Boas’ Formula and Sampling Theorem

Boas' Formula and Sampling Theorem

In 1937, Boas gave a smart proof for an extension of the Bernstein theorem for trigonometric series. It is the purpose of the present note (i) to point out that a formula which Boas used in the proof is related with the Shannon sampling theorem; (ii) to present a generalized Parseval formula, which is suggested by the Boas’ formula; and (iii) to show that this provides a very smart derivation o...

Heron's Formula and Ptolemy's Theorem

We adopt the following rules: p1, p2, p3, p4, p5, p6, p, p7 denote points of E2 T and a, b, c, r, s denote real numbers. Next we state four propositions: (1) If sin](p1, p2, p3) = sin](p4, p5, p6) and cos](p1, p2, p3) = cos](p4, p5, p6), then ](p1, p2, p3) = ](p4, p5, p6). (2) sin](p1, p2, p3) = −sin](p3, p2, p1). (3) cos](p1, p2, p3) = cos](p3, p2, p1). (4) ](p1, p4, p2)+](p2, p4, p3) = ](p1, ...

The Sampling Theorem and the Bandpass Theorem

A q-SAMPLING THEOREM AND PRODUCT FORMULA FOR CONTINUOUS q-JACOBI FUNCTIONS

In this paper we derive a q-analogue of the sampling theorem for Jacobi functions. We also establish a product formula for the nonterminating version of the q-Jacobi polynomials. The proof uses recent results in the theory of q-orthogonal polynomials and basic hypergeometric functions.

Sum Formula for Maximal Abstract Monotonicity and Abstract Rockafellar’s Surjectivity Theorem

In this paper, we present an example in which the sum of two maximal abstract monotone operators is maximal. Also, we shall show that the necessary condition for Rockafellar’s surjectivity which was obtained in ([19], Theorem 4.3) can be sufficient.