# Download Applied functional analysis: numerical methods, wavelets, by Abul Hasan Siddiqi PDF

By Abul Hasan Siddiqi

ISBN-10: 0824740971

ISBN-13: 9780824740979

Advisor covers the most up-tp-date analytical and numerical tools in infinite-dimensional areas, introducing contemporary leads to wavelet research as utilized in partial differential equations and sign and picture processing. For researchers and practitioners. contains index and references.

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Extra resources for Applied functional analysis: numerical methods, wavelets, image processing

Example text

2. Here is an example of using SAGE to compute preimages. roots() [(2*I, 1), (-2, 1), (-2*I, 1), (2, 1)] In other words, f −1 (0) = {2i, −2i, 2, −2}, each with multiplicity one, where √ i = −1 is denoted I by SAGE. 1. 2. If f : S = R → T = R is the map f (x) = x4 − 81 then ﬁnd f −1 (0). The Cartesian product (or direct product) of two sets S, T is the set of pairs of elements taken from these sets: S × T = {(s, t) | s ∈ S, t ∈ T }. An element of S × T is simply a list of two things, the ﬁrst one from S and the second one from T .

In other words, f is an injection if the condition f (s1 ) = f (s2 ) (for some s1 , s2 ∈ S) always forces s1 = s2 . 5. Suppose that S, T are ﬁnite sets and |S| > |T |. Is there an injective function f : S → T ? Explain. 6. Suppose that |S| = |T | < ∞. Show that a function f : S → T is surjective if and only if it is injective. 6. A function f : S → T is called a bijection if it is both injective and surjective. Equivalently, a bijection from S to T is a function f : S → T for which each t ∈ T is the image of exactly one s ∈ S.

Any k of the k + 1 horses have an odd number of feet, so all k + 1 do as well; hence P (k + 1) holds as well. As a more serious example, let P (k) be the logical statement |S1 ∪ . . ∪ Sk | = |S1 | + . . + |Sk |, 1 ≤ k ≤ n. , prove P (1). (2) Assuming the truth of the case k = n − 1, prove P (n). Proof: Let P (k) be the logical statement |S1 ∪ . . ∪ Sk | = |S1 | + . . + |Sk |, 1 ≤ k ≤ n. Case k = 1. P (1) is the statement |S1 | = |S1 |, which is of course true. Case k = n − 1. Assume |S1 ∪ .