Orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide. We show here that under fairly general conditions, exact reconstruction schemes with synthesis filters different from the analysis filters give rise: to two dual Riesz bases of compactly supported wavelets. We give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions, and we present a sufficient condition for the decay of their Fourier transforms. We study the regularity of these biorthogonal bases. We provide several families of examples, all symmetric (corresponding to "linear phase" filters). In particular we can construct symmetric biorthogonal wavelet bases with arbitrarily high preassigned regularity; we also show how to construct symmetric biorthogonal wavelet bases "close" to a (nonsymmetric) orthonormal basis.

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We shall develop the properties of an integral for Banach-valued functions. The formalism is the generalized Riemann integral introduced by Kurzweil [5] and Henstock [4]. More precisely, the presentation is close to the McShane approach [6]. Besides its simplicity of presentation, four advantages characterize this theory:(i) the definition can be used for real-valued functions, and can be generalized without modification to general real and complex Banach spaces;(ii) when a function is integrable its norm is also integrable, and the proof is straightforward from the definition;(iii) for finite dimension spaces the theory is equivalent to the Mc-Shane's theory, which is itself equivalent to the Lebesgue's theory;(iv) and lastly, for general Banach space, we can prove the equivalence to the Bochner's theory. 1 Gauges, Tagged Partitions of [a, b] and Conventions In the following, (X, ) will denote a Banach space. Let [a, b] be a real interval, (a < b). By definition, a gauge on [a, b] is a function δ from [a, b] to R * + . Following the McShane definition [6], a tagged partition (([x i−1 , x i ]) 1≤i≤n , (c i ) 1≤i≤n ), is a couple of finite sequences where the closed intervals ([x i−1 , x i ]) 1≤i≤n form a partition of [a, b] and the numbers (c i ) 1≤i≤n are called the corresponding tags.A tagged partition (() is subordinate to a gauge δ if ∀i ∈ {1, . . . , n}, c i − δ(c i ) ≤ x i−1 < x i ≤ c i + δ(c i ).

Elliptic functions are largely studied and standardized mathematical objects. The two usual approaches are due to Jacobi and Weierstrass.From a contour integral which allowed us to unify many summation formulae (Euler-MacLaurin, Poisson, Voronoï or Circle formulae), we will find the entirety of the elliptic functions, proposed either in the shape of Jacobi or Weierstrass. But with one translation which appears in their natural form.What could seem a defect will lead us to a renormalisation of the elliptic functions making it possible to determine, in a rather simple way, a Fourier series representation and a factorization of these functions.

The modular discriminant ∆ is known to structure the sequence of modular forms (M 2k (SL 2 (Z))) k∈ N * at level 1. For all positive integer N , we define a strong modular unit ∆ N at level N which enables one to structure the family (M 2k (Γ 0 (N ))) k∈ N * in an identical way. We will apply this result to the bases search for each of the spacesThis article is the first in a series of three. In the second part we will propose explicit bases of (M 2k (Γ 0 (N ))) k∈ N * for 1 N 10. Finally, in a third part, we will apply the results obtained in the first two parts to (S 2k (Γ 0 (N ))) k∈ N * .