Haar wavelet

The Haar sequence is now recognised as the first known wavelet basis and is extensively used as a teaching example.The technical disadvantage of the Haar wavelet is that it is not continuous, and therefore not differentiable., the Haar function ψn,k is defined on the real line, is contained in the lower or in the upper half of the other interval, on which the functionThe Haar system on the real line is the set of functions It is complete in L2(): The Haar system on the line is an orthonormal basis in L2(In Hilbert space terms, this Haar system on [0, 1] is a complete orthonormal system, i.e., an orthonormal basis, for the space L2([0, 1]) of square integrable functions on the unit interval.The Haar system on [0, 1] —with the constant function 1 as first element, followed with the Haar functions ordered according to the lexicographic ordering of couples (n, k)— is further a monotone Schauder basis for the space Lp([0, 1]) when 1 ≤ p < ∞.[7] There is a related Rademacher system consisting of sums of Haar functions, Notice that |rn(t)| = 1 on [0, 1).The Khintchine inequality expresses the fact that in all the spaces Lp([0, 1]), 1 ≤ p < ∞, the Rademacher sequence is equivalent to the unit vector basis in ℓ2.[10] In particular, the closed linear span of the Rademacher sequence in Lp([0, 1]), 1 ≤ p < ∞, is isomorphic to ℓ2.Next, for every integer n ≥ 0, functions sn,k are defined by the formula These functions sn,k are continuous, piecewise linear, supported by the interval In,k that also supports ψn,k.The Faber–Schauder system is a Schauder basis for the space C([0, 1]) of continuous functions on [0, 1].[6] For every f in C([0, 1]), the partial sum of the series expansion of f in the Faber–Schauder system is the continuous piecewise linear function that agrees with f at the 2n + 1 points k2−n, where 0 ≤ k ≤ 2n.Since f is uniformly continuous, the sequence {fn} converges uniformly to f. It follows that the Faber–Schauder series expansion of f converges in C([0, 1]), and the sum of this series is equal to f. The Franklin system is obtained from the Faber–Schauder system by the Gram–Schmidt orthonormalization procedure.The Franklin system is therefore an orthonormal basis for L2([0, 1]), consisting of continuous piecewise linear functions.[16] The Franklin system is also an unconditional Schauder basis for the space Lp([0, 1]) when 1 < p < ∞.[17] The Franklin system provides a Schauder basis in the disk algebra A(D).[17] This was proved in 1974 by Bočkarev, after the existence of a basis for the disk algebra had remained open for more than forty years.[18] Bočkarev's construction of a Schauder basis in A(D) goes as follows: let f be a complex valued Lipschitz function on [0, π]; then f is the sum of a cosine series with absolutely summable coefficients.Let T(f) be the element of A(D) defined by the complex power series with the same coefficients, Bočkarev's basis for A(D) is formed by the images under T of the functions in the Franklin system on [0, π].[20] If one has a sequence of length a multiple of four, one can build blocks of 4 elements and transform them in a similar manner with the 4×4 Haar matrix which combines two stages of the fast Haar-wavelet transform.measures a low frequency component of the input vector.The next two rows are sensitive to the first and second half of the input vector respectively, which corresponds to moderate frequency components.The remaining four rows are sensitive to the four section of the input vector, which corresponds to high frequency components.This transform cross-multiplies a function against the Haar wavelet with various shifts and stretches, like the Fourier transform cross-multiplies a function against a sine wave with two phases and many stretches.[22][clarification needed] The Haar transform is one of the oldest transform functions, proposed in 1910 by the Hungarian mathematician Alfréd Haar.It is found effective in applications such as signal and image compression in electrical and computer engineering as it provides a simple and computationally efficient approach for analysing the local aspects of a signal.The Haar transform can be thought of as a sampling process in which rows of the transformation matrix act as samples of finer and finer resolution.can be found as The input signal can then be perfectly reconstructed by the inverse Haar transform