.

B.N. Prasad received many honours. He was one of the earliest Fellows of the National Institute of Sciences of India (an F.N.I.), was the first mathematician to be elected General President of the Indian Science Congress, was the President and a Fellow of the National Academy of Sciences of India, and was the President of the Vijnan Parishad. The President of India conferred upon him the title of "Padma Bhushan" in the year 1963, in recognition of his meritorious services to the cause of Science and Education, and a year later the President again honoured him by nominating him to be a Member of the Rajya Sabha, the upper house of the Indian Parliament.

The guiding principle of B.N. Prasad's life was to seek perfection. He sought perfection in everything he did; in writing a research paper or a book, in giving a lecture to his class, in building a house, in doing gardening, in every situation whatsoever, it was this cardinal principle which consciously and unconsciously guided his thoughts and actions. He never did anything half-heartedly. He put his heart and soul in every undertaking of his, big or small, and never left things to chance. He was a man of refined taste and highly aesthetic outlook which permeated his whole personality. He would express his annoyance unhesitatingly to any one (even to his wife and children, to his friends and pupils) who failed to come upto his standards of perfection, and was very outspoken and frank in expressing his views and feelings.

That he was an exacting taskmaster many know, but very few know how sincerely he strove for the welfare of his pupils. He would thoughtfully and enthusiastically plan for their well-being and goad them to fulfil those plans just as he would constantly urge and inspire them to do creative research work in Mathematics. It is perhaps his pupils who miss him most.

The name B.N. Prasad has come to be associated with an extensive cross-secton of Analysis, including numerous subjects such as the convergence and summability problems of a Fourier series, its conjugate series, the derived series of a Fourier series and its cojugate series; the theory and applications of absolute summability of infinite series in general, summability factors; Fourier integrals; radial variation of analytic functions, non-summability in the sense of Abel, the theory of generalized derivatives, strong summability-especially with Riesz's typical means, the Gibbs phenomenon and the generalized jump of a function, the multiplication of Dirichlet series and the second theorem of consistency. The numerous doctoral theses prepared from time to time under his supervision and the large number of research papers published in the important research journals of the world in the wake of his pioneering contributions on these subjects, bear eloquent testimony to the intrinsic value and power of Prasad's outstanding work.

Work done in collaboration with his pupils:

Izumi and Kawata ( 1938 ) and Cheng (1947) extended Prasad's classical result on |A| summability factors of Fourier series; Pati (1954) obtained generalizations which contained as special cases both of these results. In 1957, Prasad and Bhatt gave further extensions and a number of allied results of various kinds in the same direction.

Prasad and Siddiqi (1949) extended the scope of summability of the derived Fourier series by applying Norlund means instead of Cesaro means which had been used by previous authors. Later (1950) they obtained a very general result concerning the Norlund summability of the r-th derived Fourier series, generalizing previous results by Zygmund, Astrachan and Wang.

Prasad and U.N. Singh worked on the strong summabiliity of order unity of the derived series of a Fourier series at a point, the generating function being assumed to be of bounded variation in the fundamental interval. Their result was a source of inspiration for some of the subsequent researches on the subject.

After the classical work of Hardy and Riesz, Hirst, Kuttner and Chandrasekharan on the 'second theorem of consistency', Prasad and Pati made a study of the 'second theorem' as well and developed his idea of the unification of the two theorems of consistency of Riesz means. They examined the question of relative inclusion between Riesz methods in which 'type' and 'order' were both changed. In a later paper, Prasad and Pati studied the multiplication of absolutely summable Dirichlet series.

While Prasad and Pramila Srivastava (1960) have studied the strong Riesz summability of Dirichlet series, Prasad and D.P. Gupta (1965) have studied the convergence of ultraspherical lacunary series.

All these papers were written in the forties, fifties and sixties and one can discern the stamp of the master on their style. But this is not all. During this period more than 120 valuable research papers have been produced by the research students of B.N. Prasad and they have been printed in almost all the important scientific and mathematical research journals being published from different parts of the world.

