This paper presents a chronological overview of the developments in interpolation theory, from the earliest times to the present date. It brings out the connections between the results obtained in different ages, thereby putting the techniques currently used in signal and image processing into historical perspective. A summary of the insights and recommendations that follow from relatively recent theoretical as well as experimental studies concludes the presentation. Keywords—Approximation, convolution-based interpolation, history, image processing, polynomial interpolation, signal processing, splines. “It is an extremely useful thing to have knowledge of the true origins of memorable discoveries, especially those that have been found not by accident but by dint of meditation. It is not so much that thereby history may attribute to each man his own discoveries and others should be encouraged to earn like commendation, as that the art of making discoveries should be extended by considering noteworthy examples of it. ” 1 I.

A course in discrete mathematics is a relatively recent addition, within the last 30 or 40 years, to the modern American undergraduate curriculum, born out of a need to instruct computer science majors in algorithmic thought. The roots of discrete mathematics, however, are as old as mathematics itself, with the notion of counting a discrete operation, usually cited as the first mathematical development

binary arithmetic for the digital computers of his day. Vacuum tubes afforded these machines a speed of computation unmatched by other calculational devices, with von Neumann writing: “Vacuum tube aggregates... have been found reliable at reaction times as short as a microsecond... ” [7, p. 188]. Predating this, in 1938 Claude Shannon (1916–2001) published a ground-breaking paper “A Symbolic Analysis of Relay and Switching Circuits ” [4] in which he demonstrated how electronic circuits can be used for binary arithmetic, and more generally for computations in Boolean algebra and logic. These relay contacts and switches performed at speeds slower than vacuum tubes. Shannon identified an economy of representing numbers electronically in binary notation as well as an ease for arithmetic operations, such as addition. These advantages of base two arithmetic are nearly identical to those cited by von Neumann. Shannon [4] writes: A circuit is to be designed that will automatically add two numbers, using only relays and switches. Although any numbering base could be used the circuit is greatly simplified by using the scale of two. Each digit is thus either 0 or 1; the number whose digits in order are ak, ak−1, ak−2,..., a2, a1, a0 has the value ∑ k j=0 aj 2 j. 1. Explain how the base 10 number 95 can be written in base 2 using the above formula. In particular, compute ak, ak−1, ak−2,..., a2, a1, a0 for the number 95. What is k in this case? Write ∑ k j=0 aj 2 j in terms of addition symbols using the above value for k and each value of aj. Claude Elwood Shannon was a pioneer in electrical engineering, mathematics and computer science, having founded the field of information theory, and discovered key relationships between Boolean algebra and computer circuits [6]. Born in the state of Michigan in 1916, he showed an interest in mechanical devices, and studied both electrical engineering and mathematics at the University of Michigan. Having received Bachelor of Science degrees in both of these subjects, he then

for 荷 花 This article shows how Chinese ( 算 盘 , suànpan) and