List of Research Papers by Prof. B.N. Prasad
  'On non-differentiable functions which have progressive or regressive derivatives for certain values of the variable', Proc. Benaras Math. Soc., III (1921), 1.4.
  'The absolute summability (A) of Fourier series', Proc. Edinburgh Math. Soc. (2), 2 (1930), 129-134.
  'A theorem on the Cesaro summability of the Allied series of a Fourier series', Jour. London Math. Soc. 6 (1931), 274-278.
  'Sur la convergence de la série conjuguée d'une série de Fourier', Comptes Rendus de l'Académie des sciences de Paris, 193 (1931), 1159-1162.
  'Sur la summabilté de la série conjuguée dune série de Fourier', Comptes Rendus de l Academie des Sciences de Paris, 193 (1931), 1386-1387.
  'Contribution a létude de la série conjuguée d'une série de Fourier', Liovelles Jour., 9(1932), 153-205.
  'On the summability (C,1) of the conjugate series of a Fourier Series', Annali di Matematica Pura ed Applicata (4), XI (1932), 207-214.
  'A new theorem on the representation of a function by Fourier's single integral', Jour. London Math. Soc., 7(1932), 36-38
  'Non-summability of the conjugate series of a Fourier series', Annals of Math. (2) 33 (1932), 771-772.
  'Direct proof of Young's theorem for the convergence of the conjugate series of a Fourier series', Bull. Calcutta Math. Soc. 24 (1932), 135-142.
  'Remarks on infinite derivatives', Jahresbericht der deutschen Mathematiker Vereinigung, 31 (1982), 174-175.
  'On the summability of Fourier series and the bounded variation of power series', Proc. London Math. Soc. (2), 35(1933), 407-424.
  'Note on the summability of the conjugate series of a Fourier series', Tôhoku Math. Jour., 36 (1933), 223-224.
  'On the summability of Fourier series by arithmetic means', Proc. U.P. Acad. Sc., 4 (1934), 39-46.
15.   'Note on the convergence of the conjugate series of a Fourier series', Proc. U.P. Acad. Sc., 4 (1934), 125-128.
16.   'On the summability (C, 1) of Fourier series', Math. Zeit., 40(1935), 496-502.
17.   'Notes on a theorem of Hardy and Littlewood concerning the Cesáro summability of the allied series of a Fourier series', Abstracts, Indian Science Congress, 1943.
18.   'On the Norlund summability of derived Fourier series', Proc, Nat. Inst. Sc. (India), 16 (1950), 71-82 (jointly with J.A. Siddiqi).
19.   'On the Norlund summability of r-th derived Fourier series', Jour. Indian Math. Soc., 14 (1951), 159-170 (jointly with J.A. Siddiqi).
20.   'On the strong summability of the derived Fourier series and its conjugate series', Math. Zeit., 56 (1951), 280-288 (jointly with U.N. Singh).
21.   'On the theorems of consistency for Riesz summability', Abstracts, Proc. International Congress of Mathematicians, 1954, 461-462.
22.   'The summability factors of a Fourier series', Duke Math. Jour., 24 (1957), 103-117 (jointly with S.N. Bhatt).
23.   'On the second theorem of consistency in the theory of absolute Riesz summability', Trans. American Math. Soc., 85 (1957), 122-123 (jointly with T. Pati).
24.   'On the Gibbs phenomenon for Nörlund means', Indian Jour. Math., 1(1958), 21-28 (jointly with J.A. Siddiqi).
25.   The second theorem of consistency for Riesz summability", Abstracts, Proc. Inter. Cong. Mathematicians, Edinburgh 1958.
26.   'Recent researches on the second theorem of consistency', Bull. Calcutta Math. Soc., Jubilee Volume (1958), 225-233.
27.   'The second theorem of consistency in the theory of absolute Reisz summability', Math. Ann., 140 (1960), 187-197 (jointly with T. Pati).
28.   'On the multiplication of absolutely summable Dirichlet series', Jour. Indian Math. Soc., Golden Jubilee Volume (1960), 421-431 (jointly with T. Pati).
29.   'On strong Riesz summability of Dirichlet series', Proc. Nat Inst. Sc. (India), (Supplement II), 26(1960) 180-209 (jointly with Pramila Srivastava).
30.   'On convergence of ultrapherical lacunary series', Annali di Matematica Pura ed Applicata, 67 (1965), 394-404 (jointly with D.P. Gupta).

List of presidential Addresses delivered by Prof. B.N. Prasad
  'The summability of a Fourier series and its conjugate series', Presidential Address, Section of Mathematis and Statistics, 32nd Session of the Indian Science Congress, Nagpur, 1945.
  'The absolute summability of Fourier series', Presidential Address, Section of Physical Sciences, National Academy of Sciences of India, 29th Annual Session, Gorakhpur, 1960.
  'Mathematics in India (Part A: General) and 'Certain aspects of the researches on Dirichlet series' (Part B: Technical), Presidential Address, 27th Conference of the Indian Mathematical Society, Ahmedabad, 1961.
  'Hindi madhyam, Ganitiya adhyayana aur anusandhana' (Hindi medium, mathematical studies and research), Presidential Address, Vijnan Parishad, Delhi, Sept. 1963.
  'Science in India' (Part A: General) and 'Recent researches in the absolute summability of infinite series and their applications' (Part B: Technical), General Presidential Address, 53rd Session of the Indian Science Congress, Chandigarh, 1966.
